Lagrangian Mechanics and Reduction on Fibered Manifolds
This paper develops a generalized formulation of Lagrangian mechanics on fibered manifolds, together with a reduction theory for symmetries corresponding to Lie groupoid actions. As special cases, this theory includes not only Lagrangian reduction (including reduction by stages) for Lie group action...
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| Цитувати: | Lagrangian Mechanics and Reduction on Fibered Manifolds / S. Li, A. Stern, X. Tang // Symmetry, Integrability and Geometry: Methods and Applications. — 2017. — Т. 13. — Бібліогр.: 34 назв. — англ. |
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| author | Li, S. Stern, A. Tang, X. |
| author_facet | Li, S. Stern, A. Tang, X. |
| citation_txt | Lagrangian Mechanics and Reduction on Fibered Manifolds / S. Li, A. Stern, X. Tang // Symmetry, Integrability and Geometry: Methods and Applications. — 2017. — Т. 13. — Бібліогр.: 34 назв. — англ. |
| collection | DSpace DC |
| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | This paper develops a generalized formulation of Lagrangian mechanics on fibered manifolds, together with a reduction theory for symmetries corresponding to Lie groupoid actions. As special cases, this theory includes not only Lagrangian reduction (including reduction by stages) for Lie group actions, but also classical Routh reduction, which we show is naturally posed in this fibered setting. Along the way, we also develop some new results for Lagrangian mechanics on Lie algebroids, most notably a new, coordinate-free formulation of the equations of motion. Finally, we extend the foregoing to include fibered and Lie algebroid generalizations of the Hamilton-Pontryagin principle of Yoshimura and Marsden, along with the associated reduction theory.
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| first_indexed | 2025-11-24T11:04:38Z |
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Symmetry, Integrability and Geometry: Methods and Applications SIGMA 13 (2017), 019, 26 pages
Lagrangian Mechanics and Reduction
on Fibered Manifolds
Songhao LI, Ari STERN and Xiang TANG
Department of Mathematics, Washington University in St. Louis,
One Brookings Drive, St. Louis MO 63130-4899, USA
E-mail: lisonghao@gmail.com, stern@wustl.edu, xtang@wustl.edu
Received October 05, 2016, in final form March 13, 2017; Published online March 22, 2017
https://doi.org/10.3842/SIGMA.2017.019
Abstract. This paper develops a generalized formulation of Lagrangian mechanics on fibered
manifolds, together with a reduction theory for symmetries corresponding to Lie groupoid
actions. As special cases, this theory includes not only Lagrangian reduction (including
reduction by stages) for Lie group actions, but also classical Routh reduction, which we
show is naturally posed in this fibered setting. Along the way, we also develop some new
results for Lagrangian mechanics on Lie algebroids, most notably a new, coordinate-free
formulation of the equations of motion. Finally, we extend the foregoing to include fibered
and Lie algebroid generalizations of the Hamilton–Pontryagin principle of Yoshimura and
Marsden, along with the associated reduction theory.
Key words: Lagrangian mechanics; reduction; fibered manifolds; Lie algebroids; Lie groupoids
2010 Mathematics Subject Classification: 70G45; 53D17; 37J15
1 Introduction
The starting point for classical Lagrangian mechanics is a function L : TQ → R, called the
Lagrangian, where TQ is the tangent bundle of a smooth configuration manifold Q.
Yet, tangent bundles are hardly the only spaces on which one may wish to study Lagrangian
mechanics. When L is invariant with respect to certain symmetries, it is useful to perform
Lagrangian reduction: quotienting out the symmetries and thereby passing to a smaller space
than TQ. For example, if a Lie group G acts freely and properly on Q, then Q → Q/G is
a principal fiber bundle; if L is invariant with respect to the G-action, then one can define
a reduced Lagrangian on the quotient TQ/G (cf. Marsden and Scheurle [21], Cendra et al. [3]).
In particular, when Q = G, the reduced Lagrangian is defined on TG/G ∼= g, the Lie algebra
of G, and the reduction procedure is called Euler–Poincaré reduction (cf. Marsden and Ratiu [18,
Chapter 13]).
Unlike TQ, the reduced spaces TQ/G and g are not tangent bundles – but all three are
examples of Lie algebroids. Beginning with a seminal paper of Weinstein [30], and with particularly
important follow-up work by Mart́ınez [23, 24, 25], this has driven the development of a more
general theory of Lagrangian mechanics on Lie algebroids. In this more general framework,
reduction is associated with Lie algebroid morphisms, of which the quotient map TQ→ TQ/G
is a particular example. Since Lie algebroids form a category, the composition of two morphisms
is again a morphism. As an important consequence, it is almost trivial to perform so-called
reduction by stages – applying a sequence of morphisms one at a time rather than all at once –
whereas, without this framework, reduction by stages is considerably more difficult (Cendra et
al. [3], Marsden et al. [17]).
This paper is a contribution to the Special Issue “Gone Fishing”. The full collection is available at
http://www.emis.de/journals/SIGMA/gone-fishing2016.html
mailto:lisonghao@gmail.com
mailto:stern@wustl.edu
mailto:xtang@wustl.edu
https://doi.org/10.3842/SIGMA.2017.019
http://www.emis.de/journals/SIGMA/gone-fishing2016.html
2 S. Li, A. Stern and X. Tang
In this paper, we generalize the foregoing theory in a new direction, based on the observation
that reduction from TQ to TQ/G is a special case of a much more general construction, involving
Lie groupoid (rather than group) actions on fibered manifolds (rather than ordinary manifolds).
This includes not only Lagrangian reduction, but also the related theory of Routh reduction,
which we show is naturally posed in the language of fibered manifolds. In the special case
of a manifold trivially fibered over a single point, i.e., an ordinary manifold, this reduces to
the previously-studied cases. Along the way, we also develop some new results on Lagrangian
mechanics on Lie algebroids – most notably a new, coordinate-free formulation of the equations of
motion, incorporating the notion of a Lie algebroid connection due to Crainic and Fernandes [6] –
and extend this theory to the Hamilton–Pontryagin principle of Yoshimura and Marsden [33].
The paper is organized as follows:
• In Section 2, we begin by briefly reviewing the classical formulation of Lagrangian mechanics
on manifolds. We then define fibered manifolds, together with appropriate spaces of vertical
tangent vectors and paths, and show how Lagrangian mechanics may be generalized to
this setting. As an application, we show that Routh reduction is naturally posed in the
language of fibered manifolds, where the classical Routhian is understood as a Lagrangian
on an appropriate vertical bundle.
• In Section 3, we discuss Lagrangian mechanics on Lie algebroids. We call the associated
equations of motion the Euler–Lagrange–Poincaré equations, since they simultaneously
generalize the Euler–Lagrange equations on TQ, Euler–Poincaré equations on g, and
Lagrange–Poincaré equations on TQ/G. We derive a new, coordinate-free formulation of
these equations, which we show agrees with the local-coordinates expression previously
obtained by Mart́ınez [23]. Finally, we show that, since the vertical bundle of a fibered
manifold is a Lie algebroid, the theory of Section 2 can be interpreted in this light.
• In Section 4, we employ the Lie algebroid toolkit of Section 3 to study Lagrangian reduction
on fibered manifolds by Lie groupoid actions, which we call Euler–Lagrange–Poincaré
reduction. In the special case where a Lie groupoid acts on itself by multiplication, we
recover the theory of Lagrangian mechanics on its associated Lie algebroid.
• Finally, in Section 5, we generalize the Hamilton–Pontryagin variational principle of
Yoshimura and Marsden [33], together with the associated reduction theory [34], to fibered
manifolds with Lie groupoid symmetries.
2 Lagrangian mechanics on fibered manifolds
2.1 Brief review of Lagrangian mechanics
Let Q be a smooth configuration manifold and L : TQ → R be a smooth function, called the
Lagrangian, on its tangent bundle. There are three ways in which one can use L to induce
dynamics on Q.
The first, which we call the symplectic approach, begins by introducing the Legendre transform
(or fiber derivative) of L, which is the bundle map FL : TQ→ T ∗Q defined fiberwise by〈
FqL(v), w
〉
=
d
dt
L(v + tw)
∣∣∣
t=0
, v, w ∈ TqQ.
This is used to pull back the canonical symplectic form ω ∈ Ω2(T ∗Q) to the Lagrangian 2-form
ωL = (FL)∗ω ∈ Ω2(TQ). The Lagrangian is said to be regular if FL is a local bundle isomorphism;
in this case, ωL is nondegenerate, so (TQ, ωL) is a symplectic manifold. The energy function
EL : TQ→ R associated to L is
EL(v) =
〈
FL(v), v
〉
− L(v),
Lagrangian Mechanics and Reduction on Fibered Manifolds 3
and the Lagrangian vector field XL ∈ X(TQ) is the vector field satisfying
iXLωL = dEL,
where iXLωL = ωL(XL, ·) is the interior product of XL with ωL. That is, XL is the Hamiltonian
vector field of EL on the symplectic manifold (TQ, ωL). Finally, a C2 path q : I → Q is called
a base integral curve of XL if its tangent prolongation (q, q̇) : I → TQ is an integral curve of XL.
(Here and henceforth, I denotes the closed unit interval [0, 1], but there is no loss of generality
over any other closed interval [a, b].)
The second, which we call the variational approach, begins with the action functional
S : P(Q)→ R, defined by the integral
S(q) =
∫ 1
0
L
(
q(t), q̇(t)
)
dt,
where P(Q) denotes the Banach manifold of C2 paths q : I → Q. A path q ∈ P(Q) satisfies
Hamilton’s variational principle if it is a critical point of S restricted to paths with fixed
endpoints q(0) and q(1), i.e., if dS(δq) = 0 for all variations δq ∈ TqP(Q) with δq(0) = 0 and
δq(1) = 0.
The third and final approach considers the system of differential equations that a solution
to Hamilton’s variational principle must satisfy. In local coordinates, assuming δq(0) = 0 and
δq(1) = 0,
dS(δq) =
∫ 1
0
(
∂L
∂qi
(q, q̇)δqi +
∂L
∂q̇i
(q, q̇)δq̇i
)
dt =
∫ 1
0
(
∂L
∂qi
(q, q̇)− d
dt
∂L
∂q̇i
(q, q̇)
)
δqi dt.
(Here, we use the Einstein index convention, where there is an implicit sum on repeated indices.)
Hence, this vanishes for all δq if and only if q satisfies the system of ordinary differential equations
∂L
∂qi
(q, q̇)− d
dt
∂L
∂q̇i
(q, q̇) = 0,
which are called the Euler–Lagrange equations.
The equivalence of these three approaches for regular Lagrangians – and of the latter two
for arbitrary Lagrangians – is a standard result in geometric mechanics. We state it now as
a theorem for later reference.
Theorem 2.1. If L : TQ → R is a regular Lagrangian and q ∈ P(Q), then the following are
equivalent:
(i) q is a base integral curve of the Lagrangian vector field XL ∈ X(TQ).
(ii) q satisfies Hamilton’s variational principle.
(iii) q satisfies the Euler–Lagrange equations.
If regularity is dropped, then (ii)⇔ (iii) still holds.
Proof. See, e.g., Marsden and Ratiu [18, Theorem 8.1.3]. �
2.2 Fibered manifolds
We begin by giving the definition of a fibered manifold, along with its vertical and covertical
bundles. These bundles generalize the tangent and cotangent bundles of an ordinary manifold,
and they will play analogous roles in fibered Lagrangian mechanics.
