Swept volume dinamical systems and their kinetic models
We study swept-volume dynamical systems for which several hydrodynamical models are formulated. The properties of those hydrodynamical models are studied by means of the algebraic Kostant – Symes technique. A differential-geometric description of swept volumes dynamical systems are devised based on...
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nasplib_isofts_kiev_ua-123456789-1769522025-02-09T14:06:25Z Swept volume dinamical systems and their kinetic models Динамічні системи, породжені орбітами точок деякої частини фазового простору, та їхні кінетичні моделі Динамические системы, порожденные орбитами точек некоторой части фазового пространства, и их кинетические модели Bogoliubov, N.N. Prykarpatsky, A.K. Blackmore, D. We study swept-volume dynamical systems for which several hydrodynamical models are formulated. The properties of those hydrodynamical models are studied by means of the algebraic Kostant – Symes technique. A differential-geometric description of swept volumes dynamical systems are devised based on the Cartan Movina frame approach. Для динамiчних систем, породжених орбiтами точок деякої частини фазового простору, наведено декiлька гiдродинамiчних моделей, властивостi яких дослiджуються за допомогою методу Костанта – Сiмза. В рамках пiдходу Картана дано диференцiально-геометричний опис вказаних систем. 1999 Article Swept volume dinamical systems and their kinetic models / N.N. Bogoliubov, A.K. Prykarpatsky, D. Blackmore // Нелінійні коливання. — 1999. — Т. 2, № 3. — С. 291-305. — Бібліогр.: 14 назв. — англ. 1562-3076 https://nasplib.isofts.kiev.ua/handle/123456789/176952 517.9 en Нелінійні коливання application/pdf Інститут математики НАН України |
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We study swept-volume dynamical systems for which several hydrodynamical models are formulated. The properties of those hydrodynamical models are studied by means of the algebraic Kostant – Symes technique. A differential-geometric description of swept volumes dynamical systems are devised based on the Cartan Movina frame approach. |
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Bogoliubov, N.N. Prykarpatsky, A.K. Blackmore, D. |
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Bogoliubov, N.N. Prykarpatsky, A.K. Blackmore, D. Swept volume dinamical systems and their kinetic models Нелінійні коливання |
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Bogoliubov, N.N. Prykarpatsky, A.K. Blackmore, D. |
| author_sort |
Bogoliubov, N.N. |
| title |
Swept volume dinamical systems and their kinetic models |
| title_short |
Swept volume dinamical systems and their kinetic models |
| title_full |
Swept volume dinamical systems and their kinetic models |
| title_fullStr |
Swept volume dinamical systems and their kinetic models |
| title_full_unstemmed |
Swept volume dinamical systems and their kinetic models |
| title_sort |
swept volume dinamical systems and their kinetic models |
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Інститут математики НАН України |
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1999 |
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https://nasplib.isofts.kiev.ua/handle/123456789/176952 |
| citation_txt |
Swept volume dinamical systems and their kinetic models / N.N. Bogoliubov, A.K. Prykarpatsky, D. Blackmore // Нелінійні коливання. — 1999. — Т. 2, № 3. — С. 291-305. — Бібліогр.: 14 назв. — англ. |
| series |
Нелінійні коливання |
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2025-11-26T15:45:26Z |
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2025-11-26T15:45:26Z |
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| fulltext |
т. 2 •№ 3 • 1999
UDC 517 . 9
SWEPT VOLUME DINAMICAL SYSTEMS
AND THEIR KINETIC MODELS
ДИНАМIЧНI СИСТЕМИ, ПОРОДЖЕНI ОРБIТАМИ
ТОЧОК ДЕЯКОЇ ЧАСТИНИ ФАЗОВОГО ПРОСТОРУ,
ТА ЇХНI КIНЕТИЧНI МОДЕЛI
N.N. Bogoliubov (jr.)
Steklov Math. Inst. Russ. Acad. Sci.,
Russia, 117966, Moscow, Vavilova str., 42
e-mail: nickolai@bogol.mian.su
A.K. Prykarpatsky
Mining and Metallurgical Academy (AGH),
Poland, 30-059, Krakow, Al. Mickiewicza, 30
e-mail: PRIKA@mat.agh.edu.pl
and Inst. Appl. Probl. Mech. and Math. Nat. Acad. Sci. Ukraine,
Ukraine, 290601, Lviv, Naukova str., 1b
D. Blackmore
New Jersey Inst. Technology,
Newark, NJ 07102–1982 USA
e-mail: deblac@chaos.njit.edu
and Courant Inst. Math. New York Univ.,
New York, USA
We study swept-volume dynamical systems for which several hydrodynamical models are formulated.
The properties of those hydrodynamical models are studied by means of the algebraic Kostant – Symes
technique. A differential-geometric description of swept volumes dynamical systems are devised based on
the Cartan Movina frame approach.
