Nonlinear Stability of Relative Equilibria and Isomorphic Vector Fields
We present applications of the notion of isomorphic vector fields to the study of nonlinear stability of relative equilibria. Isomorphic vector fields were introduced by Hepworth [Theory Appl. Categ. 22 (2009), 542-587] in his study of vector fields on differentiable stacks. Here, we argue in favor...
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| Цитувати: | Nonlinear Stability of Relative Equilibria and Isomorphic Vector Fields / S. Klajbor-Goderich // Symmetry, Integrability and Geometry: Methods and Applications. — 2018. — Т. 14. — Бібліогр.: 25 назв. — англ. |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1860014381687898112 |
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| author | Klajbor-Goderich, S. |
| author_facet | Klajbor-Goderich, S. |
| citation_txt | Nonlinear Stability of Relative Equilibria and Isomorphic Vector Fields / S. Klajbor-Goderich // Symmetry, Integrability and Geometry: Methods and Applications. — 2018. — Т. 14. — Бібліогр.: 25 назв. — англ. |
| collection | DSpace DC |
| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | We present applications of the notion of isomorphic vector fields to the study of nonlinear stability of relative equilibria. Isomorphic vector fields were introduced by Hepworth [Theory Appl. Categ. 22 (2009), 542-587] in his study of vector fields on differentiable stacks. Here, we argue in favor of the usefulness of replacing an equivariant vector field by an isomorphic one to study the nonlinear stability of relative equilibria. In particular, we use this idea to obtain a criterion for nonlinear stability. As an application, we offer an alternative proof of Montaldi and Rodríguez-Olmos's criterion [arXiv:1509.04961] for the stability of Hamiltonian relative equilibria.
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| first_indexed | 2025-12-07T16:44:07Z |
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Symmetry, Integrability and Geometry: Methods and Applications SIGMA 14 (2018), 021, 37 pages
Nonlinear Stability of Relative Equilibria
and Isomorphic Vector Fields
Stefan KLAJBOR-GODERICH
Department of Mathematics, University of Illinois at Urbana-Champaign,
1409 W. Green Street, Urbana, IL 61801 USA
E-mail: klajbor2@illinois.edu
URL: https://faculty.math.illinois.edu/~klajbor2/
Received October 31, 2017, in final form March 09, 2018; Published online March 14, 2018
https://doi.org/10.3842/SIGMA.2018.021
Abstract. We present applications of the notion of isomorphic vector fields to the study of
nonlinear stability of relative equilibria. Isomorphic vector fields were introduced by Hep-
worth [Theory Appl. Categ. 22 (2009), 542–587] in his study of vector fields on differentiable
stacks. Here we argue in favor of the usefulness of replacing an equivariant vector field by
an isomorphic one to study nonlinear stability of relative equilibria. In particular, we use
this idea to obtain a criterion for nonlinear stability. As an application, we offer an alter-
native proof of Montaldi and Rodŕıguez-Olmos’s criterion [arXiv:1509.04961] for stability of
Hamiltonian relative equilibria.
Key words: equivariant dynamics; relative equilibria; orbital stability; Hamiltonian systems
2010 Mathematics Subject Classification: 37J25; 57R25; 37J15; 53D20
1 Introduction
Relative equilibria of equivariant vector fields and their stability have garnered much interest
in the dynamics literature, partly due to their myriad applications in the sciences (see, for
example, [6]). In this paper we present an approach to determining the stability of relative
equilibria via the notion of isomorphic vector fields introduced by Hepworth [9]. In particular,
we argue that it can be useful to replace a given equivariant vector field with an isomorphic one
for which it is easier to determine stability.
Recall that a relative equilibrium of an equivariant vector field is a point for which the vector
field is tangent to the group orbit at that point. It can be difficult to determine the stability of
relative equilibria. Even determining linear stability poses a challenge. For an equilibrium, the
Lyapunov stability criterion can guarantee linear stability if all the eigenvalues in the spectrum of
the linearization of the vector field have negative real part (see, for example, [1, Theorem 4.3.4]).
In contrast, since the vector field is not necessarily zero at a relative equilibrium, the usual notion
of a linearization does not make sense. Thus, we don’t immediately have an analogue of the
Lyapunov stability criterion.
A construction due to Krupa gives a way to linearize an equivariant vector field near a relative
equilibrium and test for linear stability [11]. Krupa’s construction involves choosing a slice for
the action through the relative equilibrium and projecting the vector field onto the slice. The
projected vector field has an equilibrium at the original vector field’s relative equilibrium, so
we can linearize the projected vector field. This construction depends on a choice of slice and
projection, but it turns out the real parts of the spectrum of the linearization are independent
of these choices [5, Lemma 8.5.2]. The Lyapunov stability criterion can then be used to test
for linear stability of the equilibrium of the projected vector field. It can be shown that if this
is linearly stable it implies the linear stability of the relative equilibrium of the original vector
mailto:klajbor2@illinois.edu
https://faculty.math.illinois.edu/~klajbor2/
https://doi.org/10.3842/SIGMA.2018.021
2 S. Klajbor-Goderich
field [2, Theorem 7.4.2]. Furthermore, since the real parts of the eigenvalues of the spectrum are
independent of the choices, we can choose any slice and projection to determine linear stability;
ideally ones where the spectrum is easier to compute.
Not all stable equilibria are linearly stable, and the same is true of relative equilibria. To
use Krupa’s construction for nonlinear stability, as well as for other applications, we need to
make sense of the choices involved. Hepworth’s notion of isomorphism of vector fields is useful
for this. Hepworth introduced isomorphic vector fields to define vector fields on differentiable
stacks, a categorical generalization of differentiable manifolds [9]. Since differentiable stacks are,
in some sense, represented by Lie groupoids, it is not surprising that vector fields on a stack form
a groupoid. This gives rise to a notion of isomorphism between equivariant vector fields. Lerman
used Hepworth’s notion of isomorphism of vector fields to revisit Krupa’s construction [12]. In
particular, he showed that the choices of slice and projection lead to isomorphic vector fields.
In this paper we show how considering vector fields up to isomorphism, in the sense of
Hepworth, facilitates testing for nonlinear stability. In Theorem 3.11, which we call here the slice
stability criterion, we show that one can determine nonlinear stability of a relative equilibrium by
testing for nonlinear stability of the corresponding equilibrium of the projected vector field. This
reduces the problem to the well-studied case of equilibria on a vector space with a representation
of a compact Lie group. In fact, one can test any vector field that is isomorphic to the projected
vector field. Hence, one additionally obtains the freedom to choose a convenient slice, projection,
and isomorphism class representative to determine stability.
Hamiltonian relative equilibria are an important case where we may have nonlinear stability
but not linear stability. The integral curves of a Hamiltonian vector field do not exhibit en-
ergy dissipation, so we don’t expect the relative equilibria to be linearly stable. Lerman and
Singer [13] and Ortega and Ratiu [18], building on work of Patrick [21, 22], showed that the
definiteness of the Hessian of an augmented Hamiltonian function implies stability of the Hamil-
tonian relative equilibrium. Montaldi and Rodŕıguez-Olmos extended this criterion, allowing for
a wide choice of augmented Hamiltonians to check for stability [16, Theorem 3.6] (see also [17,
Theorem 2]). They prove this extension by building on the bundle equations in [23, 24, 25]. We
use Theorem 3.11 to provide an alternative proof of their result. Our proof is based on the fact
that the augmented Hamiltonian vector fields are isomorphic to the original Hamiltonian vector
field and that a choice of augmented Hamiltonian is equivalent to a choice of an isomorphism
class and a representative.
1.1 Organization of the paper
In Section 2, we present Hepworth’s groupoid of equivariant vector fields in the context of
Lie group actions, as well as the corresponding notion of isomorphism of equivariant vector
fields. We also present an equivalent formulation of the results in [12], and provide some general
background and results.
In Section 3, we prove a test for nonlinear stability of relative equilibria, Theorem 3.11, which
we call here the slice stability criterion. This is our main theorem on the nonlinear stability of
relative equilibria. We also show how isomorphisms of equivariant vector fields and one of the
functors involved in the slice stability criterion preserve the stability of relative equilibria.
In Section 4, we apply the slice stability criterion to obtain a proof of the result of Montaldi
and Rodŕıguez-Olmos (Theorem 4.8). We use the Marle–Guillemin–Sternberg normal form
[8, 14] in this proof. In Section 5, we reduce the general case to the normal form computation.
1.2 Notation and conventions
Throughout the paper we will assume all manifolds are Hausdorff. We will denote Lie groups
with uppercase Latin letters, their Lie algebras with the corresponding lowercase fraktur letter,
Nonlinear Stability of Relative Equilibria and Isomorphic Vector Fields 3
and the duals of these Lie algebras by adding a star superscript. The adjoint representation of
a Lie group on its Lie algebra will be denoted by Ad, while its coadjoint representation on the
dual of the Lie algebra will be denoted by Ad†. Given an action of a Lie group on a manifold,
the stabilizer subgroup of a point m will be denoted by the same letter as the group but with
the point as a subscript (e.g., Gm). The Lie algebra of the stabilizer will also carry the point as
a subscript (e.g., gm).
The vector space of smooth vector fields on a manifold M will be denoted by Γ(TM). Given
a diffeomorphism f : M → N between two manifolds, we will denote the corresponding pushfor-
ward of vector fields along f by f∗ : Γ(TM) → Γ(TN) and the pullback of vector fields along f
by f∗ : Γ(TN) → Γ(TM). We will refer to both embedded and regular submanifolds. Recall
an embedded submanifold of a manifold M is a pair consisting of a manifold N and a smooth
embedding f : N → M , whereas a regular submanifold of a manifold M consists of a subset A
of M , with smooth charts adapted from the charts of M , for which the inclusion ι : A ↪→ M is
a smooth embedding.
Given a smooth fiber bundle π : P → B, the corresponding vertical bundle is the bundle over
the manifold P with total space VP := ker dπ. The projection VP → P is the restriction of
the tangent bundle projection TP → P , and hence the vertical bundle is a subbundle of the
tangent bundle. We will also make use of associated bundles. Given a Lie group K, a manifold P
with a free and proper right action of K, and a manifold F with a proper left action of K, the
associated bundle is the bundle over the smooth orbit space P/K with total space P ×K F :=
(P × F )/K. Here, the group K acts on the space P × F by k · (p, f) := (p · k−1, k · f) in a free
and proper fashion from the left. We will denote the elements of P ×K F by [p, f ]. The bundle
projection P ×K F → P/K is defined by [p, f ] 7→ K · p, where K · p is the K-orbit of p. If the
manifold F is a product of the form M ×N , we will denote the elements of P ×K F by [p,m, n]
instead of [p, (m,n)].
2 Relative equilibria and isomorphic vector fields
In this section we define the groupoid of equivariant vector fields on a manifold with a group
action, and the corresponding notion of isomorphism of equivariant vector fields. We then
describe Krupa’s construction in this language, and Lerman’s results about the groupoids of
equivariant vector fields present in this construction. Along the way, we discuss how relative
equilibria are preserved by isomorphisms of equivariant vector fields, equivariant extension of
vector fields, pushforward and pullbacks of vector fields (when these are defined), and certain
functors between groupoids of equivariant vector fields.
We work in the following setting:
Definition 2.1 (G-manifold). A manifold M with an action of a Lie group G is called a G-
manifold. If the action of G is a proper action then we say M is a proper G-manifold.
By an equivariant vector field we mean:
Definition 2.2 (equivariant vector field). A vector field X on a manifold M is equivariant with
respect to the action of a Lie group G if for all g ∈ G we have X ◦ gM = dgM ◦ X, where
gM : M → M is the diffeomorphism m 7→ g ·m. If we need to specify the group we say X is
G-equivariant.
We next recall the definition of a relative equilibrium:
Definition 2.3 (relative equilibrium). Given an equivariant vector field X on a G-manifold M ,
a point m ∈ M is a relative equilibrium of X if the vector X(m) is tangent to the group orbit
G ·m. If we need to specify the group we say m is a G-relative equilibrium.
4 S. Klajbor-Goderich
Definition 2.4 (velocities). Let M be a proper G-manifold, let X be an equivariant vector
field on M , and let m be a point in M . A velocity for the point m is a vector ξ ∈ g such that
X(m) = ξM (m), where
ξM : M → TM, ξM (m) :=
d
dt
∣∣∣
0
exp(tξ) ·m
is the fundamental vector field generated by the vector ξ.
Remark 2.5. Velocities exist for relative equilibria since
Tm(G ·m) = {νM (m) | ν ∈ g}.
In fact, the existence of velocities at a point characterize that point as a relative equilibrium.
Furthermore, since
gm = {η ∈ g | ηM (m) = 0},
velocities are unique modulo the Lie algebra gm of the stabilizer.
The following maps are needed to define morphisms of vector fields:
Definition 2.6 (infinitesimal gauge transformations). Infinitesimal gauge transformations are
the elements of the vector space
C∞(M, g)G := {ψ : M → g |ψ(g ·m) = Ad(g)ψ(m) for all g ∈ G, m ∈M}.
Remark 2.7. If the action of G on M is free and proper, then the orbit space M/G is a mani-
fold and the orbit space map M → M/G is a principal G-bundle. In this case, the space of
infinitesimal gauge transformations C∞(M, g)G is isomorphic to the space of smooth sections of
the bundle M ×G g→M/G.
The space of infinitesimal gauge transformations C∞(M, g)G acts, as a group under pointwise
addition, on the space of equivariant vector fields Γ(TM)G by
C∞(M, g)G × Γ(TM)G → Γ(TM)G, (ψ,X) 7→ X + ψM ,
where ψM denotes the vector field on M defined by
ψM : M → TM, ψM (m) :=
d
dt
∣∣∣
0
exp (tψ(m)) ·m.
Lemma 2.8. Let M be a G-manifold and let ψ : M → g be an infinitesimal gauge transformation
on M . The induced vector field ψM is an equivariant vector field with respect to the action of G.
Proof. This is a consequence of the naturality of the exponential. Let g ∈ G and m ∈M , then
ψM (g ·m) =
d
dt
∣∣∣
0
exp(tψ(g ·m)) · g ·m
=
d
dt
∣∣∣
0
exp(tAd(g)ψ(m)) · g ·m by the equivariance of ψ
=
d
dt
∣∣∣
0
g exp(tψ(m))g−1 · g ·m by the naturality of exp
= (dgM )mψM (m).
Hence, ψM is an equivariant vector field. �
Nonlinear Stability of Relative Equilibria and Isomorphic Vector Fields 5
We can now define the groupoid of equivariant vector fields:
Definition 2.9 (groupoid of equivariant vector fields). Let M be a G-manifold. The groupoid
of equivariant vector fields X(G × M ⇒ M) is the action groupoid C∞(M, g)G n Γ(TM)G
corresponding to the action of the infinitesimal gauge transformations C∞(M, g)G on the G-
equivariant vector fields Γ(TM)G. The groupoid of G-equivariant vector fields has
objects: equivariant vector fields X ∈ Γ(TM)G,
morphisms: pairs (ψ,X) ∈ C∞(M, g)G × Γ(TM)G.