4 S. Li, A. Stern and X. Tang
Definition 2.2. A fibered manifold Q → M consists of a pair of smooth manifolds Q, M ,
together with a surjective submersion µ : Q→M .
Definition 2.3. The vertical bundle of Q → M is V Q = kerµ∗, where µ∗ : TQ → TM is the
pushforward (or tangent map) of µ. The dual of V Q is denoted V ∗Q, which we call the covertical
bundle.
Remark 2.4. Since µ is a submersion, the fiber Qx = µ−1
(
{x}
)
is a submanifold of Q for each
x ∈M . Therefore,
VqQ = TqQµ(q), V Q =
⊔
q∈Q
VqQ =
⊔
x∈M
TQx.
In other words, V Q consists of vectors tangent to the fibers, and hence is an integrable subbundle
of TQ. Similarly,
V ∗q Q = T ∗qQµ(q), V ∗Q =
⊔
q∈Q
V ∗q Q =
⊔
x∈M
T ∗Qx,
so the covertical bundle consists of covectors to the individual fibers.
Example 2.5. An ordinary smooth manifold Q can be identified with the fibered manifold
Q → •, where • denotes the space with a single point. Because µ∗ is trivial, it follows that
V Q = TQ and V ∗Q = T ∗Q.
Definition 2.6. The space of vertical vector fields on Q is XV (Q) = Γ(V Q). The space of
vertical k-forms on Q is Ωk
V (Q) = Γ(
∧k V ∗Q).
Since V Q is integrable, it follows that XV (Q) is closed under the Jacobi–Lie bracket [·, ·], i.e.,
XV (Q) is a Lie subalgebra of X(Q). Therefore, the following vertical exterior derivative operator
on Ω•V (Q) is well-defined.
Definition 2.7. Given u ∈ Ωk
V (Q), the vertical exterior derivative dV u ∈ Ωk+1
V (Q) is given by
dV u(X0, . . . , Xk) =
k∑
i=0
(−1)iXi
[
u(X0, . . . , X̂i, . . . , Xk)
]
+
∑
0≤i<j≤k
(−1)i+j
(
[Xi, Xj ], X0, . . . , X̂i, . . . , X̂j , . . . , Xk
)
,
where X0, . . . , Xk ∈ XV (Q) are arbitrary vertical vector fields, and where a hat over an argument
indicates its omission.
Remark 2.8. From the characterization of V Q in Remark 2.4, it follows that X ∈ XV (Q)
restricts to an ordinary vector field Xx ∈ X(Qx) on each fiber Qx. Likewise, u ∈ Ωk
V (Q) restricts
to an ordinary k-form ux ∈ Ωk(Qx) on each fiber Qx. Moreover, by the integrability of V Q, for
any X,Y ∈ XV (Q) and x ∈M we have [X,Y ]x = [Xx, Yx] ∈ X(Qx). Hence, the vertical exterior
derivative dV coincides with the ordinary exterior derivative dx : Ωk(Qx) → Ωk+1(Qx) on the
fiber Qx.
Note that V Q and V ∗Q are also, themselves, fibered manifolds over M . Specifically, if
τ : V Q→ Q and π : V ∗Q→ Q are the bundle projections, then we have surjective submersions
µ◦τ : V Q→M and µ◦π : V ∗Q→M ; the fibers are given by (V Q)x = TQx and (V ∗Q)x = T ∗Qx.
Now, just as there is a tautological 1-form and canonical symplectic 2-form on T ∗Q, there are
corresponding vertical forms on V ∗Q, constructed as follows.
Lagrangian Mechanics and Reduction on Fibered Manifolds 5
Definition 2.9. The tautological vertical 1-form θ ∈ Ω1
V (V ∗Q) is defined by the condition θ(v) =
〈p, π∗v〉 for v ∈ VpV ∗Q. The canonical vertical 2-form is defined by ω = −dV θ ∈ Ω2
V (V ∗Q).
Remark 2.10. Restricted to any fiber (V ∗Q)x = T ∗Qx, it follows from the preceding remarks
that θ and ω agree with the ordinary tautological 1-form θx ∈ Ω1(T ∗Qx) and canonical symplectic
2-form ωx ∈ Ω2(T ∗Qx), respectively, on the cotangent bundle of the fiber. In particular,
this implies that ω is closed (with respect to dV ) and nondegenerate, since ωx is closed and
nondegenerate for each x ∈M .
2.3 Lagrangian mechanics on fibered manifolds
In this section, we show that the three approaches to Lagrangian mechanics of Section 2.1 may
be generalized to fibered manifolds, with a corresponding generalization of Theorem 2.1. Let the
Lagrangian be a smooth function L : V Q→ R.
Definition 2.11. The Legendre transform (or fiber derivative) of L is the bundle map FL : V Q→
V ∗Q, defined for each q ∈ Q by
〈
FqL(v), w
〉
=
d
dt
L(v + tw)
∣∣∣
t=0
, v, w ∈ VqQ.
We say that L is regular if FL is a local bundle isomorphism.
Remark 2.12. Since (V Q)x = TQx, we can define a fiber-restricted Lagrangian Lx : TQx → R,
whose ordinary Legendre transform FLx : TQx → T ∗Qx coincides with the restriction FL|(V Q)x .
It is therefore useful to think of L as a smoothly varying family of ordinary Lagrangians Lx,
parametrized by x ∈M .
Now, FL maps fibers to fibers (i.e., it is a morphism of fibered manifolds over M), so its
pushforward maps vertical vectors to vertical vectors, and we may write (FL)∗ : V V Q→ V V ∗Q.
This also gives a well-defined pullback of vertical forms (FL)∗ : Ωk
V (V ∗Q) → Ωk
V (V Q), which
leads to the following vertical versions of the Lagrangian 2-form and Lagrangian vector field.
Definition 2.13. The Lagrangian vertical 2-form is ωL = (FL)∗ω ∈ Ω2
V (V Q). The Lagrangian
vertical vector field XL ∈ XV (V Q) is the vertical vector field satisfying
iXLωL = dVEL,
where the energy function EL : V Q→ R is given by EL(v) =
〈
FL(v), v
〉
− L(v).
Remark 2.14. Restricting to the fiber over x ∈M , we have
(ωL)x = (FLx)∗ωx = ωLx ∈ Ω2(TQx),
i.e., the ordinary Lagrangian 2-form for Lx on TQx, and moreover
(EL)x(v) =
〈
FLx(v), v
〉
− Lx(v) = ELx(v), v ∈ TQx,
so EL restricts to ELx . Combining these, it follows that
i(XL)xωLx = dxELx ,
so we conclude that (XL)x = XLx , i.e., XL coincides with the ordinary Lagrangian vector field
on each fiber.
6 S. Li, A. Stern and X. Tang
Next, for the variational approach, we begin by defining an appropriate space of vertical paths
on which the action functional will be defined, as well as an appropriate space of variations of
these paths.
Definition 2.15. The space of C2 vertical paths, denoted by PV (Q) ⊂ P(Q), consists of
q ∈ P(Q) whose tangent prolongation satisfies
(
q(t), q̇(t)
)
∈ V Q for all t ∈ I. The action
functional S : PV (Q)→ R is then
S(q) =
∫ 1
0
L
(
q(t), q̇(t)
)
dt,
which is well-defined for L : V Q→ R since
(
q(t), q̇(t)
)
∈ V Q.
Remark 2.16. For q ∈ PV (Q), the condition
(
q(t), q̇(t)
)
∈ V Q implies
d
dt
µ
(
q(t)
)
= µ∗
(
q(t), q̇(t)
)
= 0.
Hence, µ
(
q(t)
)
is constant in t, so q lies in a single fiber Qx, i.e., q ∈ P(Qx) for some x ∈ M .
It follows that S(q) = Sx(q), where Sx : P(Qx) → R is the ordinary action associated to the
fiber-restricted Lagrangian Lx. Moreover, since µ
(
q(t)
)
is constant in t, there is an associated
fibered (Banach) manifold structure PV (Q)→M , with PV (Q)x = P(Qx).
Definition 2.17. An element δq ∈ VqPV (Q) is called a vertical variation of q ∈ PV (Q). The
path q satisfies Hamilton’s variational principle for vertical paths if q is a critical point of S
relative to paths with fixed endpoints, i.e., if dS(δq) = 0 for all vertical variations δq with
δq(0) = 0 and δq(1) = 0.
Remark 2.18. Since PV (Q)x = P(Qx) and VqPV (Q) = TqP(Qx), this is immediately equivalent
to q ∈ PV (Q)x satisfying the ordinary form of Hamilton’s variational principle for the fiber-
restricted Lagrangian Lx.
Having defined vertical versions of the symplectic and variational approaches to Lagrangian
mechanics, we finally derive the corresponding Euler–Lagrange equations. Suppose that q =
(xσ, yi) are fiber-adapted local coordinates for Q. Since vertical variations satisfy δxσ = 0, by
definition, arbitrary fixed-endpoint variations of the action functional are given by
dS(δq) =
∫ 1
0
(
∂L
∂yi
(q, q̇)δyi +
∂L
∂ẏi
(q, q̇)δẏi
)
dt =
∫ 1
0
(
∂L
∂yi
(q, q̇)− d
dt
∂L
∂ẏi
(q, q̇)
)
δyi dt.
Therefore, a critical vertical path must have the integrand above vanish, in addition to the
vertical path condition. This motivates the following definition.
Definition 2.19. In fiber-adapted local coordinates q = (xσ, yi) on Q → M , the vertical
Euler–Lagrange equations for L : V Q→ R are
ẋσ = 0,
∂L
∂yi
(q, q̇)− d
dt
∂L
∂ẏi
(q, q̇) = 0. (2.1)
Remark 2.20. Since q = (xσ, yi) are fiber-adapted local coordinates, yi gives local coordinates
for the fiber Qx, and we may write L(q, q̇) = Lx(y, ẏ). (Note that L is defined only on vertical
tangent vectors, so ẋ is not required.) Therefore, the vertical Euler–Lagrange equations are
equivalent to the ordinary Euler–Lagrange equations,
∂Lx
∂yi
(y, ẏ)− d
dt
∂Lx
∂ẏi
(y, ẏ) = 0,
for the fiber-restricted Lagrangian Lx.
Lagrangian Mechanics and Reduction on Fibered Manifolds 7
We are now prepared to state the generalization of Theorem 2.1 to Lagrangian mechanics on
fibered manifolds.
Theorem 2.21. If L : V Q→ R is a regular Lagrangian on a fibered manifold µ : Q→M , and
q ∈ PV (Q) is a vertical C2 path over x ∈M , then the following are equivalent:
(i) q is a base integral curve of the Lagrangian vector field XL ∈ XV (V Q).
(ii) q satisfies Hamilton’s variational principle for vertical paths.
(iii) q satisfies the vertical Euler–Lagrange equations.
(i ′) q is a base integral curve of the fiber-restricted Lagrangian vector field XLx ∈ X(TQx).
(ii ′) q satisfies Hamilton’s variational principle with respect to the fiber-restricted Lagrangian
Lx.
(iii ′) q satisfies the Euler–Lagrange equations with respect to the fiber-restricted Lagrangian Lx.
If regularity is dropped, then (ii)⇔ (iii)⇔ (ii′)⇔ (iii′) still holds.