Для динамiчних систем, породжених орбiтами точок деякої частини фазового простору, на-
ведено декiлька гiдродинамiчних моделей, властивостi яких дослiджуються за допомогою ме-
тоду Костанта – Сiмза. В рамках пiдходу Картана дано диференцiально-геометричний опис
вказаних систем.
1. It is well known [1, 2] that motion planning, numerically controlled machining and robotics
are just a few of the many areas of manufacturing automation in which the analysis and
representation of swept volumes plays a crucial role. Swept volume modeling is also an
important part of task-oriented robot motion planning. In these problems a robot carrying some
objects is moved through a domain in space containing obstacles (phase space constraints) for
the purpose of reaching a desired goal. A typical motion planning problem consists of moving a
collection of solid objects around obstacles from an initial to a final position. This may include,
in particular, solving collision detection problems and obtaining optimal solution paths.
c© N.N. Bogoliubov (jr.), A.K. Prykarpatsky, D. Blackmore, 1999 291
When an object undergoes a rigid motion, the totality of points through which it passes
constitutes a region in space called the swept volume generated by the (rigid) sweep. In order
to have a mathematical framework for these sweeps in real 3-space R3, we introduce some
definitions that will prove useful in the sequel.
An Euclidean motion in (n+1)-dimensional Euclidean spaceEn+1 is a mapping σ : En+1 →
→ En+1 such that ‖σ(x)− σ(y)‖ = ‖x− y‖ for all x, y ∈ En+1, where ‖·‖ is the standard
Euclidean norm in En+1. The simplest cases of these sweeps or motions are translations and
rotations of the form
α(x) := x+ a, β(x) := gx, (1)
where x, a ∈ En+1 (= Rn+1) and g ∈ O(n + 1). The following result characterizes Euclidean
motions.
Theorem 1. Let σ be an Euclidean motion in En+1. Then there exist a unique orthogonal
mapping β ∈ O(n+ 1) and a unique translation α such that σ = α ◦ β.
Sketch of Proof. Let a := σ(0) and α be as in (1). Define β := α−1 ◦ σ. It is easy to show
that β is orthogonal. Indeed, β is a motion since
‖β(x)− β(y)‖ = ‖σ(x)− σ(y)‖ = ‖x− y‖ , (2)
and ‖β‖ = 1 because
‖β‖ = ‖β(x)− β(0)‖ = ‖x− 0‖ = ‖x‖
for all x, y ∈ En+1. We need to show that β is linear. Note first that the standard inner product
〈·, ·〉 in En+1 is preserved by the mapping β : En+1 → En+1, viz.
〈β(x), β(y)〉 =
1
2
(
‖β(x)‖2 + ‖β(y)‖2 − ‖β(x)− β(y)‖2
)
=
=
1
2
(
‖x‖2 + ‖y‖2 − ‖x− y‖2
)
= 〈x, y〉 (3)
for all x, y ∈ En+1. To prove the linearity of β, it suffices to show that
β(k1x+ k2y) = k1β(x) + k2β(y), (4)
or equivalently that the form
E(x, y) := β(k1x+ k2y)− k1β(x)− k2β(y) ≡ 0 (5)
for all x, y ∈ En+1, k1, k2 ∈ R. To prove (5) it is necessary that E(x, y) in (5) be orthogonal to
each vector of some basis in En+1. If {e1, ..., en+1} is an orthonormal basis for En+1, obviously
so is {β(e1), ..., β(en+1)} since β preserves the scalar product in view of (3). Therefore we find
that for all 1 ≤ j ≤ n+ 1,
292 ISSN 1562-3076. Нелiнiйнi коливання, 1999, т. 2, № 3
〈E(x, y), β(ej)〉 =
= 〈β(k1x+ k2y), β(ej)〉 − k1 〈β(x), ej〉 − k2 〈β(x), β(ej)〉 =
= k1 〈x, ej〉+ k2 〈y, ej〉 − k1 〈x, ej〉 − k2 〈x, ej〉 = 0. (6)
This proves the linearity of β, and the uniqueness of α and β are easily verified.
The following result is a direct consequence of the above theorem:
Corollary 1. Let σ be an Euclidean motion in En+1. Then:
(i) σ is a smooth mapping;
(ii) σ maps En+1 orthogonally onto itself; and
(iii) 〈σ′(x), σ′(y)〉 = 〈x, y〉 for all x, y ∈ En+1
p
∼= Tp(En+1), p ∈ En+1.
Indeed, for each point (p, x) ∈ En+1
p , p ∈ En+1 we have
σ′(p)x =
d
dt
σ(p+ tx) |t=0=
d
dt
α ◦ β(p+ tx) |t=0=
=
d
dt
α(β(p) + tβ(x)) |t=0=
d
dt
(β(p) + tβ(x) + σ(0)) |t=0= β(x). (7)
Whence for (p, x), (p, y) ∈ En+1
p , we obtain〈
σ′(p)x, σ′(p)y
〉
= 〈β(x), β(y)〉 = 〈x, y〉 , (8)
from which the required properties follow.
