The Hojman Construction and Hamiltonization of Nonholonomic Systems

The Hojman Construction and Hamiltonization
of Nonholonomic Systems\star\star$\star$This paper is a contribution to the
Special Issue on Analytical Mechanics and Differential Geometry in honour of
Sergio Benenti. The full collection is available at
http://www.emis.de/journals/SIGMA/Benenti.html

Ivan A. BIZYAEV †‡, Alexey V. BORISOV †§ and Ivan S. MAMAEV †

I.A. Bizyaev, A.V. Borisov and I.S. Mamaev

† Udmurt State University, 1 Universitetskaya Str., Izhevsk, 426034 Russia
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
‡ St. Petersburg State University, 1 Ulyanovskaya Str., St. Petersburg, 198504
Russia

§ National Research Nuclear University MEPhI, 31 Kashirskoe highway, Moscow,
115409 Russia

Received October 05, 2015, in final form January 26, 2016; Published online
January 30, 2016

In this paper, using the Hojman construction, we give examples of various
Poisson brackets which differ from those which are usually analyzed in
Hamiltonian mechanics. They possess a nonmaximal rank, and in the general case
an invariant measure and Casimir functions can be globally absent for them.

Hamiltonization; Poisson bracket; Casimir functions; invariant measure;
nonholonomic hinge; Suslov problem; Chaplygin sleigh

37J60; 37J05

##  1 Introduction

This paper is dedicated to Professor S. Benenti on the occasion of his 60th
birthday. He has made an essential contribution to the development of various
branches of mechanics and mathematics. In nonholonomic mechanics, which is the
subject of this paper, he proposed a new constructive form of the equations of
motion [1] and gave a new example of nonlinear nonholonomic constraint [2].

In this paper we consider the possibility of representing some well-known and
new nonholonomic systems in Hamiltonian, more precisely, Poisson form. We note
from the outset that such a representation is usually possible for reduced
(incomplete) equations [14] and after rescaling time (conformal
Hamiltonianity).

Hamiltonian systems are the best-studied class of systems from the viewpoint
of the theory of integrability, stability, topological analysis, perturbation
theory (KAM theory) etc. Therefore, the investigation of the possibility of
representing the equations of motion in conformally Hamiltonian form is an
important, but poorly studied problem, which is usually called the
Hamiltonization problem.

There are various well-known obstructions to Hamiltonization, which are
examined in detail in [7] (see also [5]). This problem has several aspects:
local, semilocal, and global. In this sense the Hamiltonization problem is in
many respects analogous to the problem of existence of analytic integrals and
a smooth invariant measure (see [32]).

In nonholonomic mechanics, the reducing multiplier method of Chaplygin [21]
(developed further in [8]) provides a powerful tool for the Hamiltonization of
equations of motion. However, this method does not apply to the problems
presented below due to the fact that it is impossible to represent the
equations of motion in the form of the Chaplygin system, or the examples
considered exhibit everywhere the absence of a smooth invariant measure with
density depending only on positional variables (as required by the Chaplygin
method). Other Hamiltonization methods are examined in [18]. We note that the
above-mentioned paper is of formal character and gives no nontrivial
mechanical examples encountered in applications.

Poisson brackets whose appearance cannot be explained by employing the
Chaplygin method were obtained in [4, 6, 10, 38, 39] with the help of an
explicit algebraic ansatz. In this paper it is shown that these brackets can
be obtained using an algorithm that we call the Hojman construction (its
foundations were laid down already by S. Lie). It is based on the existence of
conformal symmetry fields, i.e., symmetry fields obtained after rescaling
time, and was developed further in [20, 28].

The rank of the Poisson bracket obtained by using the Hojman construction is,
as a rule, smaller than the dimension of the phase space; for it a smooth (or
even singular) invariant measure and global Casimir functions may be absent in
the general case, and the system’s behavior itself may differ greatly from
Hamiltonian behavior, moreover, there may exist limit cycles. Therefore, the
resulting Poisson brackets are of rather limited utility. In practice, such a
Hamiltonian representation may turn out to be useless (especially for
infinite-dimensional systems [20]).

Indeed, the basic methods of Hamiltonian mechanics have been developed for
canonical systems. However, as shown in [3], the representation found by us
can be useful for the investigation of stability problems.

Problems of Hamiltonization and various aspects of the dynamics of
nonholonomic systems are investigated in [6, 9, 11, 15, 16, 22, 23, 38, 39].

##  2 Tensor invariants and Hamiltonization
of dynamical systems

###  2.1 Basic notions and definitions

Let a dynamical system be given on some manifold $\mathcal{M}^{n}$, a phase
space.

$\displaystyle\dot{\boldsymbol{x}}=\boldsymbol{v}(\boldsymbol{x}),$ (1)

where $\boldsymbol{x}=(x_{1},\dots,x_{n})$ are local coordinates,
$\boldsymbol{v}=(v_{1}(\boldsymbol{x}),\dots,v_{n}(\boldsymbol{x}))$.

The evolution of an arbitrary (smooth) function on $\mathcal{M}^{n}$ along the
vector field (1) is given by the linear differential operator

$\displaystyle\dot{F}=\boldsymbol{v}(F)=\sum_{i=1}^{n}v_{i}\frac{\partial
F}{\partial x_{i}}.$

As is well known, the behavior of a dynamical system is in many respects
determined by tensor invariants (conservation laws), from which one can, as a
rule, draw the main data on its dynamics. For exact formulations see [12]. We
only note that three kinds of tensor invariants are encountered particularly
frequently:

  * –

the first integral $F(\boldsymbol{x})$ for which

$\displaystyle\dot{F}=\boldsymbol{v}(F)=0;$

  * –

symmetry field $\boldsymbol{u}(\boldsymbol{x})$ for which

$\displaystyle[\boldsymbol{v},\boldsymbol{u}]=0;$

  * –

the invariant measure $\mu=\rho(\boldsymbol{x})dx_{1}\wedge\cdots\wedge
dx_{n}$ whose density $\rho(\boldsymbol{x})$ is everywhere larger than zero
and satisfies the Liouville equation

$\displaystyle{\rm div}(\rho\boldsymbol{v})=0,$ (2)

where $\wedge$ denotes the exterior product. In this paper, unless otherwise
stated, all geometric objects (first integrals, vector fields etc.) are
assumed to be analytical on $\mathcal{M}^{n}$.