Proof. We have seen, in the foregoing discussion, that (i) ⇔ (i′) for regular Lagrangians, while
(ii) ⇔ (ii′) and (iii) ⇔ (iii′) hold in general. Hence, it suffices to show (i′) ⇔ (ii′) ⇔ (iii′) for
the regular case and (ii′) ⇔ (iii′) for the general case – but this is simply Theorem 2.1 applied
to Lx. �
2.4 Application: classical Routh reduction as fibered mechanics
The technique known as Routh reduction traces its origins as far back as the 1860 treatise of
Routh [28]. Modern geometric accounts have been given by Arnold et al. [1], Marsden and
Scheurle [20], and Marsden et al. [19], with the latter two works developing a more general theory
of nonabelian Routh reduction.
The essence of Routh reduction, as we will show, is that it passes from a Lagrangian on an
ordinary manifold to an equivalent Lagrangian, known as the Routhian, on a fibered manifold.
Since the resulting dynamics are confined to the vertical components (i.e., restricted to individual
fibers), this reduces the size of the original system by eliminating the horizontal components.
Consider a configuration manifold of the form Tn ×S, where Tn denotes the n-torus and S is
a manifold called the shape space. Let θσ and yi be local coordinates for Tn and S, respectively,
and suppose the Lagrangian L : T (Tn × S)→ R is cyclic in the variables θσ, i.e., L = L(θ̇, y, ẏ)
depends only on θ̇ but not on θ itself. Then the θσ components of the Euler–Lagrange equations
imply that
d
dt
∂L
∂θ̇σ
=
∂L
∂θσ
= 0,
so xσ = ∂L/∂θ̇σ is constant in t.
Now, define the fibered manifold Q = Rn × S with M = Rn, where µ : Q→M is simply the
projection onto the Rn component, so that V Q = Rn × TS. The classical Routhian R : V Q→ R
is
R(x, y, ẏ) =
[
L(θ̇, y, ẏ)− xσ θ̇σ
]
xσ=∂L/∂θ̇σ
, (2.2)
where each θ̇σ is determined implicitly by the constraint xσ = ∂L/∂θ̇σ. Considering R as
a Lagrangian in the sense of the previous section, the vertical Euler–Lagrange equations consist
of the vertical path condition,
0 = ẋσ =
d
dt
∂L
∂θ̇σ
,
8 S. Li, A. Stern and X. Tang
and
0 =
∂R
∂yi
(x, y, ẏ)− d
dt
∂R
∂ẏi
(x, y, ẏ)
=
(
∂L
∂yi
+
∂L
∂θ̇σ
∂θ̇σ
∂yi
− xσ
∂θ̇σ
∂yi
)
− d
dt
(
∂L
∂ẏi
+
∂L
∂θ̇σ
∂θ̇σ
∂ẏi
− xσ
∂θ̇
∂ẏi
)
=
∂L
∂yi
− d
dt
∂L
∂ẏi
,
where the last step uses xσ = ∂L/∂θ̇σ to eliminate the last two terms from each parenthetical
expression.
Thus, the ordinary Euler–Lagrange equations for L are precisely equivalent to the vertical
Euler–Lagrange equations for R. This reduces the dynamics from Tn × S to those on the
individual fibers Qx ∼= S, thereby eliminating the cyclic variables θ ∈ Tn. We now summarize
this result as a theorem.
Theorem 2.22. Suppose L : T (Tn × S)→ R is an ordinary Lagrangian that is cyclic in the Tn
components, and let the classical Routhian R : V (Rn × S)→ R be the fibered Lagrangian defined
in (2.2). Then (θ, y) ∈ P(Tn × S) is a solution path for L if and only if (x, y) ∈ PV (Rn × S) is
a vertical solution path for R.
Proof. This follows from Theorem 2.21, together with the foregoing calculations. �
3 Lagrangian mechanics on Lie algebroids
In this section, we lay the groundwork for reduction theory on fibered manifolds, which will be
discussed in Section 4. In ordinary Lagrangian reduction, we pass from the tangent bundle TQ
to the quotient TQ/G, which is generally not a tangent bundle. Likewise, in Section 4, we will
pass from vertical bundles to quotients that are generally not vertical bundles. However, TQ
and TQ/G – as well as their vertical analogs, as we will show – are all examples of more general
objects called Lie algebroids, on which Lagrangian mechanics can be studied. The study of
Lagrangian mechanics on Lie algebroids was largely pioneered by Weinstein [30], and important
follow-up work was done by Mart́ınez [23, 24, 25] and several others in more recent years; see
also Cortés et al. [4], Cortés and Mart́ınez [5], Grabowska and Grabowski [9], Grabowska et
al. [10], Iglesias et al. [12, 13].
In addition to recalling some of the key results (particularly of Weinstein [30] and Mart́ınez [23])
that we will need for the subsequent reduction theory, we also develop a new, coordinate-free
formulation of the equations of motion, which we call the Euler–Lagrange–Poincaré equations
(since they simultaneously generalize the Euler–Lagrange, Euler–Poincaré, and Lagrange–Poincaré
equations). This new formulation is based on the work of Crainic and Fernandes [6], particularly
the notion of a Lie algebroid connection and its use in describing variations of paths.
3.1 Lie algebroids and A-paths
We begin by recalling the definition of a Lie algebroid A and an appropriate class of paths
in A, called A-paths. This review will necessarily be very brief, but for more information on Lie
algebroids, we refer the reader to the comprehensive work by Mackenzie [14].
Definition 3.1. A Lie algebroid is a real vector bundle τ : A→ Q equipped with a Lie bracket
[·, ·] : Γ(A) × Γ(A) → Γ(A) on its space of sections and a bundle map ρ : A → TQ, called the
anchor map, satisfying the following Leibniz rule-like compatibility condition:
[X, fY ] = f [X,Y ] + ρ(X)[f ]Y, for all X,Y ∈ Γ(A), f ∈ C∞(Q).
Lagrangian Mechanics and Reduction on Fibered Manifolds 9
Example 3.2. The tangent bundle TQ is a Lie algebroid over Q, where τ : TQ→ Q is the usual
bundle projection, [·, ·] : X(Q) × X(Q) → X(Q) is the Jacobi–Lie bracket of vector fields, and
ρ : TQ→ TQ is the identity.
Furthermore, any integrable distribution D ⊂ TQ is also a Lie algebroid over Q, where τ , [·, ·],
and ρ are just the restrictions to D of the corresponding maps for TQ. We say that D is a Lie
subalgebroid of TQ.
In particular, if Q→M is a fibered manifold, then V Q ⊂ TQ is integrable and hence a Lie
algebroid over Q. (Note that V Q is generally not a Lie algebroid over M , since it may not even
be a vector bundle over M .)
Example 3.3. Any Lie algebra g is a Lie algebroid over • (the space with one point), where the
maps τ and ρ are trivial and [·, ·] is the Lie bracket.
More generally, if Q→ Q/G is a principal G-bundle for some Lie group G, then TQ/G defines
an algebroid over Q/G called the Atiyah algebroid. The algebroid g→ • can be identified with
the special case Q = G, where G is the Lie group integrating g (which exists by Lie’s third
theorem).
Definition 3.4. A path a ∈ P(A) over the base path q = τ ◦ a ∈ P(Q) is called an A-path if
q̇(t) = ρ
(
a(t)
)
for all t ∈ I. The space of A-paths is denoted by Pρ(A).
Remark 3.5. Equivalently, a is an A-path if and only if adt : TI → A is a morphism of Lie
algebroids, where TI → I has the tangent bundle Lie algebroid structure of Example 3.2. Hence,
A-paths can be seen as “paths in the category of Lie algebroids”.
3.2 Connections and variations of A-paths
We now turn to discussing an appropriate class of variations on the space of A-paths, Pρ(A).
Crainic and Fernandes [6, Lemma 4.6] show that Pρ(A) ⊂ P(A) is a Banach submanifold.
However, we do not want to take arbitrary variations δa ∈ TaPρ(A), just as we did not want to
take arbitrary paths in P(A).
To illustrate the reasoning behind this, consider the case of a Lie algebra g. Since this is a Lie
algebroid over •, where τ and ρ are trivial, it follows that every path ξ ∈ P(g) is a g-path, i.e.,
Pρ(g) = P(g). However, the variational principle for the Euler–Poincaré equations on g considers
only variations of the form
δξ = [ξ, η] + η̇ = adξ η + η̇,
where η ∈ P(g) is an arbitrary path vanishing at the endpoints (cf. Marsden and Ratiu [18,
Chapter 13]). These constraints on admissible variations are known as Lin constraints.
To generalize these constrained variations to an arbitrary Lie algebroid A → Q, we first
discuss the notion of a connection on a Lie algebroid, of which the adjoint action (ξ, η) 7→ adξ η
of g on itself will be a special case.
Definition 3.6. If A→ Q is a Lie algebroid and E → Q is a vector bundle, then an A-connection
on E is a bilinear map ∇ : Γ(A)× Γ(E)→ Γ(E), (X,u) 7→ ∇Xu, satisfying the conditions
∇fXu = f∇Xu, ∇X(fu) = f∇Xu+ ρ(X)[f ]u,
for all X ∈ Γ(A), u ∈ Γ(E), and f ∈ C∞(Q).
Remark 3.7. A TQ-connection is just an ordinary connection. Given a TQ-connection ∇ on A,
there are two naturally-induced A-connections on A, which we write as ∇ and ∇:
∇XY = ∇ρ(X)Y, ∇XY = ∇ρ(Y )X + [X,Y ] .
10 S. Li, A. Stern and X. Tang
For example, when A = g→ •, the trivial T•-connection induces two g-connections on g:
∇XY = 0, ∇XY = [X,Y ] = adX Y.
Hence, the induced connection ∇ can be seen as a generalization of the adjoint action of a Lie
algebra.
Definition 3.8. Let a ∈ Pρ(A) be an A-path over q ∈ P(Q) and ξ ∈ P
(
Γ(A)
)
be a time-
dependent section such that a(t) = ξ
(
q(t)
)
. Suppose u ∈ P(E) has the same base path q, along
with a time-dependent section η ∈ P
(
Γ(E)
)
satisfying u(t) = η
(
q(t)
)
. Then we define
∇au(t) = ∇ξη
(
t, q(t)
)
+ η̇
(
t, q(t)
)
,
which is independent of the choice of ξ, η.
Definition 3.9. Let a ∈ Pρ(A) be an A-path over q ∈ P(Q). An admissible variation of a is
a variation of the form Xb,a ∈ TaPρ(A), where b ∈ P(A) is a path in A (but not necessarily
an A-path!) over q such that b(0) = 0 and b(1) = 0. Relative to a TQ-connection ∇, the
variation Xb,a has vertical component ∇ab and horizontal component ρ(b).
Remark 3.10. Crainic and Fernandes [6, Proposition 4.7] show that these admissible variations
form an integrable subbundle F(A) ⊂ TPρ(A), and the tangent subspaces Fa(A) ⊂ TaPρ(A) are
independent of the choice of connection ∇ in the above definition.
3.3 Lagrangian mechanics
Now that we have appropriate paths and variations, we are prepared to discuss the variational
approach to Lagrangian mechanics on Lie algebroids.
Definition 3.11. Given a Lagrangian L : A→ R, the action functional S : Pρ(A)→ R is defined
to be
S(a) =
∫ 1
0
L
(
a(t)
)
dt.
We say that a ∈ Pρ(A) satisfies Hamilton’s variational principle for A-paths if dS(Xb,a) = 0 for
all admissible variations Xb,a ∈ Fa(A).