2. Let us consider a simply-connected solid body V embedded in E3 having boundary
surface S = ∂V parametrized as follows:
S =
⋃
τ∈[0,h]
{
x(s, τ) ∈ R3 : x(s+ 2π, τ) = x(s, τ), s ∈ R
2πZ
}
. (9)
Here τ ∈ R
2πZ
is the usual parametrization via the identity 〈dx, dx〉 = ds2 of curve x(·, τ), τ ∈
∈ [0, h], obtained by cutting V straight across its diameter through a point τ ∈ [0, h]. This means
that the diameter of V is parametrized by τ and the set of curves {x(·, τ) : τ ∈ [0, h]} covers S
in a unique fashion.
To proceed further in our description of the motion of a solid body V in E3, we reformulate
it in terms of manifolds swept out by the surface S in a time interval [0, t0]. We denote these
surfaces by St0(τ); they can be represented by
St0(τ) :=
⋃
t∈[0,t0]
{x(t, τ)} . (10)
Therefore, the swept volume manifold St0(V ) of the solid body defined in [2] can be written as
St0(V ) =
⋃
τ∈[0,h]
St0(τ). (11)
ISSN 1562-3076. Нелiнiйнi коливання, 1999, т. 2, № 3 293
It is obvious that St0(V ) is equal to the compact three-dimensional submanifold with boundary
comprised of points swept by the set of curves {x(·, τ)} ⊂ S corresponding to the diameter
points τ ∈ [0, h]. This leads naturally to the problem of constructing special dynamical systems
– called swept volume dynamical systems – intimately associated with the Euclidean motion
of a solid body in space and studying their differential-geometric and differential-topological
properties [1] which are useful for applications in manufacturing automation.
Let us assume that a sweep of a solid body V in 3-space is generated by a family of Eucli-
dean motions σ(t), t ∈ [0, t0], giving rise to a swept volume such that each of the curves x(·, τ)
maintains its planarity and arc-length for all t ∈ [0, t0].This means, in particular, that the Gaussi-
an curvature of each curve x(·, τ), τ ∈ [0, h], is time independent while its torsion ξ is zero for
all t ∈ [0, h]. The invariance of planarity of the family of Euclidean motions σ(t), t ∈ [0, t0],
completely characterizes the motion of V in E3. To the above properties one needs only to add
length invariance, which can be expressed as〈
dx
ds
,K ′(t, x)
dx
ds
〉
= 0 (12)
for all t ∈ [0, t0]. Here we have postulated the evolution of curves x(·, τ, t) due to the Euclidean
motion as follows:
dx
dt
= K(t, x), (13)
whereK(t, ·) : E3 → T (E3) is a parametric family of vector fields onE3.Vector fields satisfying
(12) are called Killing vector fields. A rather complete description of such fields associated with
infinitesimal rigid body motions can be found in [3, 4]. Our next goal is to obtain an analogous
theory for swept volumes using modern differential-geometric and algebraic-topological tools.
Now suppose we are given a swept volume manifold St0(V ) as defined in (11). Over this
manifold we can define a connection Γ [5] together with the following representation of parallel
transport with respect to local spatial parameters τ ∈ [0, h], s ∈ R
2πZ
, and the temporal
parameter t ∈ [0, t0] via the covariant derivative
∇ ∂
∂yj
f :=
∂f
∂yj
+ Γjf, (14)
where y = (y1, y2, y3)T := (s, τ, t)T ∈ St0(V ) and Γj , 1 ≤ j ≤ 3, are Christoffel matrices acting
in the adjoint vector bundle over the swept volume manifold. The Christoffel matrices can be
determined uniquely by requiring that ∇ ∂
∂yj
f∗ = 0 for all horizontal [4, 5] vector fields f∗ ∈
∈ T (St0(V )). This means that parallel transport along the manifold St0(V ) must be generated
by some Euclidean motion σ(t) : E3 → E3, t ∈ [0, t0]. We first consider Cartan’s main structure
equations in the differential-geometric setting [6]:
dθ = −1
2
[ω, θ] + Θ, dω = −1
2
[ω, ω] + Ω, (15)
where ω : P (St0(V );GL(3; R))→ gl(3; R) is the connection form,
θ : P (St0(V );GL(3; R))→ E3
294 ISSN 1562-3076. Нелiнiйнi коливання, 1999, т. 2, № 3
is the canonical affine form, Ω is the corresponding curvature form and Θ is the torsion form,
all on the principal fiber bundle P (St0(V );GL(3; R)) of frames over the manifold St0(V ). In