Another important tensor invariant is the Poisson structure, which allows the
equations of motion to be represented in Hamiltonian form. In applications,
such a representation can usually be obtained only after rescaling time as
$d\tau=\mathcal{N}(\boldsymbol{x})dt$ (if the system is a priori not
Hamiltonian), i.e., the equations of motion (1) are represented in conformally
Hamiltonian form

$\displaystyle\dot{\boldsymbol{x}}=\mathcal{N}(\boldsymbol{x}){\bf
J}(\boldsymbol{x})\frac{\partial H}{\partial\boldsymbol{x}},$ (3)

where $H(\boldsymbol{x})$ is a Hamiltonian that is a first integral, and ${\bf
J}(\boldsymbol{x})=\|J_{ij}(\boldsymbol{x})\|$ is a Poisson structure, i.e., a
skew-symmetric tensor field satisfying the Jacobi identity

$\displaystyle\sum_{l=1}^{n}\left(J_{lk}\frac{\partial J_{ij}}{\partial
x_{l}}+J_{li}\frac{\partial J_{jk}}{\partial x_{l}}+J_{lj}\frac{\partial
J_{ki}}{\partial x_{l}}\right)=0,\qquad i,j,k=1,\ldots n,$ (4)

and $\mathcal{N}(\boldsymbol{x})$ is a scalar function – a reducing
multiplier. The Poisson structure ${\bf J}(\boldsymbol{x})$ allows one to
define in a natural way the Poisson bracket of the functions $f$ and $g$ by
the formula

$\displaystyle\\{f(\boldsymbol{x}),g(\boldsymbol{x})\\}=\sum_{i,j=1}^{n}J_{ij}\frac{\partial
f}{\partial x_{i}}\frac{\partial g}{\partial x_{j}}.$

######  Remark 2.1.

Time rescaling cannot be applied at points where the reducing multiplier
$\mathcal{N}(\boldsymbol{x})$ vanishes, hence, Hamiltonization is possible
only on the set
$\widetilde{\mathcal{M}}^{n}=\\{\boldsymbol{x}\in\mathcal{M}^{n}\,|\,\mathcal{N}(\boldsymbol{x})\neq
0\\}$. As a result, the behavior of the trajectories in general (on the entire
phase space $\mathcal{M}^{n}$) can considerably differ from the behavior of
the trajectory of Hamiltonian systems. In particular, not only tori, but also
two-dimensional integral manifolds can be arbitrary in this case, see, for
example, the Suslov problem [10].

In many examples the Poisson structure turns out to be

$\displaystyle\operatorname{rank}{\bf J}<n.$

As is well known, the entire phase space (in the domain of the constant rank
${\bf J}$) is foliated by symplectic leaves $\mathcal{O}$ of dimension
$\dim\mathcal{O}=\operatorname{rank}{\bf J}$. In the simplest case the
symplectic leaves are given as the level surfaces of a set of global Casimir
functions

$\displaystyle\mathcal{O}_{\boldsymbol{c}}=\big{\\{}\boldsymbol{x}\in\mathcal{M}^{n}\,|\,C_{1}(\boldsymbol{x})=c_{1},\dots,C_{m}(\boldsymbol{x})=c_{m}\big{\\}},$
$\displaystyle\sum_{j=1}^{n}J_{ij}(\boldsymbol{x})\frac{\partial
C_{k}}{\partial x_{j}}=0,\qquad i=1,\dots,n,\qquad
k=1,\dots,m=n-\operatorname{rank}{\bf J}.$ (5)

Nevertheless, in most of the examples considered below the number of global
Casimir functions turns out to be less than $m$. This may lead to unusual
(from the viewpoint of the standard theory of Hamiltonian systems) behavior of
trajectories of the system (3).

We note that there are three levels of analysis of the problem of the
existence of tensor invariants:

  * –

local level – in a neighborhood of a nonsingular point of the vector field
(1),

  * –

semilocal level – in a neighborhood of invariant sets of the system (1), such
as fixed points, periodic orbits, invariant tori etc.,

  * –

global level – on the entire phase space $\mathcal{M}^{n}$.

From the local point of view, obstructions to the existence of any tensor
invariants are not encountered by virtue of the rectification theorem for
vector fields. Therefore, in what follows it is implied that we consider the
system from the semilocal and global points of view.

###  2.2 The general Hojman construction

Consider a conformally Hamiltonian system in the form (3). From the point of
view of integrability, the case where

$\displaystyle\operatorname{rank}{\bf J}=2$

is regarded as the simplest case. As a rule, in this case it is assumed that
the system (3) admits a natural (global) restriction to the two-dimensional
symplectic leaf $\mathcal{O}^{2}$ and reduces to a Hamiltonian system with one
degree of freedom, and its trajectories are the level lines of the restriction
of the Hamiltonian $H|_{\mathcal{O}^{2}}$.

It turns out that such a conclusion cannot be drawn in the general case, when
the properties of solutions (5) are unknown to us. As already noted, a full
set of global Casimir functions can be absent in ${\bf J}$ (some of them are
defined only locally). As a result, the symplectic leaf can be immersed in the
phase space in a fairly complicated (in particular, chaotic) way, and the
above-described picture is not always realized. Nevertheless, the above
Poisson structure turns out to be useful for the investigation of stability
problems [29].

Consider a Poisson structure ${\bf J}(\boldsymbol{x})$ of rank 2 on
$\mathcal{M}^{n}$ for which there is a pair of (globally defined) vector
fields $\boldsymbol{v}(\boldsymbol{x})$ and $\boldsymbol{u}(\boldsymbol{x})$
such that

$\displaystyle{\bf
J}(\boldsymbol{x})=\boldsymbol{v}(\boldsymbol{x})\wedge\boldsymbol{u}(\boldsymbol{x}).$
(6)

######  Remark 2.2.

Every skew-symmetric bivector field of constant rank $2$ locally admits such a
representation.