We next use the notion of admissible variation from Definition 3.9, and its expression in
terms of a connection on A, to give a new, coordinate-free characterization of the solutions to
Hamilton’s variational principle for A-paths.
Theorem 3.12. An A-path a ∈ Pρ(A) satisfies Hamilton’s principle if and only if, given a TQ-
connection ∇ on A, it satisfies the differential equation
ρ∗dLhor(a) +∇∗adLver(a) = 0, (3.1)
where dLhor and dLver are the horizontal and vertical components of dL relative to ∇, and
where ρ∗ and ∇∗a are the formal adjoints of ρ and ∇a.
Proof. Given an admissible variation Xb,a ∈ Fa(A), we have
dS(Xb,a) = dS
(
Xhor
b,a
)
+ dS
(
Xver
b,a
)
=
∫ 1
0
(〈
dLhor(a), ρ(b)
〉
+
〈
dLver(a),∇ab
〉)
dt
=
∫ 1
0
〈
ρ∗dLhor(a) +∇∗adLver(a), b
〉
dt
Since b is arbitrary, it follows that dS vanishes for all Xb,a ∈ Fa(A) if and only if ρ∗dLhor(a) +
∇∗adLver(a) vanishes for all t. �
Lagrangian Mechanics and Reduction on Fibered Manifolds 11
Example 3.13. Let A = g → •, where g is a Lie algebra. Any a ∈ Pρ(g) = P(g) can be
identified with its unique time-dependent section ξ(t) = ξ(t, •) = a(t). Since ρ and ∇ are trivial,
it follows that (3.1) becomes
0 = ∇∗adL(a) =
(
adξ +
d
dt
)∗ δL
δξ
=
(
ad∗ξ −
d
dt
)
δL
δξ
,
which are precisely the Euler–Poincaré equations (cf. Marsden and Ratiu [18, Chapter 13]).
Next, we show that this coordinate-free formulation agrees with the local-coordinate expression
obtained by Weinstein [30] for regular Lagrangians and by Mart́ınez [23, 24, 25] in the more
general case.
Theorem 3.14. Let qi be local coordinates for Q, {eI} be a local basis of sections of A, and ∇
the locally trivial TQ-connection defined by ∇∂/∂qieI ≡ 0. Let ρiI and CKIJ be the local-coordinate
representations of ρ and [·, ·], where
ρ(eI) = ρiI
∂
∂qi
, [eI , eJ ] = CKIJeK .
If a ∈ P(A) has the local-coordinate representation a(t) = ξI(t)eI
(
q(t)
)
, then a is an A-path if
and only if q̇i = ρiIξ
I , and a satisfies (3.1) if and only if
ρiI
∂L
∂qi
− CKIJξJ
∂L
∂ξK
− d
dt
∂L
∂ξI
= 0. (3.2)
Proof. For the A-path condition, we have
q̇ = q̇i
∂
∂qi
, ρ(a) = ρ
(
ξIeI
)
= ρiIξ
I ∂
∂qi
,
so these are equal if and only the ∂/∂qi coefficients are equal. Next, the horizontal and vertical
components of dL are
dLhor =
∂L
∂qi
dqi, dLver =
∂L
∂ξI
eI ,
where, as usual, eI is the dual basis element satisfying eIeJ = δIJ . Moreover, extending a to the
time-dependent section ξ(t) = ξJ(t)eJ , we have
∇aηIeI = ∇ξJeJη
IeI + η̇IeI =
[
ξJeJ , η
IeI
]
+ η̇IeI = −CKIJξJηIeK + η̇IeI ,
so ∇a = −CKIJξJeIeK + d/dt. Finally,
ρ∗dLhor +∇∗adLver = ρiI
∂L
∂qi
eI − CKIJξJ
∂L
∂ξK
eI − d
dt
∂L
∂ξI
eI ,
so the left side vanishes if and only if all the eI coefficients on the right side vanish, i.e., (3.1)
holds if and only if (3.2) holds. �
Example 3.15. Suppose A = TQ → Q. Local coordinates qi on Q yield corresponding local
sections ∂/∂qi of TQ, i.e., ei = ∂/∂qi. It follows that [ei, ej ] ≡ 0 and thus Ckij ≡ 0 for all i, j, k.
Since ρ is the identity map, we have ρij = δij , so the TQ-path condition is q̇i = ξi. Putting this
all together, it follows that (3.2) yields
∂L
∂qi
− d
dt
∂L
∂q̇i
= 0,
i.e., the ordinary Euler–Lagrange equations.
12 S. Li, A. Stern and X. Tang
Remark 3.16. There is also an equivalent symplectic/pre-symplectic/Poisson approach to
Lagrangian mechanics on Lie algebroids, which has already been well studied in previous work
on the subject.
Mart́ınez [23] shows that one can define a Lie algebroid notion of differential forms (just as
we did for the vertical formalism in Section 2.2), as well as a version of the tautological 1-form
and canonical 2-form on A∗. The Legendre transform FL = dLver : A→ A∗ is then used to pull
this back to a Lagrangian 2-form on A (in the sense of forms on Lie algebroids) and to define an
energy function EL on A, which Mart́ınez [23] uses to obtain Lagrangian dynamics on A.
Weinstein [30], on the other hand, uses the canonical Poisson structure on A∗ (which generalizes
the Lie–Poisson structure on the dual of a Lie algebra), which can be pulled back along FL to A
when L is a regular Lagrangian. In this case, the Poisson structure on A induces a Lagrangian
vector field associated to EL in the usual way.
The approach of Grabowska et al. [10], Grabowska and Grabowski [9] extends Weinstein’s
approach in a different direction: instead of using the canonical Poisson structure on A∗,
which maps T ∗A∗ → TA∗, they use a related map ε : T ∗A → TA∗ to define the Tulczyjew
differential ΛL = ε ◦ dL : A → TA∗. (The map ε is related to the canonical Poisson map by
the Tulczyjew isomorphism T ∗A∗
∼=−→ T ∗A.) Using this framework, one requires that a ∈ P(A)
satisfy d
dtFL(a) = ΛL(a), which contains the Euler–Lagrange–Poincaré equations together with
the A-path condition. We remark that Grabowska et al. [10], Grabowska and Grabowski [9]
apply this approach both to Lie algebroids and to so-called “general algebroids,” for which the
map ε is taken as primitive, and where there is generally no canonical Poisson structure on the
dual.
3.4 Special case: the Lagrange–Poincaré equations
The Lagrange–Poincaré equations on a principal bundle Q→ Q/G are typically derived by the
procedure of Lagrangian reduction (cf. Marsden and Scheurle [21], Cendra et al. [3]), relative to
a particular choice of principal connection. We now discuss how these equations may instead be
obtained directly on the Atiyah algebroid A = TQ/G→ Q/G, using the framework presented
above, and how the choice of principal connection is related to the connection ∇ on A. (Note
that Q/G, not Q, is the base of this algebroid.) In particular, Example 3.13 corresponds to the
case Q = G, while Example 3.15 corresponds to the case where G is trivial.
Let L : TQ/G → R be a Lagrangian on the Atiyah algebroid. A principal connection
corresponds to a section of the anchor ρ : TQ/G→ T (Q/G), i.e., a right splitting of the Atiyah
sequence,
0→ g̃→ TQ/G
ρ−→ T (Q/G)→ 0. (3.3)
Here, following Cendra et al. [3], we use g̃ to denote the adjoint bundle Q×G g, so a left splitting
is a principal connection 1-form (cf. Mackenzie [14, Chapter 5]). This splitting lets us write
TQ/G ∼= T (Q/G)⊕ g̃; the anchor ρ is just projection onto the first component, and the bracket
of two sections ξ = (X, ξ) and η = (Y, η) is[
(X, ξ), (Y, η)
]
=
(
[X,Y ], ∇̃Xη − ∇̃Y ξ + [ξ, η]− R̃(X,Y )
)
, (3.4)
where ∇̃ is the covariant derivative and R̃ the curvature form of the principal connection (cf.
Cendra et al. [3, Theorem 5.2.4] in this particular case and Mackenzie [14, Theorem 7.3.7] in
a more general setting).
Relative to the splitting induced by the principal connection, A-paths have the form a =
(x, ẋ, v), where x is the base path in Q/G. As before, we extend a to a time-dependent section
ξ = (X, ξ), and likewise, we extend an arbitrary path b = (x, δx, w) to a time-dependent section
Lagrangian Mechanics and Reduction on Fibered Manifolds 13
η = (Y, η). To find the corresponding admissible variation δa, we calculate ρ(b) = δx and
use (3.4) to obtain
∇ab = ∇ξη + η̇ = ∇(X,ξ)(Y, η) + (Ẏ , η̇) = ∇Y (X, ξ) +
[
(X, ξ), (Y, η)
]
+ (Ẏ , η̇)
=
(
∇YX + [X,Y ] + Ẏ , ∇̃Xη + [ξ, η]− R̃(X,Y ) + η̇
)
=
(
∇XY + Ẏ , (∇̃Xη + η̇) + [ξ, η]− R̃(X,Y )
)
=
(
∇ẋ(δx), ∇̃ẋw + [v, w]− R̃(ẋ, δx)
)
.
(Here, we chose ∇ to be compatible with ∇̃, so that the ∇Y ξ and ∇̃Y ξ terms cancel.) Therefore,
admissible variations have the form δa = (δx, δẋ, δv), where
δv = ∇̃ẋw + [v, w]− R̃(ẋ, δx),
and these are precisely the admissible variations of Cendra et al. [3, Theorem 3.4.1].
Furthermore, now that we have expressions for ρ and ∇ in terms of the splitting induced by the
principal connection, it is a straightforward matter to write down the Euler–Lagrange–Poincaré
equations (3.1) in terms of their adjoints. If we write L = L(x, ẋ, v), then
ρ∗dLhor(x, ẋ, v) +∇∗(x,ẋ,v)dL
ver(x, ẋ, v)
=
(
∂L
∂x
+∇∗ẋ
∂L
∂ẋ
− (iẋR̃)∗
∂L
∂v
)
dx+
(
∇̃∗ẋ
∂L
∂v
+ ad∗v
∂L
∂v
)
dv.
Hence, this vanishes precisely when
∂L
∂x
+∇∗ẋ
∂L
∂ẋ
− (iẋR̃)∗
∂L
∂v
= 0, ∇̃∗ẋ
∂L
∂v
+ ad∗v
∂L
∂v
= 0,
which are exactly the coordinate-free Lagrange–Poincaré equations of Cendra et al. [3, Theo-
rem 3.4.1]. (The only notable difference in notation is that Cendra et al. [3] write both covariant
derivatives ∇ẋ and ∇̃ẋ as D/Dt and their adjoints as −D/Dt.)
Remark 3.17. The argument above works not only for the Atiyah algebroid of a principal
bundle, but also in the more general setting discussed in Mackenzie [14, Chapter 7], where one
can split a short exact sequence similar to (3.3) and obtain a bracket of the form (3.4). This
includes the so-called transitive Lie algebroids, of which the Atiyah algebroid is a particular
example.
Example 3.18. Wong’s equations [31] for a particle in a Yang–Mills field are a classic example
of Lagrange–Poincaré theory. Following the presentation in Cendra et al. [3, Chapter 4], we
suppose that Q→ Q/G is a principal G-bundle equipped with a Riemannian metric g on the
base Q/G and a bi-invariant Riemannian metric κ on the structure group G. Using a principal
connection to split TQ/G ∼= T (Q/G)⊕ g̃, and denoting by k the fiber metric on g̃ corresponding
to κ, we take the Lagrangian
L(x, ẋ, v) =
1
2
k(v, v) +
1
2
g(ẋ, ẋ).