component form the structure equations (15) are as follows:
dθj = −ωij ∧ θj + Θj , dωi = −ωik ∧ ωkj + Ωi
j , (16)
where
θ =
3∑
i=1
θiei, ω =
3∑
i,j=1
ωijA
j
i , Θ =
3∑
i=1
Θiei, Ω =
3∑
i,j=1
Ωi
jA
j
i , {ei : 1 ≤ i ≤ 3}
is a basis for E3 and {Aij ∈ gl(3; R) : 1 ≤ i, j ≤ 3} are in the Lie algebra gl(3; R) and satisfy the
condition Aijek = ejδ
i
k for all i, j, k = 1, 2, 3. The structure equations (16) completely describe
the swept volume dynamical system generated by a rigid sweep σ(t) : E3 → E3 of a solid body
in 3-space. The motion σ(t) considered as an affine motion in R3 must satisfy the main defining
conditions on the canonical and connection one-forms:
Ra∗ω = Ada−1ω, Ra∗θ = a−1θ
for all a ∈ GL(3; R). These conditions are obviously satisfied if the following canonical condi-
tions hold:
θ = X−1dy, ω = X−1 (dX + 〈dy,Γ(y)〉X) , (17)
where y = (s, τ, t)T ∈ St0(V ) and X = [X1, X2, X3] ∈ GL(3; R) is an arbitrary basis for the
tangent space Ty(St0). To determine the Riemannian connection matrix Γ(y), y ∈ St0 , we write
the first fundamental form as follows:
dl2 := 〈dx, dx〉 =
3∑
i,j=1
gij(y)dyidyj . (18)
If the embedding x : St0(V )→ E3 is generated by a sweep, the above Riemannian connection
matrices must have the form
Γjk,i =
1
2
3∑
s=1
gis
(
∂gis
∂yk
+
∂gsk
∂yi
− ∂gki
∂ys
)
, (19)
where
3∑
s=1
gisgsj = δji , i, j, k = 1, 2, 3; and as is well known [6], element (19) is uniquely determi-
ned by the condition ∇ ∂
∂yj
gks = 0 for all 1 ≤ j, k, s ≤ 3. From (18) we see that the above
swept volume manifold (with boundary) is not Euclidean but actually a Riemannian space with
nontrivial curvature and torsion. Thus we have arrived at the following important classification
problem: to describe an effective differential-geometric procedure for determining the mani-
folds with boundary that are generated by a Euclidean sweep of a solid object. We can simplify
this problem by employing the geometric theory of Cartan. For the case under consideration
we assume that a Lie group G acts on St0(V ) in a manner described by
dyj +
n∑
i=1
ξji (y)ω̄i(a, da) = 0, (20)
ISSN 1562-3076. Нелiнiйнi коливання, 1999, т. 2, № 3 295
where y ∈ St0(V ), ω̄i(a, da), 1 ≤ i ≤ n = dimG, are the Maurer – Cartan left-invariant forms
of the Lie group G, a ∈ G is an arbitrary element and the ξij(y) are characteristic functions on
the manifold St0(V ). The following theorem of Cartan is useful in describing a geometric object
that is invariant with respect to the group action G× St0(V )→ St0(V ).
Theorem 2. The differential system (20), with characteristic left-invariant one-forms ω̄i(a, da),
1 ≤ i ≤ n = dimG, on the Lie group G, is tantamount to invariance of the group action on
St0(V ) if and only if the following conditions hold:
(i) The coefficients ξij are analytic functions of y ∈ St0(V ).
(ii) The system (20) is completely integrable in the Frobenius – Cartan sense.
We intend to investigate the above geometric aspects of the problem more thoroughly in
a paper that is now in preparation. Here we are just going to formulate several hydrodynamic
models for swept volume dynamical systems in E3 having very rich symmetry groups and dis-
cuss some of their interesting and useful properties.
3. If St0(V ) is a swept volume manifold generated by a Euclidean motion σ(t), t ∈ [0, t0],
then there exists a set of tangent vector fields
ds
dα
= u(s, τ, t),
dτ
dα
= v(s, τ, t),
dt
dα
= w(s, τ, t), (21)
where (s, τ, t) ∈ St0(V ), α ∈ R is an evolution parameter and (u, v, w) is a smooth vector field
on St0(V ). To more effectively describe the vector field (21), let us assume that the surface
S = ∂V of the solid body satisfies the invariant equation
γ̄(x̄) = 0, (22)
where γ̄ : R3 → R is a smooth function. This means that (22) is identically satisfied for the
parametrized surface x̄ = x̄(s, τ) for all τ ∈ [0, h] and s ∈ R
2πZ
:
γ̄ (x̄(s, τ)) ≡ 0. (23)
When t 6= 0, Theorem 1 implies that the Euclidean motion can be written in the form
x(s, τ, t) = ξ(t) + a(t)x̄(s, τ) (24)
for all (s, τ, t)T ∈ St0(V ).