According to the Darboux theorem, the Jacobi identity (4) holds only in the
case where the distribution given by these vector fields is integrable

$\displaystyle[\boldsymbol{v},\boldsymbol{u}]=\mu_{1}(\boldsymbol{x})\boldsymbol{v}+\mu_{2}(\boldsymbol{x})\boldsymbol{u}.$
(7)

The Casimir functions are simultaneously the integrals of the fields
$\boldsymbol{v}$ and $\boldsymbol{u}$:

$\displaystyle\boldsymbol{u}(C_{k})=\boldsymbol{v}(C_{k})=0,\qquad
k=1,\dots,m^{\prime}.$

######  Remark 2.3.

Condition (7) was known already to S. Lie; it can also be obtained if the
Jacobi identity is rewritten by means of the Schouten bracket $[[{\bf J},{\bf
J}]]=0$. Using the properties of the Schouten bracket, for the Poisson
structure (6) we find [29]:

$\displaystyle\mbox{}[[{\bf J},{\bf
J}]]=2[\boldsymbol{v},\boldsymbol{u}]\wedge\boldsymbol{v}\wedge\boldsymbol{u}=0,$

whence we obtain condition (7).

It turns out that in many examples such decomposable Poisson structures allow
one to Hamiltonize in a natural way the dynamical systems in question. This
approach was proposed in [20, 28]. We present the necessary results and omit
the proofs, which are completely straightforward.

######  Theorem 2.4 (Hojman [28]).

Suppose that the system (1) possesses a first integral $H$ and a vector field
$\boldsymbol{u}(\boldsymbol{x})$ such that

$\displaystyle[\boldsymbol{v},\boldsymbol{u}]=\mu(\boldsymbol{x})\boldsymbol{v},\qquad
H_{u}(\boldsymbol{x})=\boldsymbol{u}(H)\not\equiv 0,$ (8)

then the initial system (1) can be represented in conformally Hamiltonian form

$\displaystyle\dot{\boldsymbol{x}}=H_{u}^{(-1)}(\boldsymbol{x}){\bf
J}^{(2)}(\boldsymbol{x})\frac{\partial H}{\partial\boldsymbol{x}},\qquad{\bf
J}^{(2)}(\boldsymbol{x})=\boldsymbol{v}\wedge\boldsymbol{u}.$ (9)

######  Remark 2.5.

The above conformally Hamiltonian representation (9) is not defined for points
$\boldsymbol{x}$ at which $H_{u}(\boldsymbol{x})=0$, and the symbol
$\not\equiv$ means that $H_{u}(\boldsymbol{x})$ must not be equal to zero for
any $\boldsymbol{x}$.

It follows from conditions (8) that the vector field
$\boldsymbol{u}(\boldsymbol{x})$ is not tangent to the level surface
$H(\boldsymbol{x})={\rm const}$ and, moreover,

$\displaystyle\boldsymbol{v}(H_{u})=0,$

hence, in this case the reducing multiplier $H_{u}(\boldsymbol{x})$ is the
first integral of the system. If, in addition, the system possesses additional
tensor invariants, then a natural generalization of this result holds (see
[28] for details).

######  Proposition 2.6.

Suppose that the system (1) satisfies the conditions of Theorem 2.4 and,
moreover, possesses the symmetry fields $\boldsymbol{u}_{1}$ and
$\boldsymbol{u}_{2}$ which define the integrable distribution and preserve the
Hamiltonian

$\displaystyle[\boldsymbol{u}_{1},\boldsymbol{v}]=[\boldsymbol{u}_{2},\boldsymbol{v}]=0,\qquad\boldsymbol{u}_{1}(H)=\boldsymbol{u}_{2}(H)=0,$
$\displaystyle[\boldsymbol{u}_{1},\boldsymbol{u}_{2}]=\lambda_{1}(\boldsymbol{x})\boldsymbol{u}_{1}+\lambda_{2}(\boldsymbol{x})\boldsymbol{u}_{2},$

then the system (1) admits the conformally Hamiltonian representation

$\displaystyle\dot{\boldsymbol{x}}=H^{(-1)}_{u}(\boldsymbol{x}){\bf
J}^{(4)}(\boldsymbol{x})\frac{\partial H}{\partial\boldsymbol{x}},$
$\displaystyle{\bf
J}^{(4)}(\boldsymbol{x})=\boldsymbol{v}\wedge\boldsymbol{u}+\boldsymbol{u_{1}}\wedge\boldsymbol{u_{2}},\qquad\operatorname{rank}{\bf
J}^{(4)}=4.$

Below we consider some nonholonomic mechanics problems illustrating the
applicability of the above theorem and allowing one to obtain Poisson brackets
with various (unusual) properties.

The following construction shows that any dynamical system admits a rank 2
Hamiltonization in an extended phase space.

Let $\boldsymbol{v}$ denote the vector field that defines the dynamics of a
nonholonomic system on some manifold $\mathcal{M}$. Let
$E\colon\mathcal{M}\rightarrow\mathbb{R}$ be the energy of the system, so that
$\boldsymbol{v}(E)=0$. Consider the following rank two Poisson structure in
$\mathcal{M}\times\mathbb{R}$:

$\displaystyle{\bf J}=\boldsymbol{v}\wedge\frac{\partial}{\partial y},$

where $y$ is the coordinate on the factor $\mathbb{R}$ of the Cartesian
product $\mathcal{M}\times\mathbb{R}$. The fact that ${\bf J}$ satisfies the
Jacobi identity is obvious from the fact that
${[\boldsymbol{v},\frac{\partial}{\partial y}]=0}$.

It is obvious that the Hamiltonian vector field of $y$ with respect to is
$\boldsymbol{v}$. Moreover, $E$ is a Casimir of ${\bf J}$.

This example shows how the symplectic leaves of a rank two Poisson structure
can be immersed in the phase space in a rather complicated way. Moreover, it
also illustrates that usually the construction of brackets in an extended
phase space cannot add anything new to understand the properties of a given
system.

###  2.3 Invariant measure in Hamiltonian systems

Let us briefly discuss the problem of existence of an invariant measure in
Hamiltonian systems on $\mathcal{M}^{n}$ with a Poisson structure

$\displaystyle\dot{\boldsymbol{x}}={\bf J}(\boldsymbol{x})\frac{\partial
H}{\partial\boldsymbol{x}},\qquad\operatorname{rank}{\bf J}<n.$ (10)

We recall that for a symplectic manifold the rank of ${\bf J}$ equals $n=2k$
and that, according to the Liouville theorem, an invariant measure exists for
an arbitrary Hamiltonian. By analogy with this, if for the system (10) with an
arbitrary Hamiltonian there exists the same invariant measure, it is called
the Liouville measure. Its density satisfies the system of equations on
$\mathcal{M}^{n}$

$\displaystyle\sum_{j=1}^{n}\frac{\partial}{\partial x_{j}}(\rho
J_{ij}(\boldsymbol{x}))=0,\qquad i=1,\ldots,n.$

A manifold with this Poisson structure ${\bf J}(\boldsymbol{x})$ and volume
form $\rho(\boldsymbol{x})dx_{1}\wedge\cdots\wedge dx_{n}$ is called a
unimodular Poisson manifold.