The affine connection ∇ is then chosen to agree with ∇̃ on g̃ and with the Levi-Civita connection
associated to g on the base.
With this connection in hand, we now compute the dLver components,
∂L
∂ẋ
= g(ẋ, ·) = g[(ẋ),
∂L
∂v
= k(v, ·) = k[(v),
14 S. Li, A. Stern and X. Tang
using the familiar “flat” notation for metrics. Since the fiber metric k is necessarily ad-invariant,
the term ad∗v k
[(v) vanishes, so the dv component of the Lagrange–Poincaré equations is
∇̃ẋk[(v) = 0. (3.5)
Next, since ∇ agrees with the Levi-Civita connection on Q/G, the torsion-free property implies
∇XY = ∇YX + [X,Y ] = ∇XY,
so we just have ∇ ≡ ∇. Moreover, using the metric-compatibility of ∇ along with (3.5) to
compute dLhor, it can be seen that
∂L
∂x
+∇∗ẋ
∂L
∂ẋ
= g[(∇ẋẋ),
and therefore the dx component of the Lagrange–Poincaré equations is
g[(∇ẋẋ) =
(
iẋR̃
)∗
k[(v). (3.6)
The equations (3.5) and (3.6) are precisely the coordinate-free version of Wong’s equations. For
further discussion on Wong’s equations from the perspective of Lie algebroids, see León et al. [7],
Grabowska et al. [10].
We conclude this example with some remarks on the relationship between Wong’s equations
and the generalized notion of geodesics on a Lie algebroid. Montgomery [26] called g⊕k a Kaluza–
Klein metric and related Wong’s equations to Kaluza–Klein geodesics. However, a Kaluza–Klein
metric is a particular example of a Lie algebroid metric (in this case, on A = TQ/G), for which
there is a unique Levi-Civita (torsion-free, metric-compatible) A-connection ∇, and one may
consider the corresponding geodesic equations,
∇aa = 0.
(See Crainic and Fernandes [6], Cortés and Mart́ınez [5], Cortés et al. [4].) Grabowska et al. [10]
pointed out that Wong’s equations may in fact be considered a special case of the generalized
geodesic equations on a Lie algebroid; this correspondence is hidden slightly by the fact that
Wong’s equations are written relative to an A-connection obtained from ∇̃ rather than the
Levi-Civita A-connection.
3.5 Fibered manifolds revisited
The results of Section 2 for fibered manifolds are, in fact, a special case of Lagrangian mechanics
on the Lie algebroid V Q.
Recall from Example 3.2 that, whenever Q→M is a fibered manifold, the vertical bundle V Q
is a Lie algebroid over Q; in particular, it is a Lie subalgebroid of TQ, from which it inherits
the bracket [·, ·], projection ρ, and (identity) anchor ρ. Now, by Definition 3.4, a ∈ P(V Q)
over q ∈ P(Q) is a V Q-path if and only if it satisfies q̇ = a. Since a(t) ∈ V Q for each t ∈ I,
this means that V Q-paths are precisely the tangent prolongations of vertical paths q ∈ PV (Q).
Hence, we may identify Pρ(V Q) with PV (Q).
Suppose now that L : V Q → R is a Lagrangian in the sense of Section 3.3. If (xσ, yi) are
fiber-adapted local coordinates for Q → M , then ei = ∂/∂yi defines a basis of local sections
of V Q. Since an A-path is just a tangent prolongation of a vertical path, it follows that the
A-path conditions are ẋσ = 0 and ẏi = ξi. Furthermore, as in Example 3.15, we have ρij = δij ,
ρiσ ≡ 0, and Ckij ≡ 0, so (3.2) becomes
∂L
∂yi
− d
dt
∂L
∂ẏi
= 0.
Together with the A-path condition, this agrees precisely with the vertical Euler–Lagrange
equations (2.1).
Lagrangian Mechanics and Reduction on Fibered Manifolds 15
3.6 Lie algebroid morphisms and reduction
Finally, we give a brief review of Lagrangian reduction on Lie algebroids. Weinstein [30] and
Mart́ınez [25] showed that, whenever Φ: A→ A′ is a Lie algebroid morphism, then one can relate
Lagrangian dynamics on A to those on A′.
Informally, a Lie algebroid morphism is a mapping that “preserves” the Lie algebroid structure
in an appropriate sense. More precisely, if A → M and A′ → M ′ are Lie algebroids (possibly
over different base manifolds), then a bundle mapping Φ: A→ A′ is a Lie algebroid morphism if
the dual comorphism Φ∗ : A′∗ → A∗ is a Poisson relation with respect to the canonical Poisson
structures on A∗ and A′∗. (See also Remark 3.16.)
Theorem 3.19. Let Φ: A→ A′ be a morphism of Lie algebroids, and suppose L : A→ R and
L′ : A′ → R are Lagrangians such that L = L′ ◦Φ. If a ∈ Pρ(A) is such that a′ = Φ ◦ a ∈ Pρ′(A′)
is a solution path for L′, then a is a solution path for L. Moreover, the following converse
holds when Φ: A → A′ is fiberwise surjective: If a ∈ Pρ(A) is a solution path for L, then
a′ = Φ ◦ a ∈ Pρ′(A′) is a solution path for L′.
Proof. See Mart́ınez [25, Theorems 5–6]. This generalized results by Weinstein [30, Theorems 4.8
and 4.5, respectively] for regular Lagrangians, where the converse also required the stronger
assumption that Φ be a fiberwise isomorphism. �
For example, if G is a Lie group acting freely and properly on Q, then the quotient morphism
TQ→ TQ/G is a Lie algebroid morphism, and the corresponding reduction theory is just classical
Lagrangian reduction. However, there is a much more general class of quotient morphisms –
for fibered manifolds – that bear directly on reduction theory, and this is the topic of the next
section.
4 Lie groupoid symmetries and reduction on fibered manifolds
In this section, we recall the definition of a Lie groupoid G ⇒ M and of a free, proper Lie
groupoid action on a fibered manifold Q → M over the same base manifold. We then show
that there is a quotient morphism V Q → V Q/G, which is a Lie algebroid morphism, and
hence applying Theorem 3.19 yields a reduction theory for fibered Lagrangian mechanics. This
generalizes the special case M = •, in which G is a Lie group acting on an ordinary manifold Q
and the quotient morphism TQ→ TQ/G is the one used in ordinary Lagrangian reduction.
4.1 Lie groupoids
Just as it is natural to consider Lie group actions on ordinary manifolds, it is natural to consider
Lie groupoid actions on fibered manifolds. We begin by recalling the definition of a Lie groupoid
and a groupoid action, as well as giving a few examples. We then prove that, just as a free and
proper Lie group action on an ordinary manifold Q lifts to TQ, so, too, does a free and proper
Lie groupoid action on a fibered manifold Q→M lift to V Q.
Definition 4.1. A groupoid is a small category in which every morphism is invertible. Specifically,
a groupoid denoted G⇒M consists of a space of morphisms G, a space of objects M , and the
following structure maps:
(i) a source map α : G→M and target map β : G→M ;
(ii) a multiplication map m : G α×β G→ G, (g, h) 7→ gh;
(iii) an identity section ε : M → G, such that for all g ∈ G,
gε(α(g)
)
= g = ε
(
β(g)
)
g;
16 S. Li, A. Stern and X. Tang
(iv) and an inversion map i : G→ G, g 7→ g−1, such that for all g ∈ G,
g−1g = ε
(
α(g)
)
, gg−1 = ε
(
β(g)
)
.
A Lie groupoid is a groupoid G ⇒ M where G and M are smooth manifolds, α and β are
submersions, and m is smooth.
Remark 4.2. A few other properties of the structure maps are immediate from this definition
of a Lie groupoid: in particular, it also follows that m is a submersion, ε is an immersion, and i
is a diffeomorphism.
Example 4.3. A Lie group is a Lie groupoid G⇒ • over a single point.
Example 4.4. If Q is a smooth manifold, then the pair groupoid Q×Q⇒ Q, defined by the
structure maps
α(q1, q0) = q0, β(q1, q0) = q1, m
(
(q2, q1), (q1, q0)
)
= (q2, q0),
ε(q) = (q, q), i(q1, q0) = (q0, q1),
is a Lie groupoid. More generally, if µ : Q→M is a fibered manifold and
Q µ×µ Q =
{
(q1, q0) ∈ Q×Q : µ(q1) = µ(q2)
}
,
then Q µ×µ Q ⇒ Q is also a Lie groupoid, and its structure maps are just the restrictions of
those above for Q×Q⇒ Q. We then say that Q µ×µQ⇒ Q is a Lie subgroupoid of Q×Q⇒ Q.
Example 4.5. Let G be a Lie group and Q→ Q/G be a principal G-bundle, i.e., G acts freely
and properly on Q. The diagonal action of G on Q×Q is also free and proper, so we may form
the quotient (Q×Q)/G. Let [q] ∈ Q/G denote the orbit of q ∈ Q and [q1, q0] ∈ (Q×Q)/G denote
the orbit of (q1, q0) ∈ Q×Q. Then the gauge groupoid (or Atiyah groupoid) (Q×Q)/G⇒ Q/G
of the principal bundle is defined by the structure maps
α
(
[q1, q0]
)
= [q0], β
(
[q1, q0]
)
= [q1], m
(
[q2, q1], [q1, q0]
)
= [q2, q0],
ε
(
[q]
)
= [q, q], i
(
[q1, q0]
)
= [q0, q1].
Notice that G ⇒ • is the special case where Q = G acts on itself by multiplication, while
Q×Q⇒ Q is the special case where G = {e} acts trivially on Q.
Definition 4.6. A left action (or just action) of a Lie groupoid G⇒M on a fibered manifold
Q→M is a smooth map G α×µ Q→ Q, (g, q) 7→ gq, such that
(i) µ(gq) = β(g) for all (g, q) ∈ G α×µ Q,
(ii) g(hq) = (gh)q for all (g, h, q) ∈ G α×β G α×µ Q, and
(iii) ε
(
µ(q)
)
q = q for all q ∈ Q.
The action is free if gq = q implies g = ε
(
µ(q)
)
, and it is proper if its graph,
G α×µ Q→ Q×Q, (g, q) 7→ (gq, q),
is a proper map. A principal G-space is a fibered manifold endowed with a free and proper
G-action.
Remark 4.7. As with group actions, it can be shown that if G⇒M acts freely and properly
on Q → M , then the quotient Q/G consisting of G-orbits is a smooth manifold, and there is
a smooth quotient map Q → Q/G. We refer to Dufour and Zung [8, Chapter 7] for a more
detailed discussion of this and other properties of groupoid actions.
Lagrangian Mechanics and Reduction on Fibered Manifolds 17
Example 4.8. The action of a Lie group G on a manifold Q is precisely the action of the Lie
groupoid G ⇒ • on the fibered manifold Q → •. If the action is free and proper, then the
associated principal G-space corresponds to the principal G-bundle Q→ Q/G.
Example 4.9. For any smooth manifold Q, the pair groupoid Q × Q ⇒ Q acts on Q by
(q1, q0)q0 = q1. (In this case, we treat Q as the fibered manifold Q → Q, rather than Q → •.)
Since any two points q0, q1 lie in the same orbit, it follows that Q/(Q×Q) ∼= •, and the quotient
map is simply Q→ •.