Solving (24) for x̄ and substituting this in (23), we find that
γ̄ (a∗(t)(x− ξ(t)) := γ(t, x) = 0 (25)
for t ∈ [0, h] and x ∈ St0(V ). Recalling now the general form (21) of a vector field on the
manifold St0(V ), it follows directly from (25) that
w
∂γ
∂t
+
〈
gradγ, x′ · (u, v, w)T
〉
= 0, (26)
where the prime denotes the Fréchet derivative of the mapping (24). Thus (26) gives a necessary
condition for the set of vector fields (21) to belong to T (St0(V )) .
296 ISSN 1562-3076. Нелiнiйнi коливання, 1999, т. 2, № 3
If (21) satisfies (26) for the parameter α := t ∈ [0, t0], then we obtain
ds
dt
= u(s, τ, t),
dτ
dt
= v(s, τ, t), (27)
w(s, τ, t) ≡ 1 for all (s, τ, t)T ∈ St0(V ). Using once again the representation (24), a simple but
tedious calculation shows that equations (27) admit the prolongation
Du
Dt
= −∂p
∂s
,
Dv
Dt
= −∂p
∂τ
,
(28)
where
D
Dt
is the Eulerian total or material derivative [7] and p(s, τ, t) := q(s, t) + r(τ, t).
Moreover, if we use the equation
Dv
Dt
= −∂r
∂τ
, the dynamical system (28) takes the form of
the well known Navier – Stokes equations for the virtual flow of an ideal incompressible two-
dimensional fluid under external pressure p(s, τ), namely
Du
Dt
= −ρ−1∂p
∂s
,
Dv
Dt
= −ρ−1 ∂p
∂τ
,
Dp
Dt
= −ρ
(
∂u
∂s
+
∂v
∂τ
)
,
(29)
where (u, v)T is the velocity vector in the s, τ -plane and ρ > 0 is the density of the liquid.
As the liquid is incompressible, i.e.,
Dρ
Dt
=
∂ρ
∂t
+
u∂ρ
∂s
+
v∂ρ
∂τ
≡ 0, we easily establish the
condition
∂u
∂s
+
∂v
∂τ
= 0 for all s ∈ R
2πZ
and τ ∈ [0, h]. We may assume that the density is
normalized to unity, i.e., ρ ≡ 1.
Consider the Navier – Stokes equations (29) with a free surface given by the equation τ =
= h(s, t), where h(·, t), t ∈ [0, t0], is the height of the fluid above the bottom (the s-axis) at time
t. Then (29) reduces to the system
du
dt
dq
dt
dh
dt
= K[u, q, h] :=
−uus − uτ
τ∫
0
usdτ − qs
−hs
− ∂
∂s
h∫
0
udτ
, (30)
which is similar to formula (109) in [8] subject to the condition uτ = 0. Here we have also
assumed that
Dh
Dt
= v |τ=h, which ensures that the virtual fluid does not pass through the free
surface τ = h(s, t), and
dq
dt
= −∂h
∂s
, which stems from a wind pressure of unity along the s-axis.
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The system (30) generates a nonlinear integro-differential dynamical system on an infinite-
dimensional functional manifold M(u,q,h) ⊂ C∞(R2; R × R2
+). In order to better understand
this system, we will further investigate its Lax integrability, i.e., we study the existence of an
infinite hierarchy of involutive conservation laws (with respect to some Poisson bracket) and a
special operator representation of Lax type. In addition, we shall show that (30) has a natural
connection with the nonlinear kinetic Boltzmann – Vlasov equation for a one-dimensional
particle flow with a pointwise interaction potential between particles. This property of (30)
enables us to establish a physical analogy between turbulence in kinetic multiparticle systems
connected with stochastization of particle trajectories and instability and shocks in the flow of
an ideal incompressible fluid flowing over a horizontal bottom and having a free boundary.
4. The Boltzmann – Vlasov equation can be obtained from (30) by use of the representation
τ =
u(s,τ)∫
−∞
dpf(s, p; t),
where τ ∈ [0, h] and f ∈ C2(R2; R+) is the Boltzmann distribution function for the kinetics of
a one-dimensional system of particles. The equation of the free boundary in (30) is determined
by the compatibility condition for the distribution function:
h(s) =
u(s,h)∫
−∞
dpf(s, p; t), s ∈ R
2πZ
.
Note also that the above transformation of the dynamical system (30) is canonical, i.e., the
Boltzmann – Vlasov equation obtained is Hamiltonian and has a special symplectic structure
on the functional manifold M(f) ⊂ C2(R2; R+), and the same is true of (30).
In order to better understand the dynamical system (30), we introduce the moment functi-
onals
an(s) :=
h(s)∫
0
dτun(s, τ),
for all s ∈ R
2πZ
, n ∈ Z+.