The source function is given by
s : C∞(M, g)G × Γ(TM)G → Γ(TM)G, (ψ,X) 7→ X,
and the target function is given by
t : C∞(M, g)G × Γ(TM)G → Γ(TM)G, (ψ,X) 7→ X + ψM .
The composition of a composable pair of morphisms (ϕ,X + ψM ) and (ψ,X) is given by
(ϕ,X + ψM ) ◦ (ψ,X) = (ϕ+ ψ,X).
The unit function is given by
u : Γ(TM)G → C∞(M, g)G × Γ(TM)G, X 7→ (0, X),
and the inversion function is given by
− : C∞(M, g)G × Γ(TM)G → C∞(M, g)G × Γ(TM)G, (ψ,X) 7→ (−ψ,X).
Remark 2.10. Hepworth [9] defined vector fields on differentiable stacks and showed they form
a category. In the case of a quotient stack [M/G] for the action of a compact group G on a mani-
fold M , Hepworth showed that the category X ([M/G]) of vector fields on the stack [M/G] is
equivalent to the category X(G×M ⇒M) given in Definition 2.9 [9, Proposition 5.1].
In the following definition we highlight what it means for two vector fields to be isomorphic
in the groupoid X(G×M ⇒M) of equivariant vector fields:
Definition 2.11 (isomorphic vector fields). Two equivariant vector fields X and Y on a G-
manifold M are G-isomorphic if there exists an infinitesimal gauge transformation ψ in the
space C∞(M, g)G such that
Y = X + ψM .
As noted in [12, Corollary 2.8], isomorphisms of equivariant vector fields preserve relative
equilibria in the following sense:
Lemma 2.12. Let X and Y be two isomorphic equivariant vector fields on a G-manifold M . If
a point m is a relative equilibrium of X then it is a relative equilibrium of Y .
Proof. Since X and Y are isomorphic, there exists a map ψ ∈ C∞(M, g)G such that Y =
X + ψM . Note that the vector X(m) is tangent to the group orbit G ·m since the point m is
a relative equlibrium of X. The vector ψM (m) is also tangent to the group orbit G ·m since
the vector ψM (m) is defined to be the derivative of a curve on the group orbit of the point m.
Thus, we have
Y (m) = X(m) + ψM (m) ∈ Tm(G ·m),
meaning the point m is a relative equilibrium of the vector field Y . �
6 S. Klajbor-Goderich
We will use the following vector fields in our application of Theorem 3.11 to Hamiltonian
relative equilibria in Section 4:
Definition 2.13 (augmented vector fields). Let M be a proper G-manifold and X an equivariant
vector field on M . Given a vector ξ ∈ g, the corresponding vector field augmented by ξ is the
vector field
Xξ : M → TM, Xξ := X − ξM .
Remark 2.14. Given a G-equivariant vector field X on a proper G-manifold M , the correspond-
ing augmented vector field Xξ is not G-equivariant. However, it is equivariant with respect to
the Lie subgroup
Gξ := {g ∈ G | Ad(g)ξ = ξ}.
Also note, that if ξ ∈ g is a velocity for a G-relative equilibrium m of the vector field X, then
the augmented vector field Xξ has an equilibrium at the point m.
Lemma 2.15. Let M be a proper G-manifold, let X be an equivariant vector field on M , and
let ξ be a given vector in the Lie algebra g of G. The vector field X is Gξ-isomorphic to its
augmented vector field Xξ ∈ Γ(TM)Gξ .
Proof. Let gξ be the Lie algebra of the Lie subgroup Gξ. The constant map
ξ : M → gξ, m 7→ ξ
is a smooth Ad(Gξ)-equivariant map, and hence gives a morphism of the groupoid of Gξ-
equivariant vector fields X(Gξ ×M ⇒ M). Note X = Xξ + ξM by definition, so the result
follows. �
Recall we can assemble the maximal integral curves of a smooth vector field on a Hausdorff
manifold into a maximal flow:
Definition 2.16 (Flow). Let M be a Hausdorff manifold and let X be a smooth vector field
on M . For every point m ∈M , let γm : Im →M be the maximal integral curve of X such that
γm(0) = m. Let A be the open subset of R×M defined by
A :=
⋃
m∈M
Im × {m}.
The maximal flow, or just flow, of the vector field X is the smooth map
φ : A→M, φ(t,m) := γm(t).
The set A is called the flow domain of φ.
Remark 2.17. It is important to recall that we are assuming all manifolds are Hausdorff, this is
required for some of the definitions and results in this paper. From now on, we won’t explicitly
mention this hypothesis.
The following result, due to Lerman, relates the flows of isomorphic vector fields:
Theorem 2.18 (Lerman [12, Theorem 1.6]). Let M be a proper G-manifold and let X and Y
be two isomorphic equivariant vector fields on M . Then there exists a family of smooth maps
{Ft : M → G} depending smoothly on t so that the maximal flows φX and φY , of X and Y
respectively, satisfy
φX(t,m) = Ft(m) · φY (t,m)
for all (t,m) ∈ R×M in the domain of the flow φX .
Nonlinear Stability of Relative Equilibria and Isomorphic Vector Fields 7
We recall the following notion of continuous flows on topological spaces:
Definition 2.19. Let Z be a topological space, and let B be an open subset of R×Z containing
the set {0} × Z. An abstract flow on Z is a continuous map Φ: B → Z satisfying:
1) Φ(0, z) = z for all z ∈ Z;
2) Φ(t,Φ(s, z)) = Φ(s+ t, z) whenever both sides make sense.
For a given point z ∈ Z, the curve of the abstract flow starting at z is the curve γz : Iz → Z
defined by γz(t) = Φ(t, z), where Iz consists of all times t for which (t, z) ∈ B.
We recall the following standard result about G-equivariant vector fields:
Lemma 2.20. Let M be a proper G-manifold and let X be an equivariant vector field on M .
The flow φ : A → M of the vector field X induces an abstract flow Φ: B → M/G on the orbit
space M/G such that the following diagram commutes:
A M
B M/G,
φ //
id×π
��
π
��
Φ
//
(2.1)
where π : M →M/G is the orbit map.
Proof. First, define the set B := (id×π)(A) ⊆ R×M/G and the map
Φ: B →M/G, Φ(t, π(m)) := π(φ(t,m)).
We want to show that the map Φ is our desired abstract flow. Thus, we need to show that the
set B is open, that B contains {0}×M/G, that the map Φ is well-defined and continuous, that Φ
makes the diagram (2.1) commute, and that Φ satisfies properties (1) and (2) in Definition 2.19.
Observe that the action of the Lie group G on the manifold M gives an action on the product
R×M by
g · (t,m) := (t, g ·m)
for all g ∈ G and (t,m) ∈ R ×M . The orbit space of this action is the product R ×M/G and
the quotient map is id×π : R×M → R×M/G, where π : M →M/G is the quotient map of the
given action. To see that the set B is open, it suffices to check that the open set A is saturated
with respect to the quotient map id×π, or equivalently that it is G-invariant with respect to
the action of G on the product R×M . For this, let (t,m) ∈ A and note, using the equivariance
of the vector field X, that the curve given by t 7→ g · φ(t,m) is the maximal integral curve of X
starting at g ·m. In particular, it is defined for the same times t that the integral curve starting
at the point m is defined. Thus, if (t,m) ∈ A then (t, g · m) ∈ A, or equivalently the flow
domain A is G-invariant. Furthermore, note that (id×π)({0} ×M) = {0} ×M/G. Hence,
{0} ×M/G = (id×π)({0} ×M) ⊆ (id×π)(A) = B
as desired.
Next, note that the map Φ is well-defined by the equivariance of the flow φ. Furthermore,
the map Φ makes the square (2.1) commute by definition. By the characteristic property of the
quotient topology, the map Φ is continuous if and only if the map Φ◦(id×φ) is continuous. Since
8 S. Klajbor-Goderich
the diagram (2.1) commutes, we have that Φ ◦ (id× φ) = π ◦ φ. Since π ◦ φ is the composition
of continuous maps, then the map Φ is continuous.
Now, for every point π(m) ∈M/G we have
Φ(0, π(m)) = π(φ(0,m)) = π(m)
since φ(0,m) = m. Similarly, the second property follows by the corresponding property of the
flow φ
Φ
(
t,Φ(s, π(m))
)
= Φ
(
t, π(φ(s,m))
)
= π
(
φ(t, φ(s,m))
)
= π
(
φ(s+ t,m)
)
= Φ
(
s+ t, π(m)
)
.
Hence, the map Φ: B →M/G is the desired abstract flow. �
The following is a corollary of Theorem 2.18 and Lemma 2.20 (also see [12, Corollary 2.8]):
Corollary 2.21. Let M be a proper G-manifold, and let X and Y be two isomorphic equivariant
vector fields on M . Then the maximal flows of X and Y have the same domain and induce the
same abstract flow on the orbit space M/G.
Proof. Let φX an φY be the maximal flows of the vector fields X and Y respectively. Let
{Ft : M → G} be the family of maps relating the flows (see Theorem 2.18). Thus, for all pairs
(t,m) in the domain of the flow φY we have
φX(t,m) = Ft(m) · φY (t,m). (2.2)
In particular, note that any pair (t,m) in the domain of the flow φY is in the domain of the
flow φX . Reversing the role of X and Y in Theorem 2.18 gives the opposite inclusion of the flow
domains. Hence, the flows φX and φY have the same domain.
Now let ΦX : B → M/G and ΦY : B → M/G be the induced flows on the orbit space of X
and Y respectively. Using equality (2.2) and the definition of the induced orbit space flow given
in the proof of Lemma 2.20, we have that
ΦX
(
t, π(m)
)
= π
(
φX(t,m)
)
= π
(
Ft(m) · φY (t,m)
)
= π
(
φY (t,m)
)
= ΦY
(
t, π(m)
)
.
Hence, the induced flows on the orbit space are equal. �
Abstract flows can also have fixed points:
Definition 2.22 (fixed point of an abstract flow). Let φ : A → Z be an abstract flow on
a topological space Z. A fixed point of the flow φ is a point z ∈ Z such that φ(t, z) = z for all
times t with (t, z) ∈ A.
Remark 2.23. Let M be a proper G-manifold and let π : M → M/G be the quotient map.
Observe that if X is an equivariant vector field on M , then a point m ∈ M is a relative
equilibrium of X if and only if the point π(m) ∈ M/G is a fixed point of the induced abstract
flow on the orbit space.
We proceed to describe Krupa’s decomposition following Lerman [12]. We begin by recalling
saturations and equivariant extension:
Definition 2.24 (saturation). Given a G-manifold M and a subset A ⊆M , the saturation of A
is the subset of M defined by G ·A := {g · a | g ∈ G, a ∈ A}.
Nonlinear Stability of Relative Equilibria and Isomorphic Vector Fields 9
Recall that equivariant maps out of regular submanifolds of a proper G-manifold have unique
equivariant extensions to the saturation of the submanifold, provided some additional hypotheses
as in the following standard lemma (see also [5, Lemma 2.10.1]):
Lemma 2.25. Let M and N be proper G-manifolds, let K be a Lie subgroup of G, let A be
a K-invariant regular submanifold of M , and let f : A → N be a K-equivariant map. Suppose
that the map G × A → G · A given by (g, a) 7→ g · a descends to a diffeomorphism from the
associated bundle G ×K A to the saturation G · A. Then there exists a unique G-equivariant
extension of the map f given by
εf : G ·A→ N, εf(g · a) := g · f(a).
Proof. Define the G-equivariant map
G×A→ N, (g, a) 7→ g · f(a).
By using the K-equivariance of the map f , note that this map is K-invariant with respect to
the action of K on G×A. Thus, this map descends to a smooth G-equivariant map
G×K A→ N, [g, a] 7→ g · f(a).
Using the diffeomorphism between the associated bundle G ×K A and the saturation G · A,
we obtain the smooth extension εf . To see it is unique, suppose that F : G · A → N is any
other G-equivariant extension. Then note F (g · a) = g · F (a) = g · f(a) = εf(g · a), and hence
F = εf . �
We recall the definition of a slice:
Definition 2.26 (slices). Given a G-manifold M , let Gm be the stabilizer of a point m ∈ M .
A slice for the action through m is a Gm-manifold V and a Gm-equivariant embedding j : V →M
such that
1) the point m is in the image j(V );
2) the saturation G · j(V ) is open in M ;
3) the map
G× V → G · j(V ), (g, v) 7→ g · j(v)
descends to a G-equivariant diffeomorphism
G×Gm V → G · j(V ), [g, v] 7→ g · j(v),
where G×Gm V := (G× V )/Gm is the associated bundle.
For the sake of conciseness, we often write G · V instead of G · j(V ).
Remark 2.27. It is a classic theorem of Palais [20] that slices exist for points in proper G-
manifolds (see also [4, Theorem 2.3.3]). In proper G-manifolds, it is also possible and convenient
to take the slice V through a point m to be an open ball around the origin of a vector space
with a representation of the stabilizer Gm (see, for example, [7, Theorem B.24]).
The following definition will be convenient for the sake of brevity:
Definition 2.28 (proper G-manifold with slice). A proper G-manifold with slice is a quintuple
(M,G,m, V, j) consisting of a proper G-manifold M , a point m on the manifold, and a slice V
for the action through the point m with corresponding Gm-equivariant embedding j : V →M .
10 S. Klajbor-Goderich
Remark 2.29. Let (M,G,m, V, j) be a proper G-manifold with slice. The following facts will
be important:
1. The bundle G · V → G · m has typical fiber V and is G-equivariantly diffeomorphic to
the associated bundle G×Gm V → G/Gm (see, for example, [4, Theorem 2.4.1]). In other
words, the following diagram, with the canonical maps, commutes:
G×Gm V G · V
G/Gm G ·m.
∼= //
�� ��∼= //
We think of G · V as a tubular neighborhood of the group orbit G ·m and often refer to
it as a tube. Thus, the associated bundle G ×Gm V → G/Gm serves as a model for the
tubular neighborhood G · V , and we will sometimes identify G×Gm V and G · V .
2. Definition 2.26 implies that the tangent space at the point m splits in the form
TmM = Tm(G ·m)⊕ Tmj(V ),
while for any point v ∈ V we have
Tj(v)M = Tj(v)(G · j(v)) + Tj(v)j(V ).
Definition 2.26 also implies that for any point v ∈ V , if a group element g ∈ G is such that
g · j(v) ∈ j(V ) then g ∈ Gm.
Remark 2.30. By the previous remark, we can model tubes generated by slices by considering
arbitrary associated bundles of the form G×K V , where K is a compact Lie subgroup of a Lie
group G, and V is an open ball around the origin in a vector space with a representation of K.