Example 4.10. Let G be a Lie group acting freely and properly on Q, so that Q → Q/G is
a principal G-bundle. Then the gauge groupoid (Q×Q)/G acts on Q→ Q/G, in the sense of
Definition 4.6, and is uniquely defined by the condition [q1, q0]q0 = q1. (Notice that Example 4.9
is the special case where G = {e} acts trivially on Q.) Again, we see that any two points
q0, q1 ∈ Q lie in the same orbit, so Q/
(
(Q×Q)/G
) ∼= •, and the quotient map is Q→ •.
Example 4.11. For any Lie groupoid G ⇒ M , the multiplication map m is an action of G
on itself, treated as the fibered manifold β : G → M . This action is free, since gh = h
implies g = (gh)h−1 = hh−1 = ε
(
β(h)
)
. Moreover, the action is proper: (g, h) 7→ (gh, h) is
a diffeomorphism, having the inverse (g, h) 7→ (gh−1, h), so in particular it is a proper map.
The orbit of each h ∈ G is its α-fiber α−1
(
{x}
)
, where x = α(h). Identifying the fiber
α−1
(
{x}
)
with the corresponding base point x ∈M , it follows that G/G ∼= M , and the quotient
map is just α : G→M .
Example 4.12. If G ⇒ M acts on Q → M , then it also acts on V Q → M , considered as
a fibered manifold. Specifically, we have the action
G α×µ◦τ V Q→ V Q, (g, v) 7→ g∗v,
where g∗ denotes the pushforward of q 7→ gq.
Lemma 4.13. Suppose G ⇒ M has a free, proper action on Q → M . Then its diagonal
action on Q µ×µ Q→M , given by g(q1, q0) = (gq1, gq0), is also free and proper. Moreover, the
quotient can be given a natural Lie group structure (Q µ×µ Q)/G⇒ Q/G, and the quotient map
Q µ×µ Q→ (Q µ×µ Q)/G is a morphism of Lie groupoids over Q→ Q/G.
Proof. The fact that
(
g, (q1, q0)
)
7→ (gq1, gq0) is a free and proper groupoid action follows
immediately from the fact that, by assumption, (g, q) 7→ gq is. As stated in Remark 4.7, the
freeness and properness of these actions imply that Q/G and (Q µ×µQ)/G are smooth manifolds,
so it suffices to specify the groupoid structure maps for (Q µ×µ Q)/G⇒ Q/G. These may be
taken to be formally identical to those for the gauge groupoid in Example 4.5, i.e.,
α
(
[q1, q0]
)
= [q0], β
(
[q1, q0]
)
= [q1], m
(
[q2, q1], [q1, q0]
)
= [q2, q0],
ε
(
[q]
)
= [q, q], i
(
[q1, q0]
)
= [q0, q1].
As with the gauge groupoid, it is simple to check directly that these satisfy the conditions of
Definition 4.1, so this is a Lie groupoid. Finally, using α̃, β̃, . . . to denote the structure maps on
Q µ×µ Q⇒ Q, we observe that
α
(
[q1, q0]
)
=
[
α̃(q1, q0)
]
, β
(
[q1, q0]
)
=
[
β̃(q1, q0)
]
,
m
(
[q2, q1], [q1, q0]
)
=
[
m̃
(
(q2, q1), (q1, q0)
)]
,
ε
(
[q]
)
=
[
ε̃(q)
]
, i
(
[q1, q0]
)
=
[̃
ı(q1, q0)
]
,
so the quotient map preserves the structure maps and hence is a Lie groupoid morphism. �
18 S. Li, A. Stern and X. Tang
Lemma 4.14. The action of a Lie groupoid G ⇒ M on Q→ M is free (resp., proper) if and
only if the induced action on V Q→M is free (resp., proper).
Proof. If G acts freely on Q, then g∗v = v implies g
(
τ(v)
)
= τ(v), so g = ε
(
µ
(
τ(v)
))
=
ε
(
(µ ◦ τ)(v)
)
, and hence G acts freely on V Q. Conversely, if G acts freely on V Q, then gq = q
implies g∗0q = 0q so g = ε
(
(µ ◦ τ)(0q)
)
= ε
(
µ(q)
)
, and hence G acts freely on Q.
The proof of properness essentially amounts to chasing compact sets around the following
diagram:
G α×µ◦τ V Q V Q µ◦τ×µ◦τ V Q
G α×µ Q Q µ×µ Q.
id×τ τ×τid×0 0×0
First, suppose G acts properly on Q. If K ⊂ V Q µ◦τ×µ◦τ V Q is compact, then we wish to show
that the preimage,{
(g, v) ∈ G α×µ◦τ V Q : (v, g∗v) ∈ K
}
,
is also compact. Observe that
{
v ∈ V Q : (v, g∗v) ∈ K
}
is compact by the continuity of
(v, g∗v) 7→ v, and
{
g ∈ G : (v, g∗v) ∈ K
}
is compact by the continuity of (v, g∗v) 7→ (q, gq), with
q = τ(v), the properness of (g, q) 7→ (q, gq), and the continuity of (g, q) 7→ g. Hence, the preimage
in question is also compact, so G acts properly on V Q.
Conversely, supposeG acts properly on V Q. IfK ⊂ Qµ×µQ is compact, then so is
{
(0q, g∗0q) ∈
V Q µ◦τ×µ◦τ V Q : (q, gq) ∈ K
}
, and by properness, so is
{
(g, 0q) ∈ G α×µ◦τ V Q : (q, gq) ∈ K
}
.
Finally, the preimage,{
(g, q) ∈ G α×µ Q : (g, q) ∈ K
}
,
is compact by the continuity of (g, 0q) 7→ (g, q), so G acts properly on Q. �
4.2 Lie algebroid of a Lie groupoid
Before discussing reduction by an arbitrary free and proper groupoid action, we first consider
the important special case where a groupoid acts on itself by left multiplication. (This can be
thought of as the “groupoid version” of Euler–Poincaré reduction, which is the special case of
Lagrange–Poincaré reduction where Q = G is a Lie group.)
Recall from Example 4.11 that a Lie groupoid G⇒M acts freely and properly on itself (as the
fibered manifold β : G→M) by left multiplication. Lemma 4.14 implies that this induces a free
and proper action of G on the β-vertical bundle V βG→M . (Since G can be seen as a fibered
manifold in two different ways, α : G→M and β : G→M , we denote the corresponding vertical
bundles by V αG and V βG to avoid any possible confusion.) Since the orbit of v ∈ V β
g G is
uniquely determined by its representative at the identity section, (g−1)∗v ∈ V β
ε(α(g))G, we can
identify the quotient V βG/G with the vector bundle AG = V β
ε(M)G over M .
This vector bundle AG → M is in fact a Lie algebroid, called the Lie algebroid of G. The
anchor map is given by the restriction of α∗ : TG→ TM to AG. Furthermore, the identification
of AG with V βG/G implies that sections X ∈ Γ(AG) correspond to G-invariant, β-vertical
vector fields
←−
X ∈ Xβ(G), with
←−
X (g) = g∗X
(
α(g)
)
. The bracket [X,Y ] of X,Y ∈ Γ(AG) is then
defined so that
←−−−
[X,Y ] = [
←−
X,
←−
Y ], where the bracket on the right-hand side of this expression is
just the Jacobi–Lie bracket of vector fields on G. (See Mackenzie [14].)
Lagrangian Mechanics and Reduction on Fibered Manifolds 19
Example 4.15. Let G be a Lie group, so that G ⇒ • is a Lie groupoid. Since β is trivial,
we have V βG = TG, and hence AG = TeG = g → •, where g is the Lie algebra of G and
e = ε(•) ∈ G is the identity element of G.
Example 4.16. For the pair groupoid Q×Q⇒ Q, we have V β(Q×Q) = TQ×Q, and hence
A(Q×Q) = TQ τ×Q ∼= TQ→ Q.
More generally, if we consider the groupoid Q µ×µ Q ⇒ Q for a fibered manifold Q → M ,
then V β(Q µ×µ Q) = V Q µ◦τ×µ Q, and hence A(Q µ×µ Q) = V Q τ×Q ∼= V Q→ Q.
Example 4.17. For the gauge groupoid (Q×Q)/G⇒ Q/G of a principal bundle Q→ Q/G, we
have V β
(
(Q×Q)/G
)
= (TQ×Q)/G, and hence A
(
(Q×Q)/G
)
= (TQτ×Q)/G ∼= TQ/G→ Q/G.
This is called the gauge algebroid (or Atiyah algebroid) of the principal bundle.
More generally, considering the groupoid (Q µ×µ Q)/G⇒ G of a principal G-space, we have
V β
(
(Qµ×µQ)/G
)
= (V Qµ◦τ×µQ)/G, and hence A
(
(Qµ×µQ)/G
)
= (V Q τ×Q)/G ∼= V Q/G→
Q/G.
Remark 4.18. The relationship between a groupoid G and its algebroid AG has an interesting
application to the discretization of Lagrangian mechanics, which can be used to develop structure-
preserving numerical integrators. In this approach, pioneered by Weinstein [30] (see also Marrero
et al. [15, 16], Stern [29]), one replaces the Lagrangian L : AG → R by a discrete Lagrangian
Lh : G→ R, replaces AG-paths by sequences of composable arrows in G, and uses a variational
principle to derive discrete equations of motion. In particular, using G = Q×Q⇒ Q to discretize
AG = TQ → Q gives the framework of variational integrators (cf. Moser and Veselov [27],
Marsden and West [22]).
4.3 Reduction by a groupoid action
Recall from Lemma 4.14 that if G⇒M acts freely and properly on Q→M , then it also acts
freely and properly on V Q→M . In other words, V Q is also a principal G-space, equipped with
a quotient map V Q→ V Q/G. We have seen that V Q is also a Lie algebroid, and moreover, in
Example 4.16, that it is the Lie algebroid of the Lie groupoid Q µ×µ Q ⇒ Q. Similarly, from
Example 4.17, we have that V Q/G is the Lie algebroid of the Lie groupoid (Q µ×µQ)/G⇒ Q/G.
Therefore, in order to perform reduction using Theorem 3.19, it suffices to show that the
quotient map V Q→ V Q/G is in fact a Lie algebroid morphism.
Lemma 4.19. Let G ⇒ M be a Lie groupoid and Q → M a principal G-space. Then the
quotient map V Q→ V Q/G is a Lie algebroid morphism covering Q→ Q/G.
Proof. We can use a result stated in Mackenzie [14, Proposition 4.3.4], which says that a mor-
phism of Lie groupoids G→ G′ induces a corresponding morphism of Lie algebroids AG→ AG′.
This defines the so-called Lie functor between the categories of Lie groupoids and Lie algebroids,
taking objects G 7→ AG and morphisms (G→ G′) 7→ (AG→ AG′).
Now, we have already proved in Lemma 4.13 that the quotient map Qµ×µQ→ (Qµ×µQ)/G is
a morphism of Lie groupoids, so applying the Lie functor to this morphism proves the result. �
Theorem 4.20. Let G ⇒ M be a Lie groupoid and Q → M a principal G-space. Suppose
the Lagrangian L : V Q→ R is G-invariant, i.e., that it factors through the quotient morphism
Φ: V Q → V Q/G as L = ` ◦ Φ, where ` : V Q/G → R is called the reduced Lagrangian. Then
a ∈ PV (Q) is a solution path for L if and only if Φ ◦ a ∈ Pρ(V Q/G) is a solution path for `.