Then by direct calculation we find that (30) is equivalent to the following infinite-
dimensional system of moment equations on the functional manifold
M(Z+) :=
{
an ∈ C2(R; R) : n ∈ Z+, sup
n
nk |an| <∞, k ∈ Z+
}
:
(31)
dan
dt
dq
dt
= K[a, q] :=
(
−nan−1qx − an+1,x
−a0,x
)
.
298 ISSN 1562-3076. Нелiнiйнi коливання, 1999, т. 2, № 3
We first establish the complete integrability of the dynamical system (31) on M(Z+). For this
purpose we consider the Lie algebra G0 of symbols
l(ξ) :=
∑
j�∞
aj(s)ξ−(j+1),
where {aj(s)} ∈M(Z+), with bracket defined by the formula [8]:
[l1(ξ), l2(ξ)]0 =
∂l1
∂ξ
∂l2
∂x
− ∂l2
∂ξ
∂l1
∂x
.
As shown in [9], the bracket [·, ·]0 is a natural hydrodynamic limit of the standard bracket [·, ·]
on the Lie algebra G (see formula (56) in [8]) of symbols of pseudodifferential operators on R.
The Lie algebraG0 admits a natural direct sum decomposition:G0 = G0+⊕G0−.Moreover,
the identifications G∗0+
∼= G0−, G
∗
0−
∼= G0+ hold for the space G∗0 dual to G0 with respect to the
standard invariant inner product having the form (l1, l2) := Tr (l1 ◦ l2), l1, l2,∈ G0, where
Tr l(ξ) :=
∫
R
ds res l(ξ), l ∈ G0.
Let the gradient τγ(l)G0 for given γ ∈ D(G∗0) and all m ∈ G∗0 is defined as
(∆γ(l),m) :=
d
dε
γ(l + εm)|ε=0
and let R = P+ − P−, where P+, P− are projection operators of G0 on G0+, G0− respectively.
Consider the vector field K on G∗0 defined by
dl(ξ)
dt
= K[l(ξ)] := ad∗R∇γ(l)l(ξ), (32)
which is a coadjoint action of R∇γ(l) ∈ G0 on G∗0 , where γ ∈ D(G∗0) a Casimir functional.
The isomorphism G∗0
∼= G0 obtained from the inner product on G0 implies that (31) can be
represented on G∗0 in the form
dl
dt
= K[l] := [l,R∇γ(l)]. (33)
Let us show that (33) is Hamiltonian with respect to the standard symplectic Lie – Poisson
structure on G∗0. Indeed, the Lie – Poisson bracket
{γ, µ}L(l) := (l, [∇γ(l),∇µ(l)]0)
is defined naturally on G∗0. Then it clearly follows from the properties of the scalar product (·, ·)
that (33) is equivalent to the Hamiltonian system
dl/dt = {γ, l}θ,
where θ := LR+R∗L. Define
l(a(ξ)) =
ξ2
2
+ q +A(ξ), (34)
ISSN 1562-3076. Нелiнiйнi коливання, 1999, т. 2, № 3 299
where
A(ξ) :=
∑
j∈Z+
aj(s)ξ−(j+1) ∈ G0−
is constructed from the values of the moment functions on M(Z+). According to the Kostant
– Symes theorem [10, 11], all of the functionals γj = Tr lj/2 are Casimir and in involution with
respect to the Lie – Poisson bracket {·, ·}θ on G∗0. Consequently, (33) with γ = H = Tr l2 ∈
∈ D(M(Z+)) is a completely integrable Hamiltonian flow on M(Z+)
∼= M(u,q,h), where the Lie
– Poisson bracket is given by
{γ, µ}θ :=
∫
R
ds 〈gradγ, θ(a, q)gradµ〉 . (35)
Here 〈·, ·〉 is the standard scalar product on the space of sequences l2(R) and
θ(a, q) := ‖θmn(a)‖ ⊗ θ(q), m, n ∈ Z+,
θmn(a) := mam+n−1
d
ds
+ n
d
ds
am+n−1,
θ(q) :=
d
ds
.
(36)
With the above we have proved the following result.
Theorem 3. The moment dynamical system (31) on the functional manifold M(Z+) is a
completely Lax integrable Hamiltonian flow with respect to the Lie – Poisson bracket (35), (36);
the Hamiltonian functional is H := Tr l2 and the Lax representation has the form
dl
dt
= [l,R∇H(l)], (37)
where l ∈ G∗0 is given by (34).
We now prove that the above mappings of M(Z+) onto M(u,q,h) and M(f,q) are canonical. In
the first case it is easy to verify that the mapping
an(s) =
h(s)∫
0
dτun(s, τ) ∈M(Z+),
where n ∈ Z+, s ∈
R
2πZ
, transforms the Hamiltonian structure {·, ·}θ(a,q) into {·, ·}θ(u,q,h).