For such models, note that the point m := [1, 0] has as stabilizer the Lie subgroup K acting on
G ×K V as a subgroup of G. Therefore, the K-manifold V with the K-equivariant embedding
j : V ↪→ G×K V defined by j(v) := [1, v], is a slice for the action through the point m.
Remark 2.31. A proper G-manifold with slice (M,G,m, V, j) gives rise to two action groupoids,
namely:
• the action groupoid Gm × V ⇒ V of the slice;
• the action groupoid G× (G · V ) ⇒ G · V of the tube.
Thus, the choice of slice gives rise to two groupoids of equivariant vector fields in the sense of
Definition 2.9:
• the groupoid X(Gm × V ⇒ V ) of Gm-equivariant vector fields on the slice V ;
• the groupoid X(G×G · V ⇒ G · V ) of G-equivariant vector fields on the tube G · V .
It is a theorem of Lerman that these groupoids are equivalent (see [12, Theorem 1.16]). This
theorem was stated using 2-term chains of topological vector spaces. In Theorem 2.39 we state
his result using an equivalent formulation.
Given a proper G-manifold with slice, we can use the embedding of the slice to push forward
vector fields and infinitesimal gauge transformations onto the image of the slice. We can then
extend these uniquely to the tube as in Lemma 2.25. This assembles into a canonical functor as
follows:
Nonlinear Stability of Relative Equilibria and Isomorphic Vector Fields 11
Definition 2.32 (equivariant extension functor). Let (M,G,m, V, j) be a proper G-manifold
with slice. The equivariant extension functor is the functor
E : X(Gm × V ⇒ V ) X(G×G · V ⇒ G · V ),//
(
X Y
)(ψ,X) //
(
E(X) E(Y )
)
,
(E(ψ),E(X)) //� //
where for any vector field X ∈ Γ(TV )Gm we define
E(X) : G · V → T (G · V ), E(X)
(
g · j(v)
)
:= d(g ◦ j)X(v),
and for any infinitesimal gauge transformation ψ ∈ C∞(V, gm)Gm we define
E(ψ) : G · V → g, E(ψ)(g · j(v)) := Ad(g)ψ(v).
Remark 2.33. The equivariant extension functor makes use of push-forwards by the slice
embedding and of equivariant extension as in Lemma 2.25 at both the object and morphism
level. Let (M,G,m, V, j) be a proper G-manifold with slice. Using the notation of Lemma 2.25,
the equivariant extension functor E satisfies
E(X) = ε(j∗X) and E(ψ) = ε(j∗ψ)
for any equivariant vector field X and any infinitesimal gauge transformation ψ on the slice.
Furthermore, note that the image under the functor E of the space of Gm-equivariant vector
fields on the slice consists of the space of G-equivariant vertical vector fields on the bundle
G · V → G ·m. That the image is contained in the space of G-equivariant vertical vector fields
follows from the definition. That the functor on objects is surjective onto the vertical vector
fields can be shown by using the functor of Definition 2.37.
The functor E is only part of the equivalence stated in Remark 2.31. For a functor in the
opposite direction we first need to obtain a connection via a choice of Lie algebra splitting, as
follows:
Lemma 2.34. Let G be a Lie group with Lie algebra g, let K be a Lie subgroup with Lie
algebra k, and let A be a proper K-manifold. Then a choice of K-equivariant splitting g = k⊕ q
gives rise to a G-equivariant connection on the associated bundle G×K A→ G/K.
Proof. We show that the given splitting of the Lie algebra gives rise to a bundle projection from
the tangent bundle T (G ×K A) = TG ×K TA to the vertical bundle V(G ×K A) = G ×K TA.
Here, recall that the vertical bundle V(G ×K A) is a bundle over the total space G ×K A of
the associated bundle G ×K A → G/K. Thus, we show that the Lie algebra splitting induces
a connection Φ̃ ∈ Ω1
(
G×K A;V(G×K A)
)
.
The Lie algebra splitting gives rise to a K-equivariant projection P : g→ k. The projection P
in turn gives rise to a principal connection Φ ∈ Ω1(G;V(G)) on the principal K-bundle G →
G/K, where the subgroup K acts on G by right-multiplication. For any g ∈ G and X ∈ TgG,
this principal connection is given by
Φg(X) := (dLg)1P
((
dLg−1
)
g
(X)
)
.
The remaining part of the argument consists of showing that the principal connection Φ induces
a connection on the associated bundle G×K A→ G/K. This part of the argument is standard.
However, we include an overview here so that we can refer to the construction in the sequel (for
more details see, for example, [10, Section 11.8]).
12 S. Klajbor-Goderich
Consider the quotient map $ : TG× TA→ TG×K TA and the G-equivariant map
Φ× id : TG× TA→ TG× TA, (Φ× id)(X,Y ) = (Φ(X), Y ).
Since the composition $ ◦ (Φ× id) is K-invariant with respect to the action of the subgroup K
on the product TG× TA, there exists a unique smooth map Φ̃ such that the following diagram
commutes:
TG× TA TG× TA
TG×K TA TG×K TA.
Φ×id //
$
��
$
��
Φ̃
//
(2.3)
The map Φ̃ is idempotent since the map Φ is idempotent. Also, the image of the map Φ̃ is the
vertical bundle V(G ×K A) = G ×K TA. Hence, Φ̃ is a projection, so it gives a connection on
the associated bundle G ×K A → G/K. Furthermore, the map Φ̃ is G-equivariant since the
map Φ× id is G-equivariant and the G-action commutes with the quotient map $. Hence, the
map Φ̃ gives the desired G-equivariant connection. �
Definition 2.35 (connection induced by a splitting). Let (M,G,m, V, j) be a proper G-manifold
with slice and let g = gm ⊕ q be a Gm-equivariant splitting. The connection induced by the
splitting is the G-equivariant connection Φ̃ ∈ Ω1
(
G · V ;V(G · V )
)
obtained from Lemma 2.34
by setting K = Gm, setting A = V , and using the canonical G-equivariant diffeomorphism
G×Gm V ∼= G · V . The vertical projection of vector fields induced by the splitting is the map:
ν : Γ(T (G · V ))→ Γ(V(G · V )), ν(X) := Φ̃ ◦X.
Remark 2.36. Since the connection of Definition 2.35 is equivariant, the vertical projection of
vector fields maps G-equivariant vector fields to G-equivariant vector fields. Hence, we may also
take the vertical projection ν to be a map Γ(T (G · V ))G → Γ(V(G · V ))G. In fact, if H is any
Lie subgroup of G, the vertical projection takes H-equivariant vector fields to H-equivariant
vector fields. Hence, we may also take the vertical projection of vector fields to be a map
Γ(T (G · V ))H → Γ(V(G · V ))H .
Thus, we obtain the following functor that generalizes Krupa’s decomposition from [11]:
Definition 2.37 (projection functor). Let (M,G,m, V, j) be a proper G-manifold with slice, let
g = gm ⊕ q be a Gm-equivariant splitting, let P : g → gm be the corresponding Gm-equivariant
projection, and let ν : Γ(T (G · V ))G → Γ(V(G · V ))G be the vertical projection of equivariant
vector fields induced by the splitting (see Definition 2.35). The projection functor corresponding
to the Lie algebra splitting g = gm ⊕ q is the functor
P : X(G×G · V ⇒ G · V ) X(Gm × V ⇒ V ),//
(
X Y
)(ψ,X) //
(
P (X) P (Y )
)
,
(P (ψ),P (X)) //� //
where for any vector field X ∈ Γ(T (G · V ))G we define
P (X) : V → TV, P (X) := j∗(ν(X)),
and for any map ψ ∈ C∞(G · V, g)G we define
P (ψ) : V → gm, P (ψ) := j∗(P ◦ ψ).
Nonlinear Stability of Relative Equilibria and Isomorphic Vector Fields 13
Remark 2.38. Recall that given a smooth embedding we can pull back those vector fields on
the target manifold that are tangent to the image of the embedding. Hence, if (M,G,m, V, j) is
a proper G-manifold; the vertical vector fields on the bundle G · V → G ·m can be pulled-back
by the embedding j. Consequently, the functor P of Definition 2.37 is well-defined on objects.
We can now state the equivalence of groupoids mentioned in Remark 2.31, which is due
to Lerman. Instead of the functors of Definitions 2.32 and 2.37, Lerman used an equivalent
formulation in terms of 2-term chain complexes. We state the equivalence using the functor
formulation:
Theorem 2.39 (Lerman, [12, Theorem 4.3]). Let (M,G,m, V, j) be a proper G-manifold with
slice. The equivariant extension functor E : X(Gm × V ⇒ V ) → X(G × G · V ⇒ G · V ) (see
Definition 2.32) and the projection functor P : X(G × G · V ⇒ G · V ) → X(Gm × V ⇒ V )
corresponding to a choice of Gm-equivariant splitting g = gm ⊕ q (see Definition 2.37) form an
equivalence of categories. In particular, such functors satisfy
P ◦ E = id and E ◦ P ' id .
For a given equivariant vector field X on the tube G ·V , the natural isomorphism α : E ◦P ⇒ id
is of the form
αX =
(
ψX , E(P (X))
)
,
where ψX ∈ C∞(G ·V, g)G is an infinitesimal gauge transformation taking values in the comple-
ment q. Thus, the map ψX is such that
X = E(P (X)) + ψXG·V ,
where ψXG·V
is the vector field induced by the map ψX .
Remark 2.40. Lerman introduced this approach to Krupa’s decomposition to quantify the
result of the choices in slice and projection. The choice in slice is adressed as follows. Let M
be a proper G-manifold and m a point in M . If V1 and V2 are two slices for the action through
the point m, then the corresponding groupoids X(Gm×V1 ⇒ V1) and X(Gm×V2 ⇒ V2) of Gm-
equivariant vector fields are isomorphic groupoids [12, Lemma 3.21]. After perhaps shrinking
the slices, the isomorphism is induced by a Gm-equivariant diffeomorphism between the slices.
The choice in projection, or equivalently the choice of Lie algebra splitting, is addressed as
follows. Given a proper G-manifold with slice (M,G,m, V, j), and two choices of Gm-equivariant
splittings
g = gm ⊕ q1 = gm ⊕ q2,
the corresponding projection functors
P1, P2 : X(G×G · V ⇒ G · V )→ X(Gm × V ⇒ G× V )
are naturally isomorphic [12, Lemma 3.17].
As may be expected, the functors we have introduced preserve relative equilibria. We prepare
for the proof of this fact via Lemmas 2.41, 2.42, and 2.43.
Lemma 2.41. Let M and N be proper G-manifolds and let f : M → N be a G-equivariant
diffeomorphism. Suppose that X and Y are f -related equivariant vector fields on M and N
respectively. Then a point m ∈ M is a relative equilibrium of the vector field X if and only if
the point f(m) is a relative equilibrium of the vector field Y . Thus, pullbacks and pushforwards
of vector fields by equivariant diffeomorphisms preserve relative equilibria.
14 S. Klajbor-Goderich
Proof. The verification is a straightforward computation using the equation df ◦X = Y ◦ f .
First, suppose m is a G-relative equilibrium of the vector field X. Then
Y (f(m)) = (df)m(X(m)) ∈ (df)m(Tm(G ·m)) = Tf(m)(G · f(m)),
where (df)m(Tm(G ·m)) = Tf(m)(G ·f(m)) follows by the equivariance of the diffeomorphism f .
Thus, the point f(m) is a G-relative equilibrium of the vector field Y . The converse is completely
analogous. �
Lemma 2.42. Let M be a proper G-manifold, let K be a Lie subgroup of G, and let A be a K-
invariant regular submanifold of M satisfying the hypotheses of Lemma 2.25. Suppose that X
is a K-equivariant vector field on A and that the point a ∈ A is a K-relative equilibrium of
the vector field X. Then the point a is a G-relative equilibrium of the equivariant extension εX
of X. That is, equivariant extension preserves relative equilibria.
Proof. This is essentially a corollary of Lemma 2.41. Let ι : A ↪→ G · A be the inclusion of
the submanifold A. Note that the tube G · A is a K-manifold and that the inclusion ι is a K-
equivariant diffeomorphism onto its image A. Observe that the vector fields X and εX are
ι-related; in fact, εX restricts to X on A. Thus, by Lemma 2.41, we know that a is a K-relative
equilibrium of the vector field εX; that is, εX(a) ∈ Ta(K · a). Since A is a regular submanifold,
the tangent space Ta(K · a) is contained in the tangent space Ta(G · a). Hence, the point a is
a G-relative equilibrium of εX. �
Lemma 2.43. Let G be a Lie group, let K be a Lie subgroup, let A be a proper K-manifold,
and let g = k⊕ q be a K-equivariant splitting. Let ν : Γ(T (G×K A))G → Γ(V(G×K A))G be the
vertical projection induced by the splitting (Definition 2.35), and let X be a G-equivariant vector
field on the associated bundle G×KA. Then if the point m ∈ G×KA is a G-relative equilibrium
of the vector field X, it is also a G-relative equilibrium of the vertical projection ν(X). That is,
the vertical projection ν preserves relative equilibria.
Proof. Consider the quotient maps:
π : G×A→ G×K A, $ : TG× TA→ TG×K TA.
Let X be an equivariant vector field on the associated bundle G ×K A and suppose that the
point p = (g, a) ∈ G × A is such that the point m := π(p) ∈ G ×K A is a relative equilibrium
of the vector field X. Let Φ ∈ Ω1(G;VG) and Φ̃ ∈ Ω1
(
G×K A;V(G×K A)
)
be the connections
induced by the splitting of the Lie algebra (see Definition 2.35 and the proof of Lemma 2.34),
and let the map
ν : Γ
(
T
(
G×K A
))G → Γ
(
V
(
G×K A
))G
be the vertical projection of equivariant vector fields with respect to this connection (Defini-
tion 2.35). We want to show that the point m is a G-relative equilibrium of the vector field ν(X);
that is, we want to show that
ν(X)(m) ∈ Tm(G ·m).
Since the action of G commutes with the quotient maps π and $, observe that
$(Tp(G · p)) = Tm(G ·m). (2.4)
Furthermore, using that Tp(G · p) = TgG× {0}, it is clear that
(Φ× id)(Tp(G · p)) ⊆ Tp(G · p). (2.5)
Nonlinear Stability of Relative Equilibria and Isomorphic Vector Fields 15
Therefore:
Φ̃
(
Tm(G ·m)
)
= Φ̃
(
$(Tp(G · p)
)
by (2.4)
= $ ◦ (Φ× id)
(
Tp(G · p)
)
by (2.3)
⊆ $(Tp(G · p)) by (2.5)
= Tm(G ·m).
Consequently, since X(m) ∈ Tm(G ·m), the vertical projection of the vector X(m) is such that
ν(X)(m) = Φ̃(X(m)) ∈ Tm(G ·m).
Hence, the point m is a relative equilibrium of the vector field ν(X). �
Now we can prove that the functors of Definition 2.32 and Definition 2.37 also preserve
relative equilibria. Parts (3) and (4) of the following proposition are especially relevant to the
following section’s main theorem (Theorem 3.11).