######  Remark 2.7.

For the case of linear Poisson structures (Lie–Poisson brackets) the
unimodular Poisson manifolds correspond to the well-known unimodular Lie
algebras (see, e.g., [33]).

Below we distinguish between three cases:

  * –

there is an invariant measure (without singularities) which exists on the
entire manifold $\mathcal{M}^{n}$ and is defined by an explicit solution of
the Liouville equation (2);

  * –

the Liouville equation (2) admits an explicit solution for density
$\rho(\boldsymbol{x})$ which has singularities at some points of the phase
space; we shall call such a measure singular;

  * –

it is impossible to obtain an explicit Liouville solution, and in the phase
space of the system there exist attracting invariant sets – attractors111As a
rule, attractors can be detected only by numerical investigations. (fixed
points, limit cycles, strange attractors etc.); in this case we shall say that
there exists no invariant measure.

Further, we illustrate the above constructions and considerations by various
nonholonomic systems. Nonholonomic systems are characterized by the presence
of nonintegrable constraints, resulting in fairly general systems of
differential equations, which in the case of homogeneous (in velocities)
constraints possess a first energy integral (see, e.g., [15] for details). The
problem of existence of an invariant measure for nonholonomic systems is
discussed, for example, in [15, 17, 24, 30].

In all the examples of Hamiltonizable systems considered here the dimension of
the phase space $\mathcal{M}^{n}$ is equal to five ($n=5$). Moreover,
depending on the existence of an invariant measure and global Casimir
functions, a large number of types of Hamiltonian systems can arise (with a
Poisson bracket of nonmaximal rank). Nevertheless, as a rule, it is impossible
to detect all these types in practice. We list here combinations that are
encountered in the examples considered.

  * 1)

There exist an invariant measure and three global Casimir functions: a
nonholonomic hinge, one of the bodies is a plate (see Section 3.3);

  * 2)

There exist an invariant measure and two global Casimir functions: a
nonholonomic hinge in the general case (see Section 3.4);

  * 3)

There exist an invariant measure and one global Casimir function: the Suslov
problem under the condition that the constraint is imposed along the principal
axis of inertia (see Section 4.2);

  * 4)

There exist a singular invariant measure and three global Casimir functions:
the Chaplygin sleigh on a horizontal plane (see Section 5.2);

  * 5)

There are no invariant measure and no global Casimir functions: the Suslov
problem in a gravitational field (see Section 4.3) and the Chaplygin sleigh on
an inclined plane (see Section 5.3).

##  3 Nonholonomic hinge

###  3.1 Equations of motion

Consider the problem of the free motion of a system of two bodies coupled by a
nonholonomic hinge. The outer body is a homogeneous spherical shell inside
which a rigid body moves; the rigid body is connected with the shell by means
of sharp wheels in such a way as to exclude relative rotations about the
vector $\boldsymbol{e}$ fixed in the inner body (see Fig. 1).

Figure 1: A system of two bodies coupled by a nonholonomic hinge.

In what follows we shall call this system a nonholonomic hinge (this problem
was previously studied in [3, 4]).

Choose a moving coordinate system $Cx_{1}x_{2}x_{3}$ attached to the inner
body. Then, if we denote the angular velocities of the inner body and the
spherical shell by $\boldsymbol{\omega}$ and $\boldsymbol{\Omega}$, the
constraint takes the form

$\displaystyle\omega_{3}-\Omega_{3}=0.$

In the absence of external forces the evolution of the angular velocities is
governed by the following equations:

$\displaystyle\dot{\Omega}_{1}=\omega_{3}(\Omega_{2}-\omega_{2}),\qquad\dot{\Omega}_{2}=\omega_{3}(\omega_{1}-\Omega_{1}),$
$\displaystyle I_{1}\dot{\omega}_{1}=(I_{2}-I_{3})\omega_{2}\omega_{3},\qquad
I_{2}\dot{\omega}_{2}=(I_{3}-I_{1})\omega_{1}\omega_{3},$
$\displaystyle(I_{\text{s}}+I_{3})\dot{\omega}_{3}=I_{\text{s}}(\Omega_{1}\omega_{2}-\Omega_{2}\omega_{1})+(I_{1}-I_{2})\omega_{1}\omega_{2},$
(11)

where ${\bf I}=\operatorname{diag}(I_{1},I_{2},I_{3})$ is the tensor of
inertia of the inner body and $I_{\text{s}}$ is the tensor of inertia of the
shell.

###  3.2 First integrals and the Poisson bracket

The system (11) possesses a standard invariant measure and conserves the
energy:

$\displaystyle d\Omega_{1}d\Omega_{2}d\omega_{1}d\omega_{2}d\omega_{3},\qquad
E=\frac{1}{2}I_{\text{s}}(\Omega_{1}^{2}+\Omega_{2}^{2})+\frac{1}{2}\big{(}I_{1}\omega_{1}^{2}+I_{2}\omega_{2}^{2}+(I_{3}+I_{\text{s}})\omega_{3}^{2}\big{)}.$

In order to use Theorem 2.4, we choose

$\displaystyle\boldsymbol{u}=\frac{1}{\omega_{3}}\frac{\partial}{\partial\omega_{3}}$

as $\boldsymbol{u}$. Then, if we denote the initial vector field (11) by
$\boldsymbol{v}$, we obtain

$\displaystyle[\boldsymbol{v},\boldsymbol{u}]=-\omega_{3}^{-1}\boldsymbol{v},\qquad\boldsymbol{u}(E)=2(I_{3}+I_{\text{s}}).$

Thus, Theorem 2.4 holds and the system (11) can be represented in Hamiltonian
form with a Poisson bracket in the variables
$\boldsymbol{x}=(\Omega_{1},\Omega_{2},\omega_{1},\omega_{2},\omega_{3})$ of
the form

$\displaystyle{\bf
J}^{(2)}=\boldsymbol{v}\wedge\boldsymbol{u}=\begin{pmatrix}0&0&0&0&\Omega_{2}-\omega_{2}\\\
0&0&0&0&-(\Omega_{1}-\omega_{1})\vspace{1mm}\\\
0&0&0&0&\frac{I_{2}-I_{3}}{I_{1}}\omega_{2}\vspace{1mm}\\\
0&0&0&0&-\frac{I_{1}-I_{3}}{I_{2}}\omega_{1}\vspace{1mm}\\\
*&*&*&*&0\end{pmatrix},$

where the asterisks denote nonzero matrix entries resulting from the skew-
symmetry condition ${\bf J}^{(2)}$. We note that even though the vector field
$\boldsymbol{u}$ has a singularity when $\omega_{3}=0$, the resulting Poisson
structure for the system is smooth.