Moreover
d(u, q, h)T
dt
= −θ(u, q, h)gradH = K[u, q, h], (38)
where θ(u, q, h) := antidiag
(
d
ds
,
d
ds
,
d
ds
)
is the canonical structure on M(u,q,h). To prove that
the mapping of M(u,q,h) onto M(f) is canonical, we use the mapping
τ =
u(s,t)∫
−∞
dpf(s, p; t)
300 ISSN 1562-3076. Нелiнiйнi коливання, 1999, т. 2, № 3
assuming that the free surface is given by
h(s, t) =
u(s,h)∫
−∞
dpf(s, p; t), (s, t) ∈ R
2πZ
× [0, T ],
and consider the space F of smooth functions on T ∗(R) with canonical Poisson bracket
{f, g}(s, p) :=
∂g
∂s
∂f
∂p
− ∂f
∂s
∂g
∂p
, (s, p) ∈ R
2πZ
×R,
which transforms it into a Lie algebra. If we introduce Hilbert space structure on F via the
inner product (·, ·), then by the Riesz theorem we can identify F ∗ with F. Note that
(f, g) = (g, f) =
∫
R
ds
∫
R
dpf(s, p)g(s, p)
so that the scalar product is invariant with respect to the Lie – Poisson bracket on F, i.e,
(f, {g, h}) = ({f, g}, h) for all f, g, h ∈ F.
We now consider the gradient mapping ∇ : D(F ∗) → F for a functional γ ∈ D(F ∗) by
(∇γ(f), g) :=
d
dε
γ(f + εg) |ε=0+ . Then∇γ(f) =
δγ
δf
is an ordinary Euler variational derivative
of γ at f ∈ F ∗. Clearly {{γ, µ}} := (f, [∇γ,∇µ]) defines a canonical Hamiltonian structure
on F ∗.
Consider a functional H ∈ D(F ∗) and its gradient ∇H(f) ∈ F. Then the vector field
df
dt
=
= ad∗∇H(f)f on F ∗ is generated by the coadjoint action of the Lie algebra F on F ∗. It follows
from the properties of the inner product on F that this vector field is equivalent to the following
Lax type equation:
df/dt = {f,∇H(f)}, (39)
which coincides with the Hamiltonian equation df/dt = {{H, f}} on the manifold F ∗.
We now make the following identification:
M(Z+) 3 an(s) =
∫
R
dppnf(s, p), s ∈ R
2πZ
,
consistent with the mapping ofM(Z+) intoM(f) introduced above. As a result, the Lie – Poisson
bracket {{·, ·}} on F ∗ transforms into the Lie – Poisson bracket (35) on G∗0, i.e., the mapping of
M(Z+,q) into M(f,q) is canonical.
Using now the Hamiltonian
H =
∫
R
ds(a2 + 2qa0),
we find the kinetic Boltzmann – Vlasov equation for the distribution function f ∈M(f) and the
field function q ∈M(q) :
df
dt
= −pfs + qsfp,
dq
dt
= −
∫
R
dpfs(s, p, t).
(40)
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The consistent system of evolution equations (40) is important and has interesting appli-
cations to the theory of kinetic processes. Consider a system of interacting particles on an axis
R, and assume that its density ρ is a constant (= 1) and the potential of particle interaction has
the form Φ(s− s′), s, s′ ∈ R
2πZ
. Then the distribution function satisfies the kinetic Boltzmann –
Vlasov equation
df
dt
=
2π∫
0
ds′
∫
R
dp′Φ(s− s′)f(s′, p′; t), (41)
provided that there is no multiparticle correlation. Comparing (40) with (41), we see that the
following identification obtains:
q(s, t) =
2π∫
0
ds′
∫
R
dp′Φ(s− s′)f(s′, p′; t), (42)
which can be used to reduce the second equation in (40) to the linear integral equation
2π∫
0
ds′
∫
R
dp′Φ(s− s′)pf(s′, p′)−
∫
dpf(s, p) = const (43)
being valid for all f ∈ M(f) satisfying (40). But owing to the definition of the distribution
function f, (29) and the fact that the density
ρ(s, t) =
∫
R
dpf(s, p; t) ≡ 1
for all t ∈ [0, T ], we obtain that
2π∫
0
ds′
∫
R
dpΦ(s− s′)pf(s′, p)− 1 = const (44)
for all s ∈ R
2πZ
. Since, in general, Φ(s − s′) → 0 as |s− s′| → ∞, it follows from (43) and (44)
that const = 1; hence, for all s ∈ R
2πZ∫
R
dppf(s, p) = 0.
Thus, f(s, p) = f(s,−p) for all (s, p) ∈ R
2πZ
× R, and the solution (42) is consistent with the
initial dynamical system (30). A complete description of all possible solutions to (30) and (40)
will be presented elsewhere.