Proposition 2.44. Let (M,G,m, V, j) be a proper G-manifold with slice, let E : X(Gm × V ⇒
V ) → X(G × G · V ⇒ G · V ) be the equivariant extension functor (Definition 2.32), and let
P : X(G×G ·V ⇒ G ·V )→ X(Gm×V ⇒ V ) be the projection functor corresponding to a choice
of Gm-equivariant splitting g = gm ⊕ q. Then the following are true:
1. If a point v ∈ V is a relative equilibrium of an equivariant vector field X on the slice V ,
the point j(v) is a relative equilibrium of the vector field E(X) on the tube G · V .
2. If a point j(v) ∈ j(V ) is a relative quilibrium of an equivariant vector field X on the tube
G · V , the point v ∈ V is a relative equilibrium of the vector field P (X) on the slice V .
3. If the point m, through which the slice was chosen, is a relative equilibrium of an equivariant
vector field X on the tube G · V , then the point j−1(m) is an equilibrium of the vector
field P (X) on the slice V .
4. Let the point m be a G-relative equilibrium of a G-equivariant vector field X on the tube
G · V , let H be a Lie subgroup of the stabilizer Gm, and let Y be an H-equivariant vector
field on the slice V that is H-isomorphic to the vector field P (X). Then the point j−1(m)
is an equilibrium of the vector field Y .
Proof. Parts (1) and (2) are a consequence of the fact that pullbacks and pushforwards (when
these are defined), equivariant extension of vector fields, and the vertical projection of Defini-
tion 2.35 preserve relative equilibria (see Lemmas 2.41, 2.42, and 2.43).
For part (3), let the map
ν : Γ
(
T
(
G×K A
))G → Γ
(
V
(
G×K A
))G
be the vertical projection of equivariant vector fields induced by the Lie algebra splitting (Defi-
nition 2.35). Recall that the tangent space at the point m splits as
Tm(G · V ) = Tm(G ·m)⊕ Tmj(V ),
because V is a slice through the point m (see Remark 2.29). Hence, the vector ν(X)(m) is zero
because X(m) ∈ Tm(G · m) since the point m is a relative equilibrium of the vector field X.
Now note that the vector fields P (X) and ν(X) are j-related by definition. Thus
(dj)P (X)
(
j−1(m)
)
= ν(X)(m) = 0.
16 S. Klajbor-Goderich
Since the map j is an embedding, the tangent map dj : TV → T (G · V ) is fiberwise injective.
Consequently, P (X)(j−1(m)) = 0, so the point j−1(m) is an equilibrium of the vector field P (X).
For part (4), let Y be an H-equivariant vector field on the slice V and let ψ ∈ C∞(V, h)H
be a map such that Y = P (X) + ψV . Recall that there exists a slice V ′ through the point m,
with corresponding Gm-equivariant embedding j′ : V ′ →M , such that V ′ is an open ball around
the origin j′−1(m) in a vector space with a linear representation of the stabilizer Gm (see Re-
mark 2.27). After perhaps shrinking the slices, there exists a Gm-equivariant diffeomorphism
φ : V → V ′ taking j−1(m) to j′−1(m) (see Remark 2.40). Note that the vector fields φ∗Y and
φ∗P (X) on V ′ are H-isomorphic (the isomorphism is given by the map ψ ◦ φ−1). Furthermore,
the point j′−1(m) is an equilibrium of φ∗P (X) since the point j−1(m) is an equilibrium of P (X)
by part (3). If the vector field φ∗Y has an equilibrium at j′−1(m), then the vector field Y has
an equilibrium at the point j−1(m). Therefore, it is of no loss of generality to suppose that the
slice V is an open ball around the origin j−1(m) in a vector space with a linear representation
of Gm. With this assumption, note that
Y
(
j−1(m)
)
= P (X)
(
j−1(m)
)
+ ψV
(
j−1(m)
)
= ψV
(
j−1(m)
)
by part (3)
=
d
dt
∣∣∣
0
exp(tψ(0)) · 0 since j−1(m) = 0
=
d
dt
∣∣∣
0
0 since the action is linear
= 0.
Hence, the point j−1(m) is an equilibrium of the vector field Y . �
Remark 2.45. Let (M,G,m, V, j) be a proper G-manifold with slice. In this paper, we will
sometimes consider vector fields that are equivariant only with respect to a Lie subgroup H of
the full symmetry group G. We view these as objects of the groupoid
X(H ×G · V ⇒ G · V ) := C∞(G · V, h)H n Γ(T (G · V ))H
of H-equivariant vector fields in the tube G · V . Given a Gm-equivariant splitting g = gm ⊕ q,
the corresponding G-equivariant connection Φ̃ ∈ Ω1(G · V ;V(G · V )) is also H-equivariant. As
stated in Remark 2.36, the vertical projection of Definition 2.35 takes H-equivariant vector
fields to H-equivariant vector fields. However, we need to generalize the projection functor of
Definition 2.37 to handle the morphisms of the groupoid X(H × G · V ⇒ G · V ). For this, we
make a choice of Hm-equivariant splitting h = hm ⊕ p. This gives an Hm-equivariant projection
map PH : h→ hm. With this we can generalize the projection functor of Definition 2.37.
Definition 2.46 (projection functor with respect to a subgroup). Let (M,G,m, V, j) be a proper
G-manifold with slice, let H be a Lie subgroup of G, and suppose you are given splittings
g = gm ⊕ q and h = hm ⊕ p that are Gm-equivariant and Hm-equivariant respectively. Let
ν : Γ(T (G · V ))H → Γ(V(G · V ))H be the vertical projection of H-equivariant vector fields
induced by the splitting of g and let PH : h → hm be the projection induced by the splitting
of h. The projection functor with respect to the subgroup H is the functor
PH : X(H ×G · V ⇒ G · V ) X(Hm × V ⇒ V ),//
(
X Y
)(ψ,X), // (PH(X) PH(Y )),
(PH(ψ),PH(X)) //� //
where for any vector field X ∈ Γ(T (G · V ))G we define
PH(X) : V → TV, PH(X) := j∗(ν(X)),
Nonlinear Stability of Relative Equilibria and Isomorphic Vector Fields 17
and for any map ψ ∈ C∞(G · V, h)H we define
PH(ψ) : V → hm, PH(ψ) := j∗(PH ◦ ψ).
3 Stability of relative equilibria
In this section we show that stability is preserved by isomorphisms of equivariant vector fields
(Proposition 3.4), opening the door to replacing the given vector field with an isomorphic one
that is potentially easier to work with. The main result of this section (Theorem 3.11) is
a stability test that involves passing from the category of equivariant vector fields on a tube to
the category of equivariant vector fields on a slice, which is also easier to work with.
We begin by recalling the following definition of nonlinear stability in a proper G-manifold
due to Patrick [21, 22]:
Definition 3.1 (stability modulo a subgroup). Let M be a G-manifold, let X be a G-equivariant
vector field on M , and let H ≤ G be a Lie subgroup of G. A G-relative equilibrium m ∈ M of
the vector field X is H-stable, or stable modulo H, if for any H-invariant neighborhood U ⊆M
of the point m there exists a neighborhood O ⊆ U of the point m for which all maximal integral
curves of the vector field X starting at points in the neighborhood O stay in the neighborhood U
for all times for which they are defined.
Remark 3.2. Let X be a smooth vector field on a G-manifold M , and let φ : A → M be its
flow (Definition 2.16). Stability modulo a subgroup H (Definition 3.1) can be rephrased as
saying that the relative equilibrium m of the vector field X is H-stable if for all H-invariant
neighborhoods U of the point m, there exists a neighborhood O ⊆ U , containing the point m,
for which the flow φ of the vector field X satisfies φ(t, q) ∈ U for all pairs (t, q) ∈ A with q ∈ O.
The following fact about G-stability will be useful later:
Lemma 3.3. Let M be a G-manifold, let X be a G-equivariant vector field on the manifold M ,
let m ∈ M be a G-relative equilibrium of X, and let H ≤ K be Lie subgroups of G. If the
G-relative equilibrium m is H-stable, then it is K-stable.
Proof. Any K-invariant neighborhood U of the point m is in particular H-invariant since
H ≤ K. Hence, we can find the required neighborhood O ⊆ U by using the H-stability of the
point m. �
We now show that the stability of relative equilibria is preserved by morphisms of equivariant
vector fields:
Proposition 3.4. Let M be a proper G-manifold and let X and Y be two isomorphic equivariant
vector fields on M . If a point m ∈ M is a G-stable relative equilibrium of the vector field X,
then it is a G-stable relative equilibrium of the vector field Y .
Proof. Let φX and φY be the maximal flows of the vector fields X and Y respectively. By
Theorem 2.18 we know there exists a family of smooth maps {Ft : M → G}, depending smoothly
on t, such that
φY (t, q) = Ft(q) · φX(t, q)
for all pairs (t, q) for which φX is defined. Recall that this also shows the flows φX and φY have
the same domain (see Corollary 2.21).
Now let U ⊆M be a G-invariant open neighborhood of the relative equilibrium m. We seek
a neighborhood O ⊆ U of the point m such that all maximal integral curves of Y starting at
18 S. Klajbor-Goderich
points in O stay in U for all times in their domain. Since the point m is G-stable for the vector
field X, we know there exists a neighborhood O ⊆ U of the point m for which all integral curves
of X starting at points of O stay in U for all time. This means that for any point q ∈ O and all
times t for which (t, q) is in the domain of φY , we have that
φY (t, q) = Ft(q) · φX(t, q) ∈ Ft(q) · U = U,
where the last equality holds since the neighborhood U is G-invariant. Thus, the relative equi-
librium m is G-stable for the vector field Y . �
There is also a notion of stability for fixed points of abstract flows:
Definition 3.5. Let z be a fixed point of an abstract flow φ : A→ Z on a topological space Z.
The point z is stable if for all neighborhoods U ⊆ Z of the point z, there exists a neighborhood
O ⊆ U , containing the point z, such that for all pairs (t, q) ∈ A with q ∈ O, we have that
φ(t, q) ∈ U .
Next, we relate G-stability on a proper G-manifold with stability on the orbit space:
Lemma 3.6. Let M be a proper G-manifold, let π : M →M/G be the quotient map, and let X
be an equivariant vector field on M with the point m as a relative equilibrium. Let φ : A → M
be the maximal flow of X and Φ: B → M/G the induced abstract flow on the orbit space
(Lemma 2.20). Then the relative equilibrium m ∈ M is G-stable for the vector field X (in the
sense of Definition 3.1) if and only if the fixed point π(m) ∈ M/G is stable for the induced
flow Φ on the orbit space (in the sense of Definition 3.5).
Proof. First, suppose that the point m is a G-stable relative equilibrium of the vector field X
and let U ⊆M/G be an open neighborhood of the point π(m). We seek an open neighborhood
O ⊆ U of π(m) such that the curves of the abstract flow starting at points in O stay in U
for all times in their domain. Note that π−1(U) ⊆ M is a G-invariant open neighborhood of
the point m. Since the relative equilibrium m of the vector field X is G-stable, there exists a
neighborhood V ⊆ π−1(U) such that all maximal integral curves of X starting at points of V
stay in π−1(U) for all times in their domain. We need a G-invariant, and hence saturated,
neighborhood of the point m with the same properties as V . Define
W :=
⋃
g∈G
gM (V ).
As desired, this set is open, it is contained in π−1(U), and all maximal integral curves of X
starting at points in W stay in π−1(U) for all times in their domain. This set is open since it
is the union of the sets gM (V ), each of which are in turn open because the group translations
gM : M → M are diffeomorphisms. We know that the neighborhood W is contained in the
neighborhood π−1(U) because V ⊆ π−1(U) and the neighborhood π−1(U) is G-invariant. To
verify the last property, let q ∈ V and g ∈ G, so that g · q ∈ W is an arbitrary point in W .
By the choice of V , the maximal integral curve φ(·, q) of X starting at q stays in π−1(U) for
all times for which it is defined. Thus, by the G-equivariance of the flow and the G-invariance
of π−1(U)
φ(t, g · q) = g · φ(t, q) ∈ gM
(
π−1(U)
)
= π−1(U)
for all times t such that (t, g · q) ∈ A. Hence, W is as claimed.
Now consider the set O := π(W ) ⊆ π(π−1(U)) ⊆ U . The set O is an open neighborhood
of π(m) since the set W is a G-invariant, and hence saturated, open neighborhood of m in M .
Nonlinear Stability of Relative Equilibria and Isomorphic Vector Fields 19
It is contained in U since π(W ) ⊆ π(π−1(U)) ⊆ U . Furthermore, the curves of the abstract flow
starting at points in O stay in U for all times in their domain. To verify this last statement, let
q ∈ W , so that π(q) ∈ O is an arbitrary point in O. Observe that, by diagram (2.1), we have
that
Φ(t, π(q)) = π(φ(t, q))
for all t such that (t, π(q)) ∈ B. Hence, by the choice of W , for all times t such that (t, π(q)) ∈ B,
we have that
Φ(t, π(q)) = π(φ(t, q)) ∈ π
(
π−1(U)
)
⊆ U.
Hence, the point π(m) is stable in the sense of Definition 3.5.
Conversely, let the point π(m) be stable in the sense of Definition 3.5 and let U ⊆ M be an
open G-invariant neighborhood of the relative equilibrium m. We seek an open neighborhood
O ⊆ U of the point m such that the maximal integral curves of X starting at points in O stay
in U for all times in their domain. Since the open set U is G-invariant, and hence saturated, the
set π(U) ⊆ M/G is an open neighborhood of the point π(m). The stability of the point π(m)
implies that there exists a neighborhood V ⊆ π(U) of the point π(m) such that the curves of
the abstract flow starting at points of in V stay in π(U) for all times in their domain.
Now consider the open neighborhood O := π−1(V ) of the point m. Note that the neigh-
borhood O is contained in U since O = π−1(V ) ⊆ π−1(π(U)) = U ; where we use that U is
G-invariant. Furthermore, for all points q ∈ O the maximal integral curve φ(·, q) is such that
π(φ(t, q)) = Φ(t, π(q)) ∈ π(U)
for all times t with (t, q) ∈ A. Thus, by the G-invariance of the neighborhood U , we know that
φ(t, q) ∈ π−1(π(U)) = U
for all times t with (t, q) ∈ A. Hence, the relative equilibrium m of the vector field X is G-stable
in the sense of Definition 3.1. �
Remark 3.7. Lemma 3.6 says that stability of a relative equilibrium reduces to stability of the
fixed point of the induced flow on the orbit space. If the orbit space is a manifold, for example
when the action is free and proper, then one can appeal to the vast literature on stability of
fixed points to test for stability. However, if the action is not free, the orbit space is in general
not a manifold. In that case we must appeal to other arguments like the ones presented in this
paper. Proposition 3.4 is key in doing this.