The found Poisson bracket corresponds to the solvable Lie algebra. According
to the classification of [37], this is the algebra $A^{spq}_{5,17}$ with
$p=q=0$ (see [4] for details).

###  3.3 A flat inner body ($\boldsymbol{I_{3}=I_{1}+I_{2}}$)

In the case where the inner body is flat, i.e., $I_{3}=I_{1}+I_{2}$, for ${\bf
J}^{(2)}$ we find all $k=n-\operatorname{rank}{\bf J}^{(2)}=3$ Casimir
functions

$\displaystyle C_{1}(\boldsymbol{x})=\omega_{1}^{2}+\omega_{2}^{2},\qquad
C_{2}(\boldsymbol{x})=\Omega_{1}^{2}+(\omega_{2}-\Omega_{2})^{2},\qquad
C_{3}(\boldsymbol{x})=\Omega_{2}^{2}+(\omega_{1}-\Omega_{1})^{2},$

i.e., we obtain a case that is the simplest from the point of view of
integrability.

Indeed, let us restrict the system (11) to the symplectic leaf by making the
change of variables

$\displaystyle\omega_{1}=c_{1}\cos\varphi,\qquad\omega_{2}=c_{1}\sin\varphi,$
$\displaystyle\Omega_{1}=c_{2}\cos(\varphi+c_{3}),\qquad\Omega_{2}=c_{1}\sin\varphi-
c_{2}\sin(\varphi+c_{3}),$

where $\varphi\in[0,2\pi)$ is the angular coordinate, and $c_{1}$, $c_{2}$,
and $c_{3}$ parameterize the symplectic leaf

$\displaystyle C_{1}(\boldsymbol{x})=\sqrt{c_{1}},\qquad
C_{2}(\boldsymbol{x})=\sqrt{c_{2}},\qquad
C_{3}(\boldsymbol{x})=c_{1}^{2}-2c_{1}c_{2}\cos c_{3}+c_{2}^{2}.$

As a result, we obtain a system with one degree of freedom
$(\varphi,\omega_{3})$ and a canonical Poisson bracket

$\displaystyle\dot{\varphi}=\frac{\partial\widetilde{H}}{\partial\omega_{3}},\qquad\dot{\omega}_{3}=-\frac{\partial\widetilde{H}}{\partial\varphi},$
$\displaystyle\widetilde{H}=\frac{1}{2}\omega_{3}^{2}+\frac{c_{1}^{2}(I_{1}\cos^{2}\varphi+(I_{s}+I_{2})\sin^{2}\varphi)}{2(I_{s}+I_{1}+I_{2})}-\frac{I_{s}c_{1}c_{2}\sin(\varphi+c_{3})}{I_{s}+I_{s}+I_{2}}.$
(12)

Some of its typical trajectories are shown in Fig. 2.

Figure 2: Typical trajectories $\eqref{Eqb03}$ for $I_{1}=2$, $I_{2}=3$,
$I_{\text{s}}=1$, $c_{1}=1$, $c_{2}=2$, $c_{3}=\frac{\pi}{3}$.

###  3.4 The general case

In the general case the Poisson bracket possesses only two global Casimir
functions [3, 4]

$\displaystyle
C_{1}(\boldsymbol{x})=I_{1}(I_{1}-I_{3})\omega_{1}^{2}+I_{2}(I_{2}-I_{3})\omega_{2}^{2},\qquad
C_{2}(\boldsymbol{x})=(I_{1}\omega_{1}-I_{3}\Omega_{1})^{2}+(I_{2}\omega_{2}-I_{3}\Omega_{2})^{2},$

while a third one does not exist in the general case (it is defined only
locally).

Indeed, let us introduce coordinates on the symplectic leaf and assume that
$I_{3}<I_{2}<I_{1}$:

$\displaystyle\omega_{1}=\sqrt{\frac{c_{1}}{I_{1}(I_{1}-I_{3})}}\sin\varphi_{1},\qquad\omega_{2}=\sqrt{\frac{c_{1}}{I_{2}(I_{2}-I_{3})}}\cos\varphi_{1},$
$\displaystyle\Omega_{1}=\sqrt{\frac{I_{1}c_{1}}{I_{1}-I_{3}}}\frac{\sin\varphi_{1}}{I_{3}}-\sqrt{c_{2}}\frac{\sin\varphi_{2}}{I_{3}},\qquad\Omega_{2}=\sqrt{\frac{I_{2}c_{1}}{I_{2}-I_{3}}}\frac{\cos\varphi_{1}}{I_{3}}-\sqrt{c_{2}}\frac{\cos\varphi_{2}}{I_{3}},$

where $\varphi_{1},\varphi_{2}\in[0,2\pi)$ and $C_{1}(\boldsymbol{x})=c_{1}$,
$C_{2}(\boldsymbol{x})=c_{2}$.

In addition, we restrict the system to the level set of the energy integral
$H=h$ by making the substitution

$\displaystyle\omega_{3}(\varphi_{1},\varphi_{2})=\pm\sqrt{2h-\frac{2Q}{I_{3}^{2}(I_{3}+I_{\text{s}})}},$
$\displaystyle
Q=\frac{I_{\text{s}}c_{2}}{2}+\frac{c_{1}}{2}\left(\frac{I_{\text{s}}I_{1}+I_{3}^{2}}{I_{1}-I_{3}}\sin^{2}\varphi_{1}+\frac{I_{\text{s}}I_{2}+I_{3}^{2}}{I_{2}-I_{3}}\cos^{2}\varphi_{1}\right)$
$\displaystyle\hphantom{Q=}{}-I_{\text{s}}\sqrt{c_{2}}\left(\sqrt{\frac{I_{1}c_{1}}{I_{1}-I_{3}}}\sin\varphi_{1}\sin\varphi_{2}+\sqrt{\frac{I_{2}c_{1}}{I_{2}-I_{3}}}\cos\varphi_{1}\cos\varphi_{2}\right).$

As a result, the equations of motion can be represented as

$\displaystyle\dot{\varphi}_{1}=k\omega_{3}(\varphi_{1},\varphi_{2}),\qquad\dot{\varphi}_{2}=\omega_{3}(\varphi_{1},\varphi_{2}),\qquad
k^{2}=\frac{(I_{1}-I_{3})(I_{2}-I_{3})}{I_{1}I_{2}}.$ (13)

In the case where $\omega_{3}(\varphi_{1},\varphi_{2})$ vanishes nowhere and
$k$ is irrational, the trajectories (13) are rectilinear orbits on a torus
(see Fig. 3).