302 ISSN 1562-3076. Нелiнiйнi коливання, 1999, т. 2, № 3
5. Here we consider some differential-geometric aspects of a swept volume manifold St0(V )
generated by a Euclidean motion in R3. We shall use the basic notation from Theorem 2 of
Subsection 2.
We define the following system of one-forms (20) generated by a group action G× Y → Y
on a manifold Y :
βj := dyj +
n∑
i=1
ξji (y)ω̄i(a, da). (45)
For this system to be completely Frobenius integrable, the canonical one-forms {ω̄i : 1 ≤ i ≤ n}
must satisfy the Maurer – Cartan equations
dω̄j +
1
2
n∑
i,k=1
Cjikω̄
i ∧ ω̄k := Ω̄j = 0 (46)
for all 1 ≤ j ≤ n, where the Cjik are the structure constants. If the canonical one-forms {ω̄i} are
defined via the scheme
T ∗(M)
η∗←− T ∗(G)
↓ ↓
M
η−→ G
, (47)
where M is a smooth finite-dimensional manifold and η is a smooth mapping, then (46) takes
the simple form
η∗Ω̄j |M̄= 0, 1 ≤ j ≤ n, (48)
on some integral submanifold M̄ ⊂M that is diffeomorphic to St0(V ).
Let
{
αj ∈ Λ2(M) : 1 ≤ j ≤ m
}
be a basis of two-forms generating the ideal I(α) over the
two-forms (46). Using this basis we can formulate the Cartan criterion for the system of one-
forms (45) to define a group action of G on St0(V ). The ideal I(α, β) generated by both (45)
and
{
αj
}
over the prolonged locally defined manifold M × Y must be completely integrable;
therefore
dβj =
m∑
k=1
f jkα
k +
n∑
i=1
gji ∧ β
i, (49)
where f jk ∈ Λ0(M × Y ), gji ∈ Λ1(M × Y ), 1 ≤ i, j ≤ n, 1 ≤ k ≤ m. We note that the ideal
I(α, β) over the two-forms (46) should also be completely integrable via the Cartan criterion,
because it follows from the equations dΩ̄j = 0, 1 ≤ j ≤ n, that dI(α) ⊂ I(α) and I(α) |M̄= 0.
The above result enables us to interpret the locally defined manifold M × St0(V ) as the
adjoint of a principal fiber bundle P (M ;G) over the base manifold M with structure group G
acting on the fibered manifold P.By representing a point locally as (z, a) ∈ P (M ;G),we obtain
the local representation of the connection one-form as
ω(z, a) := ω̄(a, da) +Ada−1 〈Γ(z), dz〉 , (50)
ISSN 1562-3076. Нелiнiйнi коливання, 1999, т. 2, № 3 303
where Γ(z) ∈ G is the Christoffel elements, z ∈M,
ω̄(a, da) :=
n∑
i=1
ω̄i(a, da)Ai
and {Ai} is a basis for G.
Hence, the curvature two-form (defined just on M) is
Ω := dω + ω ∧ ω =
1
2
Ada−1
m∑
i,j=1
Fij(z)dzi ∧ dzj , (51)
where for all z ∈M, i, j = 1, ...,m = dimM,
Fij(z) :=
∂Γj
∂zi
− ∂Γi
∂zj
+ [Γi,Γj ]. (52)
The results obtained above for the one-forms (45) imply that
Ω |M̄= 0⇔ Fij |M̄= 0 (53)
for all 1 ≤ i, j ≤ m, and this is equivalent to
1
2
m∑
i,j=1
Fij(z)dzi ∧ dzj ∈ I(α). (54)
Thus we can find the structure constants of the Lie group G by a simple yet tedious computati-
on of the holonomy algebra of the connection (50), which is generated [12, 13] by all linear
combinations of elements Fij ,∇kFij ,∇k∇lFij , ..., i, j, k, l = 1, ...,m, where ∇k :=
∂
∂zk
− Γk,
are the appropriate covariant derivatives in T ∗(M).
Whence, we obtain the following result:
Theorem 4. Suppose that the holonomy Lie algebra G associated with a Euclidean motion
of the solid object V in E3 is generated by parallel transport along a two-dimensional integral
submanifold M̄ ⊂ M with local coordinates (s, t) ∈ R
2πZ
× [0, T ). Then owing to (48), the
following system is compatible on M̄ :
∂f
∂s
= −Γ(s)f,
∂f
∂t
= −Γ(t)f, (55)
for all f spanning a linear representation space for the Lie algebra G, where Γ(s),Γ(t) are the
nontrivial Christoffel matrices that define the curvature form Ω over M.
The above theorem gives rise to a new way of constructing exact forms of a group actions
G×Y → Y which produce Euclidean motions in R3. Some interesting applications of this new
approach are presented in [14].
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Received 17.03.99
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