Lemma 3.6 provides another way to show Proposition 3.4:
Proof. By Lemma 2.12, the relative equilibrium m of the vector field X is also a relative
equilibrium of the vector field Y . By Lemma 3.6, the G-stability of the relative equilibrium m
of Y corresponds to the stability (in the sense of Definition 3.5) of the corresponding fixed point
of the induced orbit space flow. On the other hand, by Corollary 2.21, the vector fields X
and Y induce the same abstract flow on the orbit space. Thus, the G-stability of the relative
equilibrium m for the vector field X implies the stability of the corresponding fixed point of the
abstract flow on the orbit space induced by Y . Hence, the relative equilibrium m of the vector
field Y is G-stable. �
To prove the slice stability criterion (Theorem 3.11) we need to show that the equivariant
extension functor of Definition 2.32 preserves the stability of relative equilbiria. For this, we
need the following two lemmas:
20 S. Klajbor-Goderich
Lemma 3.8. Let M and N be proper G-manifolds and let f : M → N be a G-equivariant
diffeomorphism. Suppose that X and Y are f -related equivariant vector fields on M and N
respectively. Then a point m is a G-stable G-relative equilibrium of the vector field X if and
only if the point f(m) is a G-stable G-relative equilibrium of the vector field Y . In particular, the
pushforward and pullback of vector fields by the diffeomorphism f preserve stability of relative
equilibria.
Proof. Suppose first that the relative equilibrium m is G-stable. Let U ⊆ N be a G-invariant
neighborhood of the point f(m). We seek a neighborhood O ⊆ U of the point f(m) such that
all maximal integral curves of Y starting at points in O stay in U for all times in their domain.
By the equivariance of f and the G-invariance of the set U , the open set f−1(U) is a G-invariant
neighborhood of the point m. By the G-stability of the point m, there exists a neighborhood
W ⊆ f−1(U) of the point m such that the maximal integral curves of X starting at points of W
stay in the set U for all times in their domain.
Consider the set O := f(W ). It is open since the map f is a diffeomorphism. It is contained
in the neighborhood U since f is a diffeomorphism and W ⊆ f−1(U). Consider an arbitrary
point q ∈ O and let γq be the maximal integral curve of the vector field Y starting at the
point q. Since the vector fields X and Y are f -related, the curve f−1 ◦γq is the maximal integral
curve of X starting at the point f−1(q) ∈ W , and it is defined for the same times that γq is.
By the choice of W , we know that f−1(γq(t)) ∈ f−1(U) for all times t such that the curve is
defined. Hence, γq(t) = f(f−1(γq(t))) ∈ f(f−1(U)) = U for all times t for which the curve is
defined. Therefore, the relative equilibrium f(m) is G-stable for Y . The converse is completely
analogous. �
Lemma 3.9. Let M be a proper G-manifold, let K be a Lie subgroup of G, and let A be a K-
invariant regular submanifold of M satisfying the hypotheses of Lemma 2.25. Suppose that X is
a K-equivariant vector field on A and that the point a ∈ A is a K-stable K-relative equilibrium
of X. Then the point a is a G-stable G-relative equilibrium of the equivariant extension εX
of X.
Proof. By Lemma 2.42, we know that the point a is a relative equilibrium of the equivariant
extension εX. Hence, it remains to show that the relative equilibrium is G-stable. Let ι : A ↪→
G ·A be the inclusion map of the submanifold A, and let U ⊆ G ·A be an arbitrary G-invariant
neighborhood of the point a. We seek an open neighborhood O ⊆ U of the point a such that
all maximal integral curves of the vector field εX starting at points in O stay in U for all time.
Observe that the set U ∩A is a K-invariant neighborhood in the subspace topology of A. Hence,
there exists an open set W in G · A such that W ∩ A is contained in U ∩ A, and the maximal
integral curves of X starting at points in the set W ∩ A stay in U ∩ A for all times in their
domains.
Consider the saturation O := G ·W = G · (W ∩ A). This set is open in G · A since it is the
union, over all elements g ∈ G, of the open sets gM (W ). Also note that for all g ∈ G we have
gM (W ∩ A) ⊆ gM (U ∩ A) = U , where the last equality uses the G-invariance of U . Hence, the
neighborhood O is contained in U .
Now let w ∈W ∩A and g ∈ G, so that q = g ·w ∈ O is an arbitrary point in O. Let γq and γw
be the maximal integral curves of the equivariant extension εX starting at the points q and w
respectively. Since the vector fields εX and X are ι-related, the curve γw is also the maximal
integral curve of the vector field X starting at the point w. Consequently, by the choice of W
and the K-stability of the relative equilibrium a of X, we have that γw(t) ∈ U ∩A for all times t
for which it is defined. This and the G-equivariance of the flow of εX implies that, for all times t
for which the integral curve γq is defined, we have that
γq(t) = g · γw(t) ∈ gM (U ∩A) = U,
Nonlinear Stability of Relative Equilibria and Isomorphic Vector Fields 21
where we also use the G-invariance of the set U . Therefore, the point a is G-stable for the vector
field εX. �
Proposition 3.10. Let (M,G,m, V, j) be a proper G-manifold with slice, let E : X(Gm × V ⇒
V )→ X(G×G ·V ⇒ G ·V ) be the equivariant extension functor (Definition 2.32), and let X be
a Gm-equivariant vector field on the slice V . Then if a point v ∈ V is a Gm-stable Gm-relative
equilibrium of the vector field X, the point j(v) is a G-stable G-relative equilibrium of the vector
field E(X) on the tube G · V . That is, the functor E preserves stability of relative equilibria.
Proof. We have already shown that the point j(v) is a G-relative equilibrium of the vector
field E(X) in part (1) of Proposition 2.44, so it remains to show the statement concerning
stability. Apply Lemma 3.8 to the Gm-equivariant diffeomorphism j : V → j(V ). This shows
that the point j(v) is Gm-stable for the vector field j∗X on the regular submanifold j(V ). Now
apply Lemma 3.9 with K = Gm, A = j(V ), and the vector field j∗X. This shows that the
point j(v) is G-stable for the vector field E(X) = εj∗X on the tube G · V , which is what we
wanted to prove. �
The following theorem provides a criterion for G-stability of a relative equilibrium and is the
main theorem of this section:
Theorem 3.11 (slice stability criterion). Let (M,G,m, V, j) be a proper G-manifold with slice,
let P : X(G × G · V ⇒ G · V ) → X(Gm × V ⇒ V ) be the projection functor corresponding to
a choice of Gm-equivariant splitting g = gm ⊕ q (Definition 2.37), and let X be a G-equivariant
vector field on the tube G · V with the point m as a G-relative equilibrium. Suppose there exists
a Lie subgroup H of the stabilizer Gm and an H-equivariant vector field Y on the slice V such
that
1) the vector field Y is H-isomorphic to the projected vector field P (X);
2) the point j−1(m) ∈ V is an H-stable equilibrium of the vector field Y .
Then the G-relative equilibrium m of the vector field X is G-stable.
Remark 3.12. Before proceeding with the proof of Theorem 3.11, we make some observations:
1. By Proposition 2.44, we know that the point j−1(m) is an equilibrium of the vector
field P (X). The same proposition also shows that the point j−1(m) is an equilibrium
of any vector field isomorphic to P (X) with respect to any Lie subgroup of the stabi-
lizer Gm.
2. Since we can choose the slice V to be an open ball around the origin in a finite dimensional
vector space with a representation of the stabilizer Gm (see part (2) of Remark 2.29), Theo-
rem 3.11 reduces determining nonlinear stability of a relative equilibrium on a manifold to
determining stability of an equilibrium on a vector space. Furthermore, it replaces dealing
with an action of a possibly noncompact Lie group with dealing with a representation of
a compact Lie group. This is desirable since there is a vast literature available on the
stability of equilibria on vector spaces with representations of compact Lie groups (see, for
example, Field [5] and the references therein).
3. Any choice of slice or projection leads to isomorphic projected vector fields (see Re-
mark 2.40). Hence, one may choose any convenient slice and projection to apply Theo-
rem 3.11.
4. Given a G-equivariant vector field with a G-relative equilibrium for which we want to test
for stability, Theorem 3.11 allows us to break the symmetry of the projected vector field by
considering an isomorphic vector field with respect to a subgroup of the symmetry group.
22 S. Klajbor-Goderich
Such a replacement of the given vector field by an isomorphic one will be illustrated in our
application (Theorem 4.24) in Section 4.
We now proceed with the proof of Theorem 3.11:
Proof. Let E : X(Gm × V ⇒ V )→ X(G×G · V ⇒ G · V ) be the equivariant extension functor
(see Definition 2.32). The proof uses that stability with respect to a subgroup implies stability
with respect to the larger group (Lemma 3.3), that morphisms of equivariant vector fields and
the equivariant extension functor preserve stability (Propositions 3.4 and 3.10), and that the
functors E and P form an equivalence of categories (Theorem 2.39). The argument is as follows.
Since the vector fields Y and P (X) are H-isomorphic, Proposition 3.4 implies that the
point j−1(m) is H-stable for the vector field P (X). Since H is a subgroup of the stabilizer Gm,
Lemma 3.3 implies that the point j−1(m) is Gm-stable for the vector field P (X). By Proposi-
tion 3.10, the point m is G-stable for the equivariant extension E(P (X)). Now note that the
vector fields X and E(P (X)) are G-isomorphic by Theorem 2.39. Thus, by Lemma 2.12, the
G-relative equilibrium m of the vector field X is also a G-relative equilibrium of the vector
field E(P (X)). Finally, by Proposition 3.4, the G-relative equilibrium m of the vector field X
is G-stable since it is G-stable for the isomorphic vector field E(P (X)). �
4 Stability of Hamiltonian relative equilibria
In this section we apply the slice stability criterion (Theorem 3.11) to Hamiltonian relative
equilibria. We recall the definition of an equivariant momentum map:
Definition 4.1. Let M be a symplectic manifold with an action of a Lie group G by Hamiltonian
symplectomorphisms. Suppose the symplectic form ω is G-invariant. A smooth map Φ: M → g∗
is an equivariant momentum map for the action of G on the symplectic manifold M if:
• the map Φ is equivariant with respect to the given action on M and the coadjoint repre-
sentation on g∗;
• for all vectors ξ ∈ g, the function
〈Φ, ξ〉 : M → R, m 7→ 〈Φ(m), ξ〉,
where 〈·, ·〉 : g∗×g→ R is the pairing of the Lie algebra g and its dual g∗, is a Hamiltonian
function for the fundamental vector field ξM .
Throughout this section we will work in the following settings:
Definition 4.2. A Hamiltonian G-space is a quadruple (M,ω,G,Φ) where M is a proper G-
manifold such that
1) the manifold M is symplectic with corresponding G-invariant symplectic form ω;
2) the action of the Lie group G is by Hamiltonian symplectomorphisms;
3) the map Φ: M → g∗ is an equivariant momentum map for the action.
Definition 4.3. A Hamiltonian G-system is a quintuple (M,ω,G,Φ, h) where the quadruple
(M,ω,G,Φ) is a Hamiltonian G-space and the function h : M → R is a smooth G-invariant
function called the Hamiltonian function of the system.
We are interested in the stability of G-relative equilibria of Hamiltonian vector fields. In
particular, we determine stability with respect to the following group:
Nonlinear Stability of Relative Equilibria and Isomorphic Vector Fields 23
Definition 4.4. Let (M,ω,G,Φ) be a Hamiltonian G-space and let m be a point in the mani-
fold M . The covector µ := Φ(m) ∈ g∗ is called the moment of the point m, and the Lie subgroup
of G defined by
Gµ :=
{
g ∈ G | Ad†(g)µ = µ
}
is called the moment isotropy group of the point m. We denote by gµ the Lie algebra of the
moment isotropy group.
Hamiltonian relative equilibria of Hamiltonian G-systems have the following well-known cha-
racterization (see, for example, [15, Theorem 4.1]):
Lemma 4.5. Let (M,ω,G,Φ, h) be a Hamiltonian G-system, let Ξh ∈ Γ(TM)G be the Hamil-
tonian vector field of the function h, and let m ∈ M be a point with moment µ = Φ(m). The
following are equivalent:
1) the point m is a G-relative equilibrium of the Hamiltonian vector field Ξh;
2) there exists a velocity vector ξ ∈ gµ of the point m; that is, ξ ∈ gµ is such that Ξh(m) =
ξM (m);
3) the point m ∈M is a critical point of the function hξ := h− 〈Φ, ξ〉.
Definition 4.6. Let (M,ω,G,Φ, h) be a Hamiltonian G-system. Given a vector ξ in the Lie
algebra g, the Hamiltonian function augmented by ξ, or simply the augmented Hamiltonian, is
the function
hξ : M → R, hξ := h− 〈Φ, ξ〉.
The corresponding Hamiltonian vector field Ξhξ = Ξh−ξM is called the Hamiltonian vector field
augmented by ξ, or simply the augmented Hamiltonian vector field.
Remark 4.7. Given a smooth function f ∈ C∞(M) on a manifold M , the Hessian of f is only
well-defined at critical points of f . If m ∈M is a critical point of f , the Hessian d2f(m) : TmM×
TmM → R of f at the point m behaves well under change of coordinates and pull-backs. That
is, if j : N →M is a smooth map with j(n) = m, then
d2(j∗f)(n) = j∗
(
d2f(m)
)
.
In particular, if N is a submanifold of M , then(
d2f
)
(m)|TmN = d2(f |N t)(m).
Lemma 4.5 guarantees that if m is a G-relative equilibrium of the Hamiltonian vector field Ξh,
then the augmented hamiltonian hξ has a well-defined Hessian at m.
Lerman and Singer [13] and Ortega and Ratiu [18], building on work of Patrick [21, 22], proved
a criterion for Gµ-stability of a Hamiltonian G-relative equilibrium involving the Hessian of the
augmented Hamiltonian hξ. Their work required an orthogonality condition on the velocity ξ.
Montaldi and Rodŕıguez-Olmos were able to eliminate this condition in a generalized criterion;
first, for the case of compact moment isotropy in [17, Theorem 2]; and then, more generally,
for the case of possibly noncompact moment isotropy in [16, Theorem 3.6]. They also provide
an example where this criterion predicts stability, while previous criteria were inconclusive [16,
Remark 3.7]. Their result is as follows:
24 S. Klajbor-Goderich
Theorem 4.8 (Montaldi and Rodŕıguez-Olmos [16, Theorem 3.6]). Let (M,ω,G,Φ, h) be a Ha-
miltonian G-system. Suppose m ∈ M is a G-relative equilibrium of the Hamiltonian vector
field Ξh of the function h, and let µ be the moment of the point m. Suppose further that
1) the moment isotropy subgroup Gµ acts properly on the manifold M ;
2) there exists an Ad(Gµ)-invariant inner product on the Lie algebra g of G;
3) there exists a velocity vector ξ ∈ gµ of the point m such that the Hessian d2hξ(m) is definite
and nondegenerate on a Gm-invariant complement W to the tangent space Tm(Gµ ·m) in
ker dΦm.