Figure 3: Trajectory (13) for $I_{1}=5.2$, $I_{2}=4.3$, $I_{3}=3.9$,
$I_{\text{s}}=3.6$, $c_{1}=10$, $c_{2}=40$, $h=10$.

Such behavior is partially due to the fact that in the general case the
missing solution (5) in the chosen (local) coordinates is represented as

$\displaystyle\varphi_{1}-k\varphi_{2}={\rm const},$

whence it follows that the projection of the symplectic leaf on
($\varphi_{1},\varphi_{2}$) is multi-valued.

A topological analysis of the system (11) is presented in [3]. In particular,
it is shown that in addition to a torus there are two other types of integral
surfaces: a sphere and a sphere with three handles (two-dimensional orientable
surface of genus 3).

##  4 The Suslov problem

###  4.1 Equations of motion and first integrals

If we assume the spherical shell from the preceding example to be fixed, we
obtain the so-called Suslov problem (in Vagner’s interpretation, see Fig. 4),
which describes the motion of a rigid body with a fixed point subject to the
nonholonomic constraint

$\displaystyle(\boldsymbol{\omega},\boldsymbol{e})=0,$

where $\boldsymbol{\omega}$ is the angular velocity of the body and
$\boldsymbol{e}$ is the vector fixed in the body.

Figure 4: Realization of the Suslov problem.

Choose a moving coordinate system attached to the inner body, with one of the
axes directed along $\boldsymbol{e}$. Then the tensor of inertia of the moving
body can be represented as

$\displaystyle{\bf I}=\begin{pmatrix}I_{11}&0&I_{13}\\\ 0&I_{22}&I_{23}\\\
I_{13}&I_{23}&I_{33}\\\ \end{pmatrix}.$

In the axisymmetric potential field $U=U(\boldsymbol{\gamma})$ the problem
reduces to the investigation of the following closed system of equations [10]:

$\displaystyle
I_{11}\dot{\omega}_{1}=-\omega_{2}(I_{13}\omega_{1}+I_{23}\omega_{2})+\gamma_{2}\frac{\partial
U}{\partial\gamma_{3}}-\gamma_{3}\frac{\partial U}{\partial\gamma_{2}},$
$\displaystyle
I_{22}\dot{\omega}_{2}=\omega_{1}(I_{13}\omega_{1}+I_{23}\omega_{2})+\gamma_{3}\frac{\partial
U}{\partial\gamma_{1}}-\gamma_{1}\frac{\partial U}{\partial\gamma_{3}},$
$\displaystyle\dot{\gamma}_{1}=-\gamma_{3}\omega_{2},\qquad\dot{\gamma}_{2}=\gamma_{3}\omega_{1},\qquad\dot{\gamma}_{3}=\gamma_{1}\omega_{2}-\gamma_{2}\omega_{1},$
(14)

where $\boldsymbol{\gamma}=(\gamma_{1},\gamma_{2},\gamma_{3})$ is the unit
vector of the fixed coordinate system referred to the moving axes.

The system (14) conserves the energy and possesses the geometrical integral

$\displaystyle
E=\frac{1}{2}\big{(}I_{11}\omega_{1}^{2}+I_{22}\omega_{2}^{2}\big{)}+U,\qquad
F=\gamma_{1}^{2}+\gamma_{2}^{2}+\gamma_{3}^{2}=1.$

We now consider successively several particular cases (14).

###  4.2 Case $\boldsymbol{I_{13}=I_{23}=0}$ and
$\boldsymbol{U=U(\gamma_{1},\gamma_{2})}$

Suppose that $I_{13}=I_{23}=0$, i.e., the constraint is imposed along the
principal axis of inertia and the potential field has the form
$U=U(\gamma_{1},\gamma_{2})$. Then the system (14) possesses a standard
invariant measure.

Further, we introduce the vector field

$\displaystyle\boldsymbol{u}=\frac{1}{\gamma_{3}}\frac{\partial}{\partial\gamma_{3}},$

for which

$\displaystyle[\boldsymbol{v},\boldsymbol{u}]=-\gamma_{3}^{-1}\boldsymbol{v},\qquad\boldsymbol{u}(F)=2.$

Consequently, it follows from Theorem 2.4 that the initial system (14) can be
represented in Hamiltonian form with the Poisson bracket

$\displaystyle{\bf
J}^{(2)}=\boldsymbol{v}\wedge\boldsymbol{u}=\begin{pmatrix}0&0&0&0&-\frac{1}{I_{11}}\frac{\partial
U}{\partial\gamma_{2}}\vspace{1mm}\\\ 0&0&0&0&\frac{1}{I_{22}}\frac{\partial
U}{\partial\gamma_{1}}\\\ 0&0&0&0&-\omega_{2}\\\ 0&0&0&0&\omega_{1}\\\
*&*&*&*&0\end{pmatrix},$

where the asterisks denote nonzero matrix entries resulting from the skew-
symmetry condition ${\bf J}^{(2)}$.

######  Proposition 4.1.

In the general case, the bracket ${\bf J}^{(2)}$ possesses one global Casimir
function – the energy integral $E$.