Proof. Apply Theorem 3.19 to the (fiberwise-surjective) Lie algebroid morphism defined in
Lemma 4.19. �
20 S. Li, A. Stern and X. Tang
Example 4.21. When G⇒ • is a Lie group acting freely and properly on Q→ •, Theorem 4.20
corresponds to ordinary Lagrangian reduction from TQ to TQ/G, yielding the Lagrange–Poincaré
equations of Section 3.4. In the special case where Q = G acts on itself by multiplication, this
gives Euler–Poincaré reduction from TG to TG/G ∼= g.
Example 4.22. Suppose G ⇒ M is a Lie groupoid acting on itself by multiplication, so that
the quotient morphism is Φ: V βG → V βG/G = AG. If L : V βG → R and ` : AG → R are
Lagrangians satisfying L = ` ◦ Φ, then Theorem 4.20 implies that the vertical Euler–Lagrange
equations (Section 2) on V βG reduce to the Euler–Lagrange–Poincaré equations (Section 3) for
the Lie algebroid AG. (This special case appears in Weinstein [30, Theorem 5.3].) The even
more special case where G⇒ • is a Lie group again gives Euler–Poincaré reduction on the Lie
algebra g.
5 The Hamilton–Pontryagin principle and reduction
In this section, we extend the foregoing theory to the Hamilton–Pontryagin variational principle
introduced by Yoshimura and Marsden [33] as a generalization of Hamilton’s variational principle.
This principle is especially useful for the study of “implicit Lagrangian systems” that arise in
mechanical and control systems with nonholonomic or Dirac constraints. (See also Yoshimura
and Marsden [32] for the non-variational approach to such systems, as well as Yoshimura and
Marsden [34] for the associated reduction theory.)
We begin, in Section 5.1, with a brief review of the Hamilton–Pontryagin principle for ordinary
manifolds. We then generalize it, in Section 5.2, to fibered manifolds and their (co)vertical
bundles, as we did for Hamilton’s principle in Section 2. In Section 5.3, we generalize the
Hamilton–Pontryagin principle even further to mechanics on Lie algebroids and their duals, as
was done for Hamilton’s principle in Section 3. Finally, in Section 5.4, we discuss reduction of
the Hamilton–Pontryagin principle by Lie algebroid morphisms, as in the Weinstein–Mart́ınez
reduction theorem (Theorem 3.19), and apply this to the special case of groupoid symmetries for
a fibered manifold, as in Theorem 4.20.
5.1 Hamilton–Pontryagin principle for ordinary manifolds
We begin with a quick review of the Hamilton–Pontryagin principle for ordinary (non-fibered)
manifolds, as introduced in Yoshimura and Marsden [33].
Let L : TQ → R be a Lagrangian. The Hamilton–Pontryagin action is the functional
S : P(TQ⊕ T ∗Q)→ R defined, in fiber coordinates, by
S(q, v, p) =
∫ 1
0
(
L
(
q(t), v(t)
)
+
〈
p(t), q̇(t)− v(t)
〉)
dt.
Here, (q, v, p) is an arbitrary path in the Pontryagin bundle TQ⊕ T ∗Q. We emphasize that no
restrictions are placed on this path – in particular, the second-order curve condition q̇ = v is not
a priori required.
The path (q, v, p) satisfies the Hamilton–Pontryagin principle if dS(δq, δv, δp) = 0 for all
variations (δq, δv, δp) ∈ T(q,v,p)P(TQ⊕ T ∗Q) such that δq(0) = 0 and δq(1) = 0. (That is, the
endpoints of q are fixed, while the endpoints of v and p are unrestricted.) In local coordinates,
we have
dS(δq, δv, δp) =
∫ 1
0
(
∂L
∂qi
(q, v)δqi +
∂L
∂vi
(q, v)δvi + pi(δq̇
i − δvi) + δpi(q̇
i − vi)
)
dt
=
∫ 1
0
[(
∂L
∂qi
(q, v)− ṗi
)
δqi +
(
∂L
∂vi
(q, v)− pi
)
δvi + δpi(q̇
i − vi)
]
dt.
Lagrangian Mechanics and Reduction on Fibered Manifolds 21
Hence, this vanishes when (q, v, p) satisfies the differential-algebraic equations
∂L
∂qi
(q, v)− ṗi = 0,
∂L
∂vi
(q, v)− pi = 0, q̇i − vi = 0,
which Yoshimura and Marsden [33] call the implicit Euler–Lagrange equations. The three systems
of equations correspond, respectively, to the Euler–Lagrange equations, the Legendre transform,
and the second-order curve condition. (Note that the conjugate momentum p acts like a “Lagrange
multiplier” enforcing the second-order curve condition q̇ = v.)
In this sense, the Hamilton–Pontryagin approach generalizes and unifies the symplectic and
variational approaches to Lagrangian mechanics.
5.2 Hamilton–Pontryagin for fibered manifolds
Suppose, more generally, that L : V Q→ R is a Lagrangian on the vertical bundle of a fibered
manifold Q→M . Recall that V Q and V ∗Q can both be viewed as fibered manifolds over M ,
and thus so can V Q⊕ V ∗Q, which we call the vertical Pontryagin bundle. It follows that we may
define a Banach manifold of vertical paths PV (V Q⊕ V ∗Q) and its bundle of vertical variations
V PV (V Q⊕ V ∗Q).
Definition 5.1. Given a Lagrangian L : V Q→ R, the Hamilton–Pontryagin action S : PV (V Q⊕
V ∗Q)→ R is defined, in fiber coordinates, by
S(q, v, p) =
∫ 1
0
(
L
(
q(t), v(t)
)
+
〈
p(t), q̇(t)− v(t)
〉)
dt.
A vertical path (q, v, p) ∈ PV (V Q⊕ V ∗Q) is said to satisfy the Hamilton–Pontryagin principle
if dS(δq, δv, δp) = 0 for all vertical variations (δq, δv, δp) ∈ V(q,v,p)PV (V Q ⊕ V ∗Q) such that
δq(0) = 0 and δq(1) = 0.
Theorem 5.2. A vertical path (q, v, p) ∈ PV (V Q ⊕ V ∗Q) satisfies the Hamilton–Pontryagin
principle if and only if, in fiber-adapted local coordinates q = (xσ, yi), it satisfies the implicit
vertical Euler–Lagrange equations,
ẋσ = 0, ṗi =
∂L
∂yi
(q, v), pi =
∂L
∂vi
(q, v), ẏi = vi. (5.1)
Proof. The equations ẋσ = 0 are simply the vertical path condition. Given a vertical variation
(δq, δv, δp) ∈ V(q,v,p)PV (V Q⊕ V ∗Q) satisfying δq(0) = 0 and δq(1) = 0,
dS(δq, δv, δp) =
∫ 1
0
(
∂L
∂yi
(q, v)δyi +
∂L
∂vi
(q, v)δvi + pi(δẏ
i − δvi) + δpi(ẏ
i − vi)
)
dt
=
∫ 1
0
[(
∂L
∂yi
(q, v)− ṗi
)
δyi +
(
∂L
∂vi
(q, v)− pi
)
δvi + δpi(ẏ
i − vi)
]
dt.
This vanishes for arbitrary (δq, δv, δp) if and only if each of the components in the integrand
vanishes, which completes the proof. �
5.3 Hamilton–Pontryagin for arbitrary Lie algebroids
We next generalize the Hamilton–Pontryagin principle to a Lagrangian L : A→ R, where A→ Q
is an arbitrary Lie algebroid. The previous subsections will then correspond to the special cases
A = TQ and A = V Q, respectively.
One might expect that the appropriate generalization of paths in TQ⊕ T ∗Q or V Q⊕ V ∗Q
would be paths in A ⊕ A∗. However, these generally do not contain sufficient information to
recover the A-path condition (the generalization of the second-order curve condition). Instead,
we consider an alternative class of paths that we call (A,A∗)-paths.
22 S. Li, A. Stern and X. Tang
Definition 5.3. An (A,A∗)-path consists of the following components:
(i) an A-path a ∈ Pρ(A) over some base path q ∈ P(Q);
(ii) a path v ∈ P(A), not necessarily an A-path, over q;
(iii) a path p ∈ P(A∗) over q.
We denote this by (a, v, p) ∈ P(A,A∗).
Example 5.4. Any path (q, v, p) ∈ P(TQ⊕ T ∗Q) can be identified with the (TQ, T ∗Q)-path
(q̇, v, p) ∈ P(TQ, T ∗Q). More generally, (q, v, p) ∈ PV (V Q ⊕ V ∗Q) can be identified with
(q̇, v, p) ∈ P(V Q, V ∗Q). Thus, P(V Q, V ∗Q) ∼= PV (V Q⊕ V ∗Q).
In this special case, the base path has a unique A-path prolongation, so it suffices to consider
paths in A⊕A∗ – but this is not the case in general.
Example 5.5. Let g be a Lie algebra. Since all paths in g → • are g-paths, it follows that
a (g, g∗) path (a, v, p) ∈ P(g, g∗) consists of two (generally distinct) paths a, v ∈ P(g) and a path
p ∈ P(g∗). Thus, P(g, g∗) ∼= P(g⊕ g⊕ g∗).
Definition 5.6. An admissible variation of (a, v, p) ∈ P(A,A∗) consists of an admissible variation
Xb,a ∈ Fa(A) of the A-path a, together with arbitrary variations δv ∈ TvP(A) and δp ∈ TpP(A∗),
such that all agree on the horizontal component δq = ρ(b) ∈ Pq(Q). That is, if τ : A→ Q and
π : A∗ → Q are the bundle projections, we require τ∗(v) = π∗(p) = ρ(b). Following Remark 3.10,
we denote this subbundle of admissible variations by F(A,A∗) ⊂ TP(A,A∗).
Remark 5.7. Given a TQ-connection ∇, the admissible variation (Xb,a, δv, δp) ∈ F(a,v,p)(A,A
∗)
has components Xver
b,a = ∇ab and Xhor
b,a = δvhor = δphor = ρ(b), while δvver and δpver are arbitrary
paths in A and A∗, respectively.
Example 5.8. Continuing Example 5.5, let us consider the case where g is a Lie algebra. Given
a (g, g∗) path (a, v, p), we identify a with its time-dependent section ξ(t) = ξ(t, •) = a(t). Then an
admissible variation of (a, v, p) has the form (ξ̇ + [ξ, η], δv, δp), where η, δv, and δp are arbitrary
paths in g.
Equivalently, assuming g is the Lie algebra of a Lie group G, let g ∈ P(G) be a path
integrating ξ, i.e., g(t) = g(0) exp(tξ), so that ξ = (g−1)∗ġ. It follows that arbitrary variations
(δg, δv, δp) ∈ T(g,v,p) ∈ P(G×g×g∗) correspond precisely to admissible variations (ξ̇+[ξ, η], δv, δp)
of (ξ, v, p) ∈ P(g, g∗), where η = (g−1)∗δg. This special case corresponds to the approach of
Yoshimura and Marsden [34] and Bou-Rabee and Marsden [2] for Hamilton–Pontryagin mechanics
on Lie algebras, where one considers paths in P(G× g× g∗) and then left-reduces by G.