Then the G-relative equilibrium m is Gµ-stable.
Remark 4.9. The stability criteria in [13, 16, 17, 18] can be seen from the point of view of the
isomorphic relationship between the Hamiltonian vector field and its augmented counterparts
(the precise isomorphic relationship follows from Lemma 2.15). The main goal of this section
is to provide an alternative proof of Theorem 4.8 from this point of view. Recall that the
slice stability criterion (Theorem 3.11) uses isomorphisms of equivariant vector fields to obtain
a criterion for stability of relative equilibria. In Lemma 4.15, and Corollaries 4.16 and 4.17,
which we will prove next, we describe a relationship between the invariant constants of motion
of isomorphic vector fields. We use this relationship in order to apply Theorem 3.11 in our proof
of Theorem 4.8.
Remark 4.10. To prove Theorem 4.8, one can reduce to the case when the momentum of the
relative equilibrium is fixed by the coadjoint representation. Equivalently one can reduce to
the case when the momentum is zero. This can be achieved by using symplectic cross-sections
exactly like in [13, Section 2.3]. This assumption simplifies several arguments in the proof of
Theorem 4.8, so we will suppose it holds throughout the rest of the paper. One may also assume
without loss of generality that the hamiltonian function is such that h(m) = 0.
Remark 4.11. We will need several norms on finite-dimensional Lie algebras and their duals.
Let G be a Lie group, let g be its Lie algebra, and let g∗ be the dual of g. Recall that a G-
invariant inner product (·, ·)g on the Lie algebra g induces a G-invariant norm || · ||g on g given
by
||ν||g :=
√
(ν, ν)g
for all ν ∈ g. Let (·, ·)g∗ be the inner product on g∗ dual to the inner product on g. Then we
also get a G-invariant norm || · ||g∗ on g∗ given by
||ρ||g∗ :=
√
(ρ, ρ)g∗
for all ρ ∈ g∗. The inner product on g also induces a G-invariant sup norm || · ||∞ on the dual g∗
given by
||ρ||∞ := sup
||ν||g=1
|〈ρ, ν〉|
for all ρ ∈ g∗, where 〈·, ·〉 : g∗×g→ R is the pairing between g and g∗. Recall that the sup norm
satisfies |〈ρ, ν〉| ≤ ||ρ||∞||ν||g for all ρ ∈ g∗ and all ν ∈ g.
We recall the following definition:
Definition 4.12 (constants of motion). Let M be a manifold and let X be a smooth vector
field on M . A smooth function f ∈ C∞(M) on the manifold M is a constant of motion of X if
df(X) = 0.
Nonlinear Stability of Relative Equilibria and Isomorphic Vector Fields 25
Remark 4.13. Let X be a smooth vector field on a manifold M , and let f be a smooth function
on M . A straightforward check shows that f is a constant of motion of X if and only if f is
conserved along each integral curve of X. Here we say f is conserved, or is constant, along an
integral curve α of X, if for every time t such that α is defined we have that f(α(t)) = f(α(0)).
Remark 4.14. Recall that if (M,ω,G,Φ, h) is a Hamiltonian G-system, then the Hamiltonian h
is a constant of motion of the Hamiltonian vector field Ξh. Furthermore, it is Noether’s theo-
rem that the momentum Φ is constant along the integral curves of Ξh (see, for example, [15,
Theorem 2.2]). That is, for all vectors ξ ∈ g, the function 〈Φ, ξ〉 is a constant of motion of the
Hamiltonian vector field Ξh. Suppose we are given an inner product on the Lie algebra g of G,
and let || · ||g∗ be the dual norm on the dual g∗. Then the function ||Φ||2g∗ is also a constant of
motion of the Hamiltonian vector field Ξh.
We will make use of the following relationship between the invariant constants of motion of
isomorphic vector fields:
Lemma 4.15. Let M be a proper G-manifold, let X and Y be smooth vector fields on M , and
let f be a G-invariant constant of motion of X. Let H be a Lie subgroup of G with Lie algebra h,
and suppose that there exists an infinitesimal gauge transformation ψ ∈ C∞(M, h)H such that
Y = X + ψM . Then f is a constant of motion of the vector field Y . In particular, if X and Y
are G-isomorphic vector fields, then they have the same G-invariant constants of motion.
Proof. We verify that df(Y ) = 0. Since f is a constant of motion of X, observe that
df(Y ) = df(X + ψM ) = df(X) + df(ψM ) = df(ψM ).
Hence, it suffices to show that df(ψM ) = 0. Let m be a point in M , then
(df)m(ψM (m)) = (df)m
(
d
dt
∣∣∣
0
exp(tψ(m)) ·m
)
=
d
dt
∣∣∣
0
f
(
exp(tψ(m)) ·m
)
=
d
dt
∣∣∣
0
f(m) since f is G-invariant
= 0.
Since the point m is arbitrary, we have that df(ψM ) = 0. Hence, f is a constant of motion
of Y . �
Lemma 4.15 has the following two corollaries:
Corollary 4.16. Let (M,ω,G,Φ, h) be a Hamiltonian G-system, let η be a vector in the Lie
algebra g of G, and let Ξhη be the Hamiltonian vector field augmented by η. Suppose || · ||g∗ is the
G-invariant norm on the dual g∗ that is dual to a given G-invariant inner product on the Lie
algebra g of G. Then the Hamiltonian function h and the squared-norm of the momentum ||Φ||2g∗
are constants of motion of the augmented Hamiltonian vector field Ξhη .
Proof. We want to apply Lemma 4.15. Observe that the functions h and ||Φ||2g∗ are G-invariant
constants of motion of the Hamiltonian vector field Ξh (Remark 4.14). Let H be the Lie
subgroup Gη := {g ∈ G | Ad(g)η = η}, and let h be its Lie algebra. Let ψ : M → h be the
constant map ψ = −η. Then ψ is an infinitesimal gauge transformation in C∞(M, h)H such
that Ξhη = Ξh+ψM . Hence, we obtain the result by Lemma 4.15 with Ξh in place of X and Ξhη
in place of Y . �
26 S. Klajbor-Goderich
Corollary 4.17. Let G be a Lie group with Lie algebra g, let K be a Lie subgroup of G with
Lie algebra k, let V be a K-manifold, and let M be the associated bundle G×K V . Suppose that
(M,ω,G,Φ, h) is a Hamiltonian G-system. Furthermore:
1. Let g = k ⊕ q be a K-equivariant splitting, and let ν : Γ(TM) → Γ(TV ) be the vertical
projection of vector fields with respect to this splitting (Definition 2.35).
2. Let || · ||g∗ be the G-invariant norm on the dual g∗ that is dual to a given G-invariant inner
product on the Lie algebra g of G.
3. Let η be a vector in the Lie algebra k, and let Ξhη be the Hamiltonian vector field augmented
by η.
Then the Hamiltonian function h and the squared-norm of the momentum ||Φ||2g∗ are constants
of motion of the vertical projection ν(Ξhη).
Proof. By Corollary 4.16, the functions h and ||Φ||2g∗ are G-invariant constants of motion of
the augmented Hamiltonian vector field Ξhη (Remark 4.14). We want to apply Lemma 4.15.
Observe that the group K, acting on M as a subgroup of the Lie group G, is the stabilizer of
the point m := [1, 0] ∈ M . Furthermore, the K-manifold V , together with the K-equivariant
embedding j : V ↪→ M defined by j(w) := [1, w], is a global slice of the action through the
point m (Remark 2.30). Let E : X(K×V ⇒ V )→ X(G×M ⇒M) be the canonical equivariant
extension functor (Definition 2.32) and let P : X(G × M ⇒ M) → X(K × V ⇒ V ) be the
projection functor corresponding to the splitting g = k⊕ q (Definition 2.37). By Theorem 2.39,
there exists a map ψ ∈ C∞(M, g)G such that
Ξh = E(P (Ξh)) + ψM . (4.1)
By definition of the functors E and P , the vector field Ξh is such that E(P (Ξh)) = ν(Ξh).
Since the vector η is in k, the vector field ηM is vertical in the bundle M → G/K, and thus
ν(ηM ) = ηM . Consequently, we have that
Ξhη = Ξh − ηM
= E(P (Ξh)) + ψM − ηM
= ν(Ξh) + ψM − ν(ηM ) by (4.1)
= ν(Ξhη) + ψM .
That is, the same map ψ relates the augmented Hamiltonian vector field Ξhη and its vertical
part. As a side note, observe that this doesn’t mean the vector field Ξhη is isomorphic to
its vertical part since neither vector field is necessarily G-equivariant; and while both vector
fields are equivariant with respect to the subgroup Gη := {g ∈ G | Ad(g)η = η}, the map ψ
doesn’t necessarily land in the Lie algebra of Gη. However, applying Lemma 4.15 with the
vector field Ξhη in place of X, the vector field ν (Ξhη) in place of Y , and the infinitesimal gauge
transformation −ψ ∈ C∞(M, g)G, one obtains that the functions h and ||Φ||2g∗ are constants of
motion of the vector field ν(Ξhη). �
We will need the following application of the Morse lemma for families, which generalizes
a similar application in [13, Proposition 3.4]:
Proposition 4.18. Let U and W be normed finite-dimensional vector spaces. Furthermore:
1. Let f ∈ C∞(U × W ) be a smooth function such that f(0, 0) = 0, dW f(0, 0) = 0, and
d2
W f(0, 0) is nondegenerate and positive definite, where dW and d2
W denote the differential
and Hessian in the W variables respectively.
Nonlinear Stability of Relative Equilibria and Isomorphic Vector Fields 27
2. Let ϕ ∈ C∞(U ×W ) be a smooth function such that ϕ(0, 0) = 0 and ϕ(ρ, w) ≥ |ρ|2 for all
(ρ, w) ∈ U ×W .
3. Let θ ∈ C0(U ×W ) be a nonnegative continuous function with θ(0, 0) = 0.
Let the norm on the product space U ×W be the sum of the squares of the norms on U and W .
Then for all ε > 0, there exists δ > 0 such that if γ(t) = (ρ(t), w(t)) defines a curve in U ×W
with
1) |γ(0)| < δ;
2) ϕ(γ(t)) ≤ ϕ(γ(0)) for all t;
3) |f(γ(t))| ≤ θ(γ(0)) for all t.
Then |γ(t)| < ε for all t.
Proof. Since 0 is a critical point of f(0, ·) and d2
W f(0, 0) is nondegenerate, by the Morse lemma
for families [3, Lemma 1.2.2], there exists a neighborhood Ũ of 0 in U , a neighborhood W̃ of 0
in W , and a smooth map σ : Ũ → W̃ implicitly defined by the equation
dW f(ρ, σ(ρ)) = 0
for all ρ ∈ Ũ . Additionally, there is also a smooth map y : Ũ × W̃ → W implicitly defined by
the equation
f(ρ, v) = f(ρ, σ(ρ)) +
1
2
d2
W f(ρ, σ(ρ))
(
y(ρ, w), y(ρ, w)
)
(4.2)
for all (ρ, w) ∈ Ũ × W̃ .
Now let ε > 0 be given. We can always choose ε′ > 0 such that if |ρ|2 < ε′ and |y(ρ, w)|2 < ε′
then |(ρ, w)| = |ρ|2 + |w|2 < ε. Thus, if γ(t) = (ρ(t), w(t)) is a curve in U ×W satisfying the
hypotheses, it suffices to show that there exists a δ > 0 such that if |γ(0)| < δ then
|ρ(t)|2 < ε′ and |y(ρ(t), w(t))|2 < ε′.
Since d2
W f(0, 0) is positive definite, there exists β1 > 0 and C > 0 such that if |ρ|2 < β1 then
C|w|2 < d2
W f(ρ, σ(ρ))(w,w) (4.3)
for every w ∈W . By the continuity of f and σ there exists β2 > 0 such that if |ρ|2 < β2 then
|f(ρ, σ(ρ))| < ε′C
4
. (4.4)
By the continuity of ϕ there exists α1 > 0 such that if |(ρ, w)| < α1 then
|ϕ(ρ, w)| = ϕ(ρ, w) < min(β1, β2, ε
′). (4.5)
By the continuity of θ there exists α2 > 0 such that if |(ρ, w)| < α2 then
|θ(ρ, w)| < ε′C
4
. (4.6)
Now set δ := min(α1, α2) then note that for γ(t) = (ρ(t), w(t)) satisfying the hypotheses of the
proposition
|ρ(t)|2 ≤ ϕ(ρ(t), w(t))
28 S. Klajbor-Goderich
≤ ϕ(ρ(0), w(0))
< min(β1, β2, ε
′)
≤ ε′,
where we use inequality (4.5). This was the first inequality we needed to prove.
Now, since |ρ(t)|2 < β1 for all t, inequality (4.3) gives
C|w|2 < d2
W f
(
ρ(t), σ(ρ(t))
)
(w,w) (4.7)
for all w ∈W and all t. Similarly, since |ρ(t)|2 < β2 for all t, inequality (4.4) gives∣∣f(ρ(t), σ(ρ(t))
)∣∣ < ε′C
4
(4.8)
for all t. Finally, note that∣∣y(ρ(t), σ(ρ(t))
)∣∣2 ≤ 1
C
d2
W f
(
ρ(t), σ(ρ(t))
)(
y(ρ(t), σ(t)), y(ρ(t), σ(t))
)
by (4.7)
≤ 2
C
(∣∣f(ρ(t), w(t)
)∣∣+
∣∣f(ρ(t), σ(ρ(t))
)∣∣) by (4.2)
≤ 2
C
(
θ
(
ρ(0), w(0)
)
+
∣∣f(ρ(t), σ(ρ(t))
)∣∣) by assumption (3) on γ
<
2
C
(
ε′C
4
+
ε′C
4
)
by (4.6) & (4.8)
= ε′.
This was the second inequality we needed to prove. Thus, we have that |γ(0)| < δ implies
|γ(t)| < ε for all t as desired. �
We will also need the following standard construction:
Definition 4.19. Let (M,ω,G,Φ) be a Hamiltonian G-space and let m ∈M be a point in the
manifold M . The symplectic slice at the point m is the vector space
W := ker dΦm/
(
Tm(G ·m) ∩ ker dΦm
)
.
Remark 4.20. Let (M,ω,G,Φ) be a Hamiltonian G-space and let W be the symplectic slice
at a point m ∈ M . The symplectic slice inherits a canonical Hamiltonian representation of
the stabilizer subgroup Gm (see, for example, [19, Theorem 7.1.1(iii)]). Furthermore, let µ be
the moment of the point m and observe that Tm(G · m) ∩ ker dΦm = Tm(Gµ · m). Thus, if
the moment is fixed by the coadjoint representation, then the symplectic slice takes the form
W = ker dΦm/Tm(G ·m).