######  Proof 4.2.

First of all, we note that the Casimir functions ${\bf J}^{(2)}$ are
simultaneously the integrals of the system

$\displaystyle\dot{\omega}_{1}=-\frac{1}{I_{11}}\frac{\partial
U}{\partial\gamma_{2}},\qquad\dot{\omega}_{2}=\frac{1}{I_{22}}\frac{\partial
U}{\partial\gamma_{1}},\qquad\dot{\gamma}_{1}=-\omega_{2},\qquad\dot{\gamma}_{2}=\omega_{1},$
(15)

which is Hamiltonian with Hamiltonian $E$ and the Poisson bracket

$\displaystyle\\{\omega_{1},\gamma_{2}\\}=-\frac{1}{I_{1}},\qquad\\{\omega_{2},\gamma_{1}\\}=\frac{1}{I_{2}}.$

Further, if we introduce new variables

$\displaystyle p_{1}=\sqrt{I_{1}}\omega_{1},\qquad
p_{2}=\sqrt{I_{2}}\omega_{2},\qquad q_{1}=\sqrt{I_{2}}\gamma_{1},\qquad
q_{2}=\sqrt{I_{1}}\gamma_{2},$

then the system (15) reduces to investigating a vector field with a canonical
Poisson bracket and a Hamiltonian of the form

$\displaystyle
H=\frac{1}{2}(p_{1}^{2}+p_{2}^{2})+U\left(\frac{q_{1}}{I_{2}},\frac{q_{2}}{I_{1}}\right).$
(16)

This is a natural system which describes the motion of a material point on a
plane and in which, as is well known, there are generally no additional first
integrals, and hence the resulting Poisson bracket has no remaining global
Casimir functions.

######  Remark 4.3.

However, the system (16) has the well-know potentials $U$ for which there
exist (one or two) additional integrals (and hence Casimir functions) of
different degrees in momenta (see, e.g., the recent papers [34, 35] or [26,
27] for the Hénon–Heiles system).

###  4.3 Case $\boldsymbol{I_{13}\neq 0}$, $\boldsymbol{I_{23}\neq 0}$ and
$\boldsymbol{U=(\boldsymbol{a},\boldsymbol{\gamma})}$

Now consider the more general situation $I_{13}\neq 0$, $I_{23}\neq 0$, but in
the potential field $U=(\boldsymbol{a},\boldsymbol{\gamma})$.

In this case, the equations of motion are invariant under the transformation

$\displaystyle\omega_{1}\rightarrow\lambda\omega_{1},\qquad\omega_{2}\rightarrow\lambda\omega_{2},\qquad\boldsymbol{\gamma}\rightarrow\lambda^{2}\boldsymbol{\gamma},\qquad
dt\rightarrow\lambda^{-1}dt,$

which is associated to the vector field

$\displaystyle\widehat{\boldsymbol{u}}=\omega_{1}\frac{\partial}{\partial\omega_{1}}+\omega_{2}\frac{\partial}{\partial\omega_{2}}+2\gamma_{1}\frac{\partial}{\partial\omega_{1}}+2\gamma_{2}\frac{\partial}{\partial\omega_{2}}+2\gamma_{3}\frac{\partial}{\partial\omega_{3}},$

for which we have

$\displaystyle[\boldsymbol{u},\boldsymbol{v}]=\boldsymbol{v},\qquad\boldsymbol{u}(E)=2E.$

Thus, the system under consideration can be represented in conformally
Hamiltonian form with the Poisson bracket

$\displaystyle{\bf J}^{(2)}=\boldsymbol{v}\wedge\boldsymbol{u}.$

In the case at hand, ${\bf J}^{(2)}$ has a rather cumbersome form, so we do
not present it here explicitly. We note that it is cubic in the velocities
$\boldsymbol{\omega}$.

######  Proposition 4.4.

In the general case, the system (14) has no invariant measure with smooth
density in the potential field $U=(\boldsymbol{a},\boldsymbol{\gamma})$.

######  Proof 4.5.

In the potential field $U=(\boldsymbol{a},\boldsymbol{\gamma})$ the system
(14) possesses a fixed point

$\displaystyle\omega_{1}=0,\qquad\omega_{2}=-\frac{a_{3}}{I_{23}},\qquad\gamma_{1}=0,\qquad\gamma_{2}=1,\qquad\gamma_{3}=0$

having a nonzero trace of the linearization matrix:

$\displaystyle\frac{I_{13}}{I_{1}}\frac{\sqrt{a_{3}}}{\sqrt{I_{23}}}.$

Consequently, the system considered has no invariant measure with smooth
density [30, 31, 32] in a neighborhood of the fixed point.

Indeed, in this case the characteristic polynomial of the linearized system is
represented as

$\displaystyle
P(\lambda)=\lambda^{2}\big{(}\lambda^{3}-c_{1}\lambda^{2}+c_{2}\lambda+c_{3}\big{)},$
$\displaystyle c_{1}=\frac{I_{13}\sqrt{a_{3}}}{I_{1}\sqrt{I_{23}}},\qquad
c_{2}=\frac{a_{2}}{I_{1}}-\frac{I_{1}I_{2}+2I_{23}^{2}}{I_{23}I_{1}I_{2}}a_{3},\qquad
c_{3}=\sqrt{a_{3}I_{23}}\left(\frac{I_{13}a_{3}}{I_{1}I_{23}^{2}}-\frac{2a_{1}}{I_{1}I_{2}}\right).$

This polynomial is not reciprocal.

In this system, an analogous result was obtained for $U=0$ previously in [30].
Later, however, a singular invariant measure [10] was found in this case.

We also note that the pair of brackets that was constructed for the Suslov
problem is defined in the extended phase space, as are the brackets described
above.

$\displaystyle\mathbb{R}^{2}\times\mathbb{R}^{3}=\\{(\omega_{1},\omega_{2},\gamma_{1},\gamma_{2},\gamma_{3})\\}.$

Indeed, for these brackets the geometric integral
$F=\gamma_{1}^{2}+\gamma_{2}^{2}+\gamma_{3}^{2}$ is not a Casimir function, so
it is not clear how to restrict them to the actual phase space of the system

$\displaystyle\mathbb{R}^{2}\times
S^{2}=\\{(\omega_{1},\omega_{2},\gamma_{1},\gamma_{2},\gamma_{3}),\gamma_{1}^{2}+\gamma_{2}^{2}+\gamma_{3}^{2}=1\\}.$

In view of the above discussion (at the end of Section 3.2), these brackets
cannot provide any insight into the system’s properties.

######  Remark 4.6.

It may seem that a singular invariant measure with density having
singularities in the considered family of fixed points is possible in this
case too (see, e.g., Section 5). However, it is impossible to obtain a
suitable solution of the Liouville equation in this case. Moreover, as
numerical investigations show, this system may also have more complicated
attracting sets – limit cycles. Numerical investigations also show that this
system has no additional real-analytical integrals.