Definition 5.9. Given a Lagrangian L : A→ R, the Hamilton–Pontryagin action S : P(A,A∗)→
R is defined by
S(a, v, p) =
∫ 1
0
(
L
(
v(t)
)
+
〈
p(t), a(t)− v(t)
〉)
dt,
and (a, v, p) ∈ P(A,A∗) is said to satisfy the Hamilton–Pontryagin principle if dS(Xb,a, δv, δp) = 0
for all admissible variations (Xb,a, δv, δp) ∈ F(a,v,p)(A,A
∗).
Theorem 5.10. An (A,A∗)-path (a, v, p) ∈ P(A,A∗) satisfies the Hamilton–Pontryagin principle
if and only if, given a TQ-connection ∇ on A, it satisfies the differential-algebraic equations,
ρ∗dLhor(v) +∇∗ap = 0, dLver(v)− p = 0, a− v = 0. (5.2)
Lagrangian Mechanics and Reduction on Fibered Manifolds 23
Proof. Given (Xb,a, δv, δp) ∈ F(a,v,p)(A,A
∗), we compute
dS(Xb,a, δv, δp) =
∫ 1
0
(〈
dLhor(v), ρ(b)
〉
+
〈
dLver(v), δvver
〉
+ 〈p,∇ab− δvver〉+ 〈δpver, a− v〉
)
dt
=
∫ 1
0
(〈
ρ∗dLhor(v) +∇∗ap, b
〉
+
〈
dLver(v)− p, δvver
〉
+ 〈δpver, a− v〉
)
dt.
The Hamilton–Pontryagin principle is satisfied if and only if each term in the integrand vanishes,
and since b, δvver, and δpver are arbitrary, the result follows. �
We call the differential-algebraic equations (5.2) the implicit Euler–Lagrange–Poincaré equa-
tions. As we did in Theorem 3.14 we can give an equivalent expression for (5.2) in local
coordinates.
Theorem 5.11. Let qi be local coordinates for Q, {eI} be a local basis of sections of A, {eI} be
the dual basis of local sections of A∗, ∇ be the locally trivial TQ-connection, and ρiI and CKIJ be
the local-coordinate representations of ρ and [·, ·]. Let (a, v, p) ∈ P(A⊕A⊕A∗) have the local-
coordinate representations a(t) = ξI(t)eI
(
q(t)
)
, v(t) = vI(t)eI
(
q(t)
)
, and p(t) = pI(t)e
I
(
q(t)
)
.
Then (a, v, p) ∈ P(A,A∗) if and only if q̇i = ρiIξ
I , and (a, v, p) satisfies the implicit Euler–
Lagrange–Poincaré equations (5.2) if and only if
ρiI
∂L
∂qi
− CKIJξJpK − ṗI = 0,
∂L
∂ξI
− pI = 0, ξI − vI = 0.
Proof. The proof is a straightforward computation, following Theorem 3.14. �
5.4 Reduction by groupoid symmetries
Finally, we consider the reduction of Hamilton–Pontryagin mechanics by a Lie algebroid morphism
Φ: A→ A′, as in Theorem 3.19. Here, though, we will require the slightly stronger assumption
that Φ be a fiberwise isomorphism. (This was actually assumed in the original Lie algebroid
reduction theorem of Weinstein [30], although Mart́ınez [25] showed that it could be relaxed.)
This stronger assumption is needed since Φ∗ : A′∗ → A∗ points in the “wrong direction” for
reduction from (A,A∗) to (A′, A′∗), so we need fiberwise invertibility to map A∗ → A′∗.
Theorem 5.12. Let Φ: A → A′ be a morphism of Lie algebroids, and suppose L : A → R
and L′ : A′ → R are Lagrangians such that L = L′ ◦ Φ. If Φ is a fiberwise isomorphism, then
(a, v, p) ∈ P(A,A∗) satisfies the Hamilton–Pontryagin principle for L if and only if (a′, v′, p′) ∈
P(A′, A′∗) satisfies the Hamilton–Pontryagin principle for L′, where a′ = Φ ◦ a, v′ = Φ ◦ v, and
p′ = (Φ∗)−1 ◦ p.
Proof. This can be shown directly from the variational principle – observing that admissible
variations in F(a,v,p)(A,A
∗) map to those in F(a′,v′,p′)(A
′, A′∗), and vice versa – but we give
an equivalent proof using the implicit Euler–Lagrange–Poincaré equations together with the
Weinstein–Mart́ınez reduction theorem (Theorem 3.19).
First, since Φ is a fiberwise isomorphism, we have a = v if and only if a′ = v′. Moreover, since
L = L′ ◦ Φ, the following diagram commutes:
A A′
A∗ A′∗.
Φ
∼=
dLver dL′ver
Φ∗
∼=
24 S. Li, A. Stern and X. Tang
It follows from this that p = dLver(v) if and only if p′ = dL′ver(v′). Finally, substituting these
expressions for v and p into the first equation in (5.2), we have
ρ∗dLhor(a) +∇∗adLver(a) = 0, ρ′∗dL′hor(a′) +∇
′∗
a′dL
′ver(a′) = 0.
But these are just the Euler–Lagrange–Poincaré equations (3.1) for L and L′, respectively. So
Theorem 3.19 implies that one holds if and only if the other does. �
Fortunately, the fiberwise isomorphism assumption is still sufficient to perform reduction
when A = V Q→ Q and A′ = V Q/G→ Q/G, since the quotient map for the groupoid action
in Lemma 4.19 is a fiberwise isomorphism. (Indeed, Higgins and Mackenzie [11] refer to Lie
algebroid morphisms with this property as action morphisms.) Intuitively, this is because the
quotient is taken both on the total space and on the base, so the dimension of the fibers remains
the same.
Theorem 5.13. Let G ⇒ M be a Lie groupoid and Q → M a principal G-space. Suppose
the Lagrangian L : V Q → R is G-invariant, i.e., that it factors through the quotient mor-
phism Φ: V Q → V Q/G as L = ` ◦ Φ, where ` : V Q/G → R is called the reduced Lagrangian.
Then (a, v, p) ∈ P(V Q, V ∗Q) satisfies the Hamilton–Pontryagin principle for L if and only if
(a′, v′, p′) ∈ P(V Q/G, V ∗Q/G) satisfies the Hamilton–Pontryagin principle for `, where a′ = Φ◦a,
v′ = Φ ◦ v, and p′ = (Φ∗)−1 ◦ p.
Proof. Apply Theorem 5.12 to the quotient morphism Φ, which is a fiberwise-isomorphic Lie
algebroid morphism from V Q to V Q/G. �
Example 5.14. As in Example 4.21, when G ⇒ • is a Lie group acting freely and properly
on Q → •, this corresponds to the case of ordinary Lagrangian reduction for the Hamilton–
Pontryagin principle. In the special case where Q = G acts on itself by multiplication, this
gives Euler–Poincaré-type reduction for the Hamilton–Pontryagin principle, as in Yoshimura and
Marsden [34], Bou-Rabee and Marsden [2].
Example 5.15. As in Example 4.22, suppose G ⇒ M is a Lie groupoid acting on itself by
multiplication, so that the quotient morphism is Φ: V βG→ V βG/G = AG. If L : V βG→ R and
` : AG→ R are Lagrangians satisfying L = ` ◦ Φ, then Theorem 5.13 implies that the implicit
vertical Euler–Lagrange equations (5.1) on V βG reduce to the implicit Euler–Lagrange–Poincaré
equations (5.2) on AG. The even more special case where G ⇒ • is a Lie group again gives
Hamilton–Pontryagin reduction from G to g, as in Yoshimura and Marsden [34], Bou-Rabee and
Marsden [2].
Acknowledgments and disclosures
The authors wish to thank Rui Loja Fernandes for his helpful feedback on this work. This paper
also benefited substantially from the suggestions of the anonymous referees, to whom we wish to
express our sincere gratitude. This research was supported in part by grants from the Simons
Foundation (award 279968 to Ari Stern) and from the National Science Foundation (award
DMS 1363250 to Xiang Tang). The authors declare that they have no conflict of interest.
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1 Introduction
2 Lagrangian mechanics on fibered manifolds
2.1 Brief review of Lagrangian mechanics
2.2 Fibered manifolds
2.3 Lagrangian mechanics on fibered manifolds
2.4 Application: classical Routh reduction as fibered mechanics
3 Lagrangian mechanics on Lie algebroids
3.1 Lie algebroids and A-paths
3.2 Connections and variations of A-paths
3.3 Lagrangian mechanics
3.4 Special case: the Lagrange–Poincaré equations
3.5 Fibered manifolds revisited
3.6 Lie algebroid morphisms and reduction
4 Lie groupoid symmetries and reduction on fibered manifolds
4.1 Lie groupoids
4.2 Lie algebroid of a Lie groupoid
4.3 Reduction by a groupoid action
5 The Hamilton–Pontryagin principle and reduction
5.1 Hamilton–Pontryagin principle for ordinary manifolds
5.2 Hamilton–Pontryagin for fibered manifolds
5.3 Hamilton–Pontryagin for arbitrary Lie algebroids
5.4 Reduction by groupoid symmetries
References
|
| id | nasplib_isofts_kiev_ua-123456789-148564 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2025-11-24T11:04:38Z |
| publishDate | 2017 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Li, S. Stern, A. Tang, X. 2019-02-18T15:53:08Z 2019-02-18T15:53:08Z 2017 Lagrangian Mechanics and Reduction on Fibered Manifolds / S. Li, A. Stern, X. Tang // Symmetry, Integrability and Geometry: Methods and Applications. — 2017. — Т. 13. — Бібліогр.: 34 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 70G45; 53D17; 37J15 DOI:10.3842/SIGMA.2017.019 https://nasplib.isofts.kiev.ua/handle/123456789/148564 This paper develops a generalized formulation of Lagrangian mechanics on fibered manifolds, together with a reduction theory for symmetries corresponding to Lie groupoid actions. As special cases, this theory includes not only Lagrangian reduction (including reduction by stages) for Lie group actions, but also classical Routh reduction, which we show is naturally posed in this fibered setting. Along the way, we also develop some new results for Lagrangian mechanics on Lie algebroids, most notably a new, coordinate-free formulation of the equations of motion. Finally, we extend the foregoing to include fibered and Lie algebroid generalizations of the Hamilton-Pontryagin principle of Yoshimura and Marsden, along with the associated reduction theory. The authors wish to thank Rui Loja Fernandes for his helpful feedback on this work. This paper also benefited substantially from the suggestions of the anonymous referees, to whom we wish to express our sincere gratitude. This research was supported in part by grants from the Simons Foundation (award 279968 to Ari Stern) and from the National Science Foundation (award DMS 1363250 to Xiang Tang). The authors declare that they have no conflict of interest. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Lagrangian Mechanics and Reduction on Fibered Manifolds Article published earlier |
| spellingShingle | Lagrangian Mechanics and Reduction on Fibered Manifolds Li, S. Stern, A. Tang, X. |
| title | Lagrangian Mechanics and Reduction on Fibered Manifolds |
| title_full | Lagrangian Mechanics and Reduction on Fibered Manifolds |
| title_fullStr | Lagrangian Mechanics and Reduction on Fibered Manifolds |
| title_full_unstemmed | Lagrangian Mechanics and Reduction on Fibered Manifolds |
| title_short | Lagrangian Mechanics and Reduction on Fibered Manifolds |
| title_sort | lagrangian mechanics and reduction on fibered manifolds |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/148564 |
| work_keys_str_mv | AT lis lagrangianmechanicsandreductiononfiberedmanifolds AT sterna lagrangianmechanicsandreductiononfiberedmanifolds AT tangx lagrangianmechanicsandreductiononfiberedmanifolds |