We recall the standard Marle–Guillemin–Sternberg normal form:
Theorem 4.21 (MGS normal form [8, 14]). Let (M,ω,G,Φ) be a Hamiltonian G-space. Let
m ∈M be a point in the manifold with moment Φ(m) = 0 and suppose the moment isotro-
py GΦ(m) acts properly on the manifold. Let K be the stabilizer of the point m ∈M , let k be the
corresponding Lie algebra, let k0 be the annihilator of k, and let W be the symplectic slice at the
point m (Definition 4.19). Given a K-equivariant embedding ι : k∗ ↪→ g∗, there exist K-invariant
open neighborhoods k0r ⊆ k0 and Wr ⊆W of the origins in the respective vector spaces k0 and W ,
a G-invariant symplectic form ωZ , and a G-equivariant map ΦZ : Z → g∗ on the associated
bundle
Z := G×K
(
k0r ×Wr
)
,
such that the quadruple (Z, ωz, G,ΦZ) is a Hamiltonian G-space. Furthermore:
Nonlinear Stability of Relative Equilibria and Isomorphic Vector Fields 29
1. There exists a G-invariant neighborhood Um ⊆M of the orbit G ·m, and a G-equivariant
symplectomorphism Ψ: Z → Um such that Ψ∗Φ = ΦZ .
2. The momentum map ΦZ : Z → g∗ is given by
ΦZ([g, ρ, w]) = Ad†(g)
(
ρ+ ι(ΦW (w))
)
,
where ΦW : W → k∗ is the momentum map for the representation of the stabilizer K on
the symplectic slice W .
Definition 4.22. Let (M,ω,G,Φ) be a Hamiltonian G-space and let m ∈ M be a point with
moment Φ(m) = 0. The Marle–Guillemin–Sternberg normal form (MGS normal form) is the
Hamiltonian G-space
(
Z := G ×K
(
k0r ×Wr
)
, ωZ , G,ΦZ
)
of Theorem 4.21. For a Hamiltonian
G-system (M,ω,G,Φ, h) with a point m ∈ M with moment Φ(m) = 0, the MGS normal form
is the Hamiltonian G-system (Z, ωZ , G,ΦZ , hZ) where (Z, ωZ , G,ΦZ) is the MGS normal form
of the underlying Hamiltonian G-space and the Hamiltonian function hZ is the pullback of the
Hamiltonian h by the symplectomorphism Ψ in part (1) of Theorem 4.21.
Remark 4.23. Let (M,ω,G,Φ) be a Hamiltonian G-space. Let m ∈M be a point with moment
Φ(m) = 0, and such that the moment isotropy GΦ(m) acts properly on M . Let K be the stabilizer
of m, and let k and k0 be the Lie algebra of K and the annihilator of k respectively. Then:
1. Since Φ(m) = 0, the moment of m is fixed by the coadjoint representation of the Lie group
G. Hence, GΦ(m) = G and thus the Lie group G acts properly on the manifold M . In this
case, a Gm-equivariant embedding ι : k∗ ↪→ g is guaranteed to exist (see [13, Remark 3.2]).
2. Let (Z, ωZ , G,ΦZ) be the MGS normal form for the Hamiltonian G-space (M,ω,G,Φ) at
the point m. The manifold V := k0r ×Wr ⊆ k0 ×W is a global slice for the action of G
on the manifold Z through the point [1, 0, 0] (see Definition 2.26). The corresponding
K-equivariant embedding j : V ↪→ Z is defined by j(ρ, w) := [1, ρ, w].
3. One can construct the MGS normal form without requiring that the moment Φ(m) be
fixed by the coadjoint representation (see, for example, [19, Theorem 7.5.5]). However,
when it is fixed, the MGS normal form, the expression for the moment map, and some of
our arguments become much simpler.
We can now prove the following, which is Theorem 4.8 for the special case where the Hamil-
tonian G-system is in MGS normal form and we’ve made the assumptions in Remark 4.10. In
Section 5, we reduce Theorem 4.8 to this special case.
Theorem 4.24. Let G be a Lie group with Lie algebra g, let K be a compact Lie subgroup of G
with Lie algebra k, let V be a K-manifold, and let Z be the associated bundle G×K V . Suppose
that (Z, ω,G,Φ, h) is a Hamiltonian G-system. Furthermore:
1. Let k0r be a K-invariant neighborhood of the origin in the annihilator k0 of the Lie algebra k.
2. Let Wr be a K-invariant neighborhood of the origin in a symplectic vector space W ,
where W has a Hamiltonian representation of the group K and corresponding equivari-
ant momentum map ΦW : W → k∗.
3. Suppose that the K-manifold V is the product k0r×Wr, and let the K-equivariant embedding
j : V ↪→ Z be defined by j(ρ, w) := [1, ρ, w].
4. Suppose there exists an Ad(G)-invariant inner product on the Lie algebra g, let q be the
orthogonal complement to k in g with respect to this inner product, and let P : g→ k be the
corresponding projection;
30 S. Klajbor-Goderich
5. Let the map ι : k∗ ↪→ g∗ be the K-equivariant embedding defined for every ρ ∈ k∗ by
ι(ρ) : g→ R, ι(ρ)(η) := ρ(P(η)).
6. For all [g, ρ, w] ∈ Z, let the momentum map Φ: Z → g∗ be given by
Φ([g, ρ, w]) = Ad†(g)
(
ρ+ ι(ΦW (w))
)
.
7. Suppose that the point m := [1, 0, 0] ∈ Z is a G-relative equilibrium of the Hamiltonian
vector field Ξh, and suppose it is such that h(m) = 0 and Φ(m) = 0.
If there exists a velocity ξ ∈ g such that the Hessian d2hξ(m) is definite and nondegenerate on
the subspace Tmj({0}×Wr) ⊆ TmZ, where hξ := h−〈Φ, ξ〉 is the augmented Hamiltonian, then
the relative equilibrium m is G-stable.
Proof. We want to prove the G-stability of the G-relative equilibrium m = [1, 0, 0] ∈ Z of the
Hamiltonian vector field Ξh on the manifold Z. Observe that the stabilizer of the point m is the
Lie subgroup K acting on Z as a subgroup of the group G. Also observe that the K-manifold V ,
together with the K-equivariant embedding j : V ↪→ Z, is a global slice for the action through
the point m (see Remark 4.23). Thus, in particular, the manifold Z is the tube G · V generated
by the slice V . Observe that the complement q gives a K-equivariant splitting g = k ⊕ q, and
hence we obtain a projection functor P : X(G × Z ⇒ Z) → X(K × V ⇒ V ) with respect to
the splitting (Definition 2.37). The proof is an application of the slice stability criterion (Theo-
rem 3.11). Thus, we need to give a Lie subgroup H of the stabilizer K, and a vector field Y on
the slice V that is H-isomorphic to the projected vector field P (Ξh). Furthermore, we need to
show that the point 0 = j−1(m) is H-stable for the vector field Y .
Use the K-equivariant projection P : g → k to define the vector η := P(ξ), which is the
projection of the velocity onto the isotropy Lie algebra. Now consider the Lie subgroup Gη :=
{g ∈ G | Ad(g)η = η} and the Lie subgroup
H := (Gη)m ≡ (Gm)η ≡ {g ∈ Gm | Ad(g)η = η}.
The subgroup H is the Lie subgroup of the stabilizer that we will use to apply the slice stability
criterion.
The inner product on g also determines an H-equivariant splitting gη = h ⊕ p of the Lie
algebra gη of the subgroup Gη. Let PGη : X(Gη × Z ⇒ Z)→ X(H × V ⇒ V ) be the projection
functor of Gη-equivariant vector fields with respect to this splitting of gη (Definition 2.46).
Let hη := h − 〈Φ, η〉 be the Hamiltonian augmented by the vector η and let Ξhη = Ξh − ηZ
be the corresponding augmented Hamiltonian vector field (Definition 4.6). Take the vector
field Y , required for the application of the slice stability criterion, to be the projected vector field
Y := PGη(Ξhη). Note that the Hamiltonian vector field Ξh is in particular Gη-equivariant and,
by Lemma 2.15, it is Gη-isomorphic to the vector field Ξhη . Thus, by the functoriality of PGη ,
the vector field Y is H-isomorphic to the vector field PGη(Ξh). Observe that PGη(Ξh) = P (Ξh),
so the vector fields Y and P (Ξh) are H-isomorphic.
It remains to show that the origin j−1(m) is H-stable for the vector field Y . By Lemma 3.3,
it is enough to show that the origin j−1(m) is {1}-stable for Y . That is, we show that for any
ε > 0, there exists a δ > 0, with 0 < δ < ε, such that all maximal integral curves of the vector
field Y starting at points in the δ-ball around the origin stay in the ε-ball around the origin
for all times for which they are defined. Proposition 4.18 guarantees we can obtain such a δ.
In particular, we apply Proposition 4.18 with the vector space k0 in place of U and the vector
space W of this theorem in place of the W in the proposition.
Nonlinear Stability of Relative Equilibria and Isomorphic Vector Fields 31
In order to apply Proposition 4.18 we need to pick norms on the spaces k0 and W . For this,
let || · ||g be the G-invariant norm on the Lie algebra induced by the given inner product on g,
let || · ||g∗ be the dual norm induced by the dual inner product on g∗, and let || · ||∞ be the
corresponding G-invariant sup norm on the dual g∗ (see Remark 4.11). The norm || · ||g∗ gives
an invariant norm on the subspace k0 of g∗, pick this norm for k0. Pick any norm on the vector
space W . This works since the slice V is contained in the vector space k0 ×W . Since all norms
on a finite-dimensional vector space are equivalent, there exists a constant A > 0 such that
|| · ||∞ ≤ A|| · ||g∗ . (4.9)
We apply Proposition 4.18 with the functions
f := j∗hη ∈ C∞(V ),
ϕ := ||j∗Φ||2g∗ ∈ C∞(V ),
θ := |j∗hη|+A||j∗Φ||g∗ ||η||g ∈ C0(V ).
(4.10)
To complete the proof we need to show that the functions in (4.10) and the maximal integral
curves of the vector field Y satisfy the hypotheses of Proposition 4.18.
Now we verify that the functions (4.10) satisfy the conditions of Proposition 4.18. For the
function f we need to show that f(0) = 0, dW f(0) = 0, and that the Hessian d2
W f(0) is definite
and nondegenerate. Here dW f(0) and d2
W f(0) are respectively the differential and the Hessian
of the function f in the W variables. For the first, note that
f(0) = hη(j(0)) = hη(m) = h(m)− 〈Φ(m), η〉 = 0− 〈0, η〉 = 0.
Now note that the image of the dual k∗ under the K-equivariant embedding ι : k∗ ↪→ g∗ is the
annihilator q0. Thus we have that
ι(ΦW (w)) ∈ q0 (4.11)
for all w ∈ W . Let ξ⊥ ∈ q be the vector in q such that ξ = η + ξ⊥. Then, using (4.11) and the
explicit form of the momentum map, we have that
〈Φ, ξ〉([1, 0, w]) = 〈Φ, η〉([1, 0, w]) +
〈
ι(ΦW (w)), ξ⊥
〉
= 〈Φ, η〉([1, 0, w])
for all w ∈ W . This in turn implies that hξ|j(Wr) = hη|j(Wr), where we identify the spaces Wr
and {0} ×Wr ⊆ V . Hence,
0 = dhξ(m)|Tmj(Wr) = d
(
hξ|j(Wr)
)
(0) = d
(
hη|j(Wr)
)
(0) = d
(
j∗hη|Wr
)
(0) = dW f(0),
which is the second condition we needed to check. This condition guarantees that the Hessian
d2
W f(0) is well-defined. The equality hξ|j(Wr) = hη|j(Wr) and the pullback properties of Hessians
(see Remark 4.7) imply that
d2hξ(m)|Tmj(Wr) = d2
(
hξ|j(Wr)
)
(0) = d2
(
hη|j(Wr)
)
(0) = d2
(
j∗hη|Wr
)
(0) = d2
W f(0).
Hence, the Hessian d2
W f(0) is definite and nondegenerate since d2hξ(m)|Tmj(Wr) is such by
assumption. It is of no loss of generality to assume that the Hessian d2hξ(m)|Tmj(Wr) is positive
definite. Hence, the Hessian d2
W f(0) is positive definite too. This was the third and last thing
we needed to verify that f satisfies.
For the function ϕ we need to show that ϕ(0) = 0 and that ϕ(ρ, w) ≥ ||ρ||2g∗ for all (ρ, w) ∈ V .
For the first, note
ϕ(0) = ||j∗Φ||2g∗(0) = ||Φ(m)||2g∗ = ||0||2g∗ = 0.
32 S. Klajbor-Goderich
For the second, note that for all (ρ, w) ∈ V we have that ρ ∈ k0 and ι(ΦW (w)) ∈ q0 = (k⊥)0 =
(k0)⊥. Thus, for all (ρ, w) ∈ V we have that
ϕ(ρ, w) = ||ρ+ ι(ΦW (w))||2g∗ = ||ρ||2g∗ + ||ι(ΦW (w))||2g∗ ≥ ||ρ||2g∗ .
Hence, ϕ satisfies the desired conditions.
For the function θ = |j∗hη| + A||j∗Φ||g∗ ||η||g we need to show that θ(0) = 0. Using that
h(m) = 0 and Φ(m) = 0, note we have that
θ(0) = |j∗hη(0)|+A||j∗Φ(0)||g∗ ||η||g
= |h(m)− 〈Φ(m), η〉|+A||Φ(m)||g∗ ||η||g
= |h(0)− 〈0, η〉|+A||0||g∗ ||η||g
= 0.
Hence, the function θ satisfies the desired condition.
We now verify the other set of hypotheses of Proposition 4.18. Let β be an arbitrary maximal
integral curve of the vector field Y . Then we need to show that
ϕ(β(t)) ≤ ϕ(β(0)) and |f(β(t))| ≤ θ(β(0)) (4.12)
for all times t for which the curve β is defined.
In place of the first inequality in (4.12), we actually prove the equality ϕ(β(t)) = ϕ(β(0))
for all times t such that the integral curve β is defined. Let ν : Γ(TZ)→ Γ(VZ) be the vertical
projection with respect to the splitting g = k⊕q (Definition 2.35). Since the vector fields ν (Ξhη)
and Y are j-related (see Definition 2.46), the curve α := j ◦β is a maximal integral curve of the
vector field ν (Ξhη). By Corollary 4.17, the function ||Φ||2g∗ is a constant of motion of the vector
field ν (Ξhη). Therefore, we have that
ϕ(β(t)) = ||j∗Φ||2g∗(β(t)) = ||Φ||2g∗(α(t)) = ||Φ||2g∗(α(0))︸ ︷︷ ︸
by Corollary 4.17
= ϕ(β(0)),
which is the desired equality.
We now verify the second inequality. Recall that the norm || · ||∞ satisfies |〈ρ, ζ〉| ≤ ||ρ||∞||ζ||g
for any ρ ∈ g∗ and ζ ∈ g. Also, as mentioned above, the curve α := j ◦ β is a maximal integral
curve of the vector field ν (Ξhη). Now, observe that
|f(β(t))| = |j∗hη(β(t))|
= |hη(α(t))|
= |h(α(t))− 〈Φ(α(t)), η〉|
≤ |h(α(t))|+ |〈Φ(α(t)), η〉|
= |h(α(0))|+ |〈Φ(α(t)), η〉| by Corollary 4.17
≤ |h(α(0))|+ ||Φ(α(t))||∞||η||g by properties of || · ||∞
≤ |h(α(0))|+A||Φ(α(t))||g∗ ||η||g by (4.9)
= |h(α(0))|+A||Φ(α(0))||g∗ ||η||g by Corollary 4.17
≤ |j∗h(β(0))|+A||j∗Φ(β(0))||g∗ ||η||g
= θ(β(0)).
This was the second inequality we needed to prove.
Thus, any integral curve β of the vector field Y satisfies the hypotheses of Proposition 4.18.
Applying Proposition 4.18, yields that the origin j−1(m) is {1}-stable for the vector field Y as
required. Consequently, the G-relative equilibrium m is G-stable for the Hamiltonian vector
field Ξh. �
Nonlinear Stability of Relative Equilibria and Isomorphic Vector Fields 33
5 Reduction to the Marle–Guillemin–Sternberg normal form
The goal of this section is to reduce Theorem 4.8 (Montaldi and Rodriguez-Olmos’s criterion)
to the special case Theorem 4.24, which we proved in the previous section. The main result of
this section is Lemma 5.3.
We need the following result from linear algebra:
Lemma 5.1. Let V be a finite dimensional vector space and let T : V × V → R be a bilinear
form on it. Let U , W , and W̃ be linear subspaces of V such that
V = U ⊕W = U ⊕ W̃ .
Furthermore, suppose that U is contained in kerT ], where T ] is the linear map
T ] : V → V ∗, T ](v) := T (v, ·).
Then the following are true:
1. If T |W is nondegenerate, then T |
W̃
is nondegenerate as well.
2. If T |W is positive (respectively negative) definite, then T |
W̃
is positive (respectively nega-
tive) definite as well.
Proof. Since we have the splittings V = U ⊕W = U ⊕W̃ , there exists a linear map L : W → U
such that W̃ is the graph of L; that is, such that
Graph(L) := {w + Lw |w ∈W} = W̃ . (5.1)
Now suppose that T |W is nondegenerate and let w̃ ∈ W̃ be a vector such that T ](w̃) = 0.
We want to show that w̃ = 0. By (5.1), there exists a vector w ∈ W such that w̃ = w + Lw.
Observe that
0 = T ](w̃) = T ](w + Lw) = T ](w) + T ](Lw) = T ](w),
where the last equality is because U is contained in kerT ]. This in turn means that w = 0 since
w ∈W and T |W is nondegenerate by hypothesis. Consequently, the given vector w̃ is 0, so T |
W̃
is nondegenerate.
For the second statement, suppose T |W is positive definite and let w̃ ∈ W̃ be an arbitrary
nonzero vector in W̃ . By (5.1) and since w̃ is nonzero, there exists a nonzero vector w ∈W such
that w̃ = w + Lw. Then note
T (w̃, w̃) = T (w + Lw,w + Lw) = T (w,w) + T (w,Lw) + T (Lw,w) + T (Lw,Lw)
= T (w,w) > 0.
The third equality follows because U is contained in kerT ], and also Lw ∈ U . The inequality
follows because T |W is positive definite. Thus, since w̃ ∈ W̃ is an arbitrary nonzero vector of W̃ ,
the map T |
W̃
is positive definite. A completely analogous argument works in the case when T |W
is negative definite. �
With this we can prove the following:
Lemma 5.2. Let (M,ω,G,Φ, h) be a Hamiltonian G-system, let the point m in the manifold M
be a relative equilibrium of the Hamiltonian vector field Ξh such that Φ(m) = 0, and let K be
34 S. Klajbor-Goderich
the stabilizer of the point m. Furthermore, let W and W̃ be K-invariant subspaces of the kernel
ker dΦm such that
ker dΦm = Tm(G ·m)⊕W = Tm(G ·m)⊕ W̃ .
Suppose there exists a velocity ξ ∈ g of m, with augmented Hamiltonian hξ := h − 〈Φ, ξ〉, such
that the Hessian d2hξ(m)|W is definite and nondegenerate. Then the Hessian d2hξ(m)|
W̃
is
definite and nondegenerate.
Proof. By Lemma 5.1, it suffices to show that
Tm(G ·m) ⊆ ker
(
d2hξ(m)|ker dΦm
)]
.
Let u ∈ Tm(G ·m) be an arbitrary vector. We need to show that the functional:(
d2hξ(m)
)]
(u) = d2hξ(m)(u, ·)
vanishes on the vector space ker dΦm. Since Tm(G ·m) = {ηM (m) | η ∈ g}, the vector u is of the
form u = ηM (m) for some η ∈ g. Thus, we want to show that for every vector v ∈ ker dΦm we
have that
d2hξ(m)(ηM (m), v) = 0.
For this, recall that for all g ∈ G, the vector Ad(g)ξ is a velocity for the G-relative equilibrium
g ·m, so that dhAd(g)ξ(g ·m) = 0 for all g ∈ G. In particular, for any s ∈ R, setting g = exp(sη)
we have
0 = dhAd(exp(sη))ξ(exp(sη) ·m)
= dh(exp(sη) ·m)− d〈Φ,Ad(exp(sη)ξ)〉(exp(sη) ·m).
(5.2)
Now let γ : (−ε, ε)× (−ε, ε)→M be a family of smooth curves such that
1) γ(0, 0) = m,
2) ∂
∂t
∣∣∣
t=0
γ(s, t) ∈ Texp(sη)·mM for all s ∈ (−ε, ε),
3) ∂
∂s
∣∣∣
s=0
γ(s, 0) = ηM (m),
4) ∂
∂t
∣∣∣
t=0
γ(0, t) = v.
Then for all s ∈ (−ε, ε), use (5.2) to get that
0 = dhAd(exp(sη))ξ(exp(sη) ·m)
(
∂
∂t
∣∣∣
t=0
γ(s, t)
)
=
(
dh(exp(sη) ·m)− d〈Φ,Ad(exp(sη)ξ)〉(exp(sη) ·m)
)( ∂
∂t
∣∣∣
t=0
γ(s, t)
)
=
∂
∂t
∣∣∣
t=0
(
h(γ(s, t))− 〈Φ(γ(s, t)),Ad(exp(sη))ξ〉
)
.
(5.3)
Now differentiate (5.3) with respect to s at s = 0 to get
0 =
∂
∂s∂t
∣∣∣
s=t=0
(
h(γ(s, t))− 〈Φ(γ(s, t)),Ad(exp(sη))ξ〉
)
=
∂
∂s∂t
∣∣∣
s=t=0
(
h(γ(s, t))− 〈Φ(γ(s, t)), ξ〉
)
− ∂
∂t
∣∣∣
t=0
〈Φ, [η, ξ]〉(γ(0, t))
= d2hξ(m)(ηM (m), v)− (d〈Φ, [η, ξ]〉)(m)(v)
= d2hξ(m)(ηM (m), v)− 〈dΦmv, [η, ξ]〉
= d2hξ(m)(ηM (m), v),
Nonlinear Stability of Relative Equilibria and Isomorphic Vector Fields 35
where the second equality follows by the product rule and the last equality follows because
v ∈ ker dΦm. This shows
(
d2hξ(m)
)]
(u) vanishes on ker dΦm, concluding the proof. �
We are now ready to prove the main result of this section. As mentioned in Remark 4.10,
to prove Theorem 4.8 it is of no loss of generality to suppose the Hamiltonian vanishes at the
relative equilibrium and that the moment is fixed. Under these assumptions, the following
lemma reduces Theorem 4.8 to Theorem 4.24 by showing that the assumptions on the Hessian
in Theorem 4.8 reduce to the MGS normal form as in Theorem 4.24.
Lemma 5.3. Let (M,ω,G,Φ, h) be a Hamiltonian G-system with the point m in the manifold M
as a relative equilibrium of the vector field Ξh. Suppose that h(m) = 0 and Φ(m) = 0. Denote
by K the stabilizer of the point m. Let
(
Z := G×K V, ωz, G,ΦZ , hZ
)
be the MGS normal form
of the given Hamiltonian G-system (see Definition 4.22). Furthermore:
1. Let V := k0r × Wr be the slice, where k0r is a K-invariant neighborhood of the origin in
the annihilator k0 of the Lie algebra k of K, and Wr is a K-invariant neighborhood of the
origin in the symplectic slice W of m.
2. Let the map j : V ↪→ Z be the K-equivariant embedding defined by j(ρ, w) := [1, ρ, w].
3. Let Ψ: Z → Um be the symplectomorphism to a neighborhood Um of the orbit G · m
guaranteed by Theorem 4.21.
Suppose that there exists a velocity ξ ∈ g of the point m and a linear subspace U of the kernel
ker dΦm such that
1. The kernel ker dΦm splits as ker dΦm = Tm(G ·m)⊕ U .
2. The Hessian d2hξ(m)|U is definite and nondegenerate.
Consider the tangent space W̃ := T[1,0,0]j ({0} ×Wr) ⊆ T[1,0,0]Z, then the Hessian
d2hξZ
(
[1, 0, 0]
)
|
W̃
is well-defined, definite, and nondegenerate.
Proof. To simplify notation throughout this proof, we identify the spaces Wr and {0} ×Wr
and write
W̃ := T[1,0,0]j(Wr) ⊆ T[1,0,0]Z and Ũ := dΨ
(
W̃
)
⊂ TmM.
We begin with three observations. First, since the map Ψ pulls back the momentum map on M
to the one on Z, we get that
dΨ
(
ker(dΦZ)[1,0,0]
)
= ker dΦm. (5.4)
Second, from the G-equivariance of the diffeomorphism Ψ we get that
dΨ
(
T[1,0,0](G · [1, 0, 0])
)
= Tm(G ·m). (5.5)
Third, note that the linear subspace W̃ is K-invariant under the K-representation it inherits
from the action of G on the manifold M . Also recall that Wr is a neighborhood in the symplectic
slice W . Thus, in particular, the linear subspace W̃ is such that
ker(dΦZ)[1,0,0] = T[1,0,0](G · [1, 0, 0])⊕ W̃ . (5.6)
36 S. Klajbor-Goderich
Using (5.4), (5.5), and (5.6) we get that Ũ is a K-invariant subspace of TmM such that
ker dΦm = Tm(G ·m)⊕ Ũ . (5.7)
Using (5.7), the assumptions on the space U , and Lemma 5.2, we obtain that the Hessian
d2hξ(m)|
Ũ
is definite and nondegenerate.
Now observe that the augmented Hamiltonian hξZ of the pullback hZ is the pullback of the
augmented Hamiltonian hξ:
hξZ = hZ − 〈ΦZ , ξ〉 = Ψ∗h− 〈Ψ∗Φ, ξ〉 = Ψ∗hξ.
Thus, since the point m is a critical point of the augmented Hamiltonian hξ we have that
dhξZ([1, 0, 0]) = d
(
Ψ∗hξ
)
([1, 0, 0]) = Ψ∗
(
dhξ
)
(m) = Ψ∗(0) = 0.
Hence, the Hessian d2hξZ([1, 0, 0]) is well-defined.
Finally, by the pullback properties of Hessians (see Remark 4.7), we have that
d2hξZ([1, 0, 0])|
W̃
= d2
(
Ψ∗hξ
)
([1, 0, 0])|
W̃
= Ψ∗
(
d2hξ(m)
)
|
Ũ
.
Since the map Ψ is a diffeomorphism, this implies that the Hessian d2hξZ([1, 0, 0])|
W̃
is definite
and nondegenerate because the Hessian d2hξ(m)|
Ũ
is definite and nondegenerate. �
Acknowledgements
The author would like to thank Eugene Lerman for providing guidance throughout this project,
for his enduring patience with my many questions, and for the many interesting conversations
on the subject. The author would also like to express their gratitude to the anonymous referee
for their helpful comments and careful review of the preprint of this paper.
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1 Introduction
1.1 Organization of the paper
1.2 Notation and conventions
2 Relative equilibria and isomorphic vector fields
3 Stability of relative equilibria
4 Stability of Hamiltonian relative equilibria
5 Reduction to the Marle–Guillemin–Sternberg normal form
References
|
| id | nasplib_isofts_kiev_ua-123456789-209443 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2025-12-07T16:44:07Z |
| publishDate | 2018 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Klajbor-Goderich, S. 2025-11-21T18:56:08Z 2018 Nonlinear Stability of Relative Equilibria and Isomorphic Vector Fields / S. Klajbor-Goderich // Symmetry, Integrability and Geometry: Methods and Applications. — 2018. — Т. 14. — Бібліогр.: 25 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 37J25; 57R25; 37J15; 53D20 arXiv: 1707.02828 https://nasplib.isofts.kiev.ua/handle/123456789/209443 https://doi.org/10.3842/SIGMA.2018.021 We present applications of the notion of isomorphic vector fields to the study of nonlinear stability of relative equilibria. Isomorphic vector fields were introduced by Hepworth [Theory Appl. Categ. 22 (2009), 542-587] in his study of vector fields on differentiable stacks. Here, we argue in favor of the usefulness of replacing an equivariant vector field by an isomorphic one to study the nonlinear stability of relative equilibria. In particular, we use this idea to obtain a criterion for nonlinear stability. As an application, we offer an alternative proof of Montaldi and Rodríguez-Olmos's criterion [arXiv:1509.04961] for the stability of Hamiltonian relative equilibria. The author would like to thank Eugene Lerman for guiding this project, for his enduring patience with my many questions, and for the many interesting conversations on the subject. The author would also like to express their gratitude to the anonymous referee for their helpful comments and careful review of the preprint of this paper. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Nonlinear Stability of Relative Equilibria and Isomorphic Vector Fields Article published earlier |
| spellingShingle | Nonlinear Stability of Relative Equilibria and Isomorphic Vector Fields Klajbor-Goderich, S. |
| title | Nonlinear Stability of Relative Equilibria and Isomorphic Vector Fields |
| title_full | Nonlinear Stability of Relative Equilibria and Isomorphic Vector Fields |
| title_fullStr | Nonlinear Stability of Relative Equilibria and Isomorphic Vector Fields |
| title_full_unstemmed | Nonlinear Stability of Relative Equilibria and Isomorphic Vector Fields |
| title_short | Nonlinear Stability of Relative Equilibria and Isomorphic Vector Fields |
| title_sort | nonlinear stability of relative equilibria and isomorphic vector fields |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/209443 |
| work_keys_str_mv | AT klajborgoderichs nonlinearstabilityofrelativeequilibriaandisomorphicvectorfields |