######  Remark 4.7.

A proof of the absence of meromorphic integrals for other (more particular)
cases of the Suslov problem was obtained in [25, 36, 41]. The methods
developed in these papers can be used in this case also, since the system (14)
has a particular periodic solution which for $I_{13}=0$, $a_{1}=0$, and
$a_{2}=0$ has the form

$\displaystyle\omega_{1}(t)=2k\operatorname{cn}(t,k),\quad\omega_{2}(t)=0,\qquad\gamma_{1}(t)=0,\qquad\gamma_{2}(t)=-\frac{2I_{1}k}{a_{3}}\operatorname{sn}(t,k)\operatorname{dn}(t,k),$
$\displaystyle\gamma_{3}(t)=\frac{I_{1}}{a_{3}}\big{(}2k^{2}\operatorname{sn}^{2}(t,k)-1\big{)},\qquad
H=I_{1}\big{(}2k^{2}-1\big{)},$

where $\operatorname{sn}(t,k)$, $\operatorname{cn}(t,k)$,
$\operatorname{dn}(t,k)$ are the elliptic Jacobi functions with parameter $k$.

######  Remark 4.8.

We note that in [10, 32] other cases of representation of the equations of the
Suslov problem were found in Hamiltonian form, with a Poisson bracket of rank
4, which cannot be obtained with the help of the Hojman construction.

##  5 The Chaplygin sleigh

###  5.1 Equations of motion

The Chaplygin sleigh is a rigid body moving on a horizontal plane and
supported at three points: two (absolutely) smooth posts and a knife edge such
that the body cannot move perpendicularly to the plane of the wheel.

As a historical remark, we note that although the Chaplygin sleigh is usually
linked to the works of Chaplygin [21] and Caratheodory [19], they were
considered slightly earlier by Brill [40] as an example of the mechanism of a
nonholonomic planimeter.

Figure 5: The Chaplygin sleigh on a plane.

In this case, the configuration space coincides with the group of motions of
the plane ${\rm SE}(2)$. To parameterize it, we choose Cartesian coordinates
$(x,y)$ of point $O_{1}$ (see Fig. 5) in the coordinate system $Oxy$ and the
angle $\varphi$ of rotation of the axes $O_{1}x_{1}x_{2}$ relative to $Oxy$.

The motion of the Chaplygin sleigh in the variables
$(v_{1},\omega,x,y,\varphi)$ is described by the equations (see [13] for
details)

$\displaystyle m\dot{v}_{1}=ma\omega^{2}-\frac{\partial U}{\partial
x}\cos\varphi-\frac{\partial U}{\partial
y}\sin\varphi,\qquad(I+ma^{2})\dot{\omega}=-ma\omega v_{1}-\frac{\partial
U}{\partial\varphi},$
$\displaystyle\dot{x}=v_{1}\cos\varphi,\qquad\dot{y}=v_{1}\sin\varphi,\qquad\dot{\varphi}=\omega,$
(17)

where $v_{1}$ is the projection of the velocity of point $O_{1}$ on the axis
$O_{1}x_{1}$, and $m$ and $I$ are, respectively, the mass and the moment of
inertia of the body, $a$ is the distance specifying the position of the knife
edge, and $U$ is the potential of the external forces.

The system (17) conserves the energy integral

$\displaystyle
E=\frac{1}{2}\big{(}mv_{1}^{2}+\big{(}I+ma^{2}\big{)}\omega^{2}\big{)}+U.$

Below we consider successively several particular cases.

###  5.2 The Chaplygin sleigh on a horizontal plane

Symmetry fields and invariant measure. In the absence of an external potential
field $U=0$ the system (17) admits the symmetry group $SE(2)$ which is
associated to the vector fields

$\displaystyle\boldsymbol{u}_{x}=\frac{\partial}{\partial
x},\qquad\boldsymbol{u}_{y}=\frac{\partial}{\partial
y},\qquad\boldsymbol{u}_{\varphi}=\frac{\partial}{\partial\varphi},$

which are symmetry fields.

Moreover, (17) possess a singular invariant measure

$\displaystyle\omega^{-1}dv_{1}d\omega dxdyd\varphi.$

Poisson bracket. We note that the system (17) is invariant under the
transformation

$\displaystyle v_{1}\rightarrow\lambda
v_{1},\qquad\omega\rightarrow\lambda\omega,\qquad
dt\rightarrow\lambda^{-1}dt,$

which is associated to the vector field

$\displaystyle\boldsymbol{u}=v_{1}\frac{\partial}{\partial
v_{1}}+\omega\frac{\partial}{\partial\omega},$

and

$\displaystyle[\boldsymbol{u},\boldsymbol{v}]=\boldsymbol{v},\qquad\boldsymbol{u}(E)=2E,$

where $\boldsymbol{v}$ is the initial vector field defined by the system (17).

Consequently, in the case $E\neq 0$ the system (17) is represented in
Hamiltonian form

$\displaystyle\dot{\boldsymbol{x}}={\bf J}^{(2)}\frac{\partial
E}{\partial\boldsymbol{x}},\qquad\boldsymbol{x}=(v_{1},\omega,x,y,\varphi),$

with the Poisson bracket

$\displaystyle{\bf
J}^{(2)}=\frac{\boldsymbol{v}\wedge\boldsymbol{u}}{2E}=\begin{pmatrix}0&\frac{a\omega}{I+ma^{2}}&-\frac{v_{1}^{2}\cos\varphi}{mv_{1}^{2}+(I+ma^{2})\omega^{2}}&-\frac{v_{1}^{2}\sin\varphi}{mv_{1}^{2}+(I+ma^{2})\omega^{2}}&-\frac{\omega
v_{1}}{mv_{1}^{2}+(I+ma^{2})\omega^{2}}\\\
*&0&-\frac{v_{1}\omega\cos\varphi}{mv_{1}^{2}+(I+ma^{2})\omega^{2}}&-\frac{v_{1}\omega\sin\varphi}{mv_{1}^{2}+(I+ma^{2})\omega^{2}}&-\frac{\omega^{2}}{mv_{1}^{2}+(I+ma^{2})\omega^{2}}\\\
*&*&0&0&0\\\ *&*&*&0&0\\\ *&*&*&*&0\end{pmatrix},$

where the asterisks denote the matrix entries resulting from the skew-symmetry
condition ${\bf J}^{(2)}$.

We examine in more detail a symplectic foliation which is defined by the
bracket ${\bf J}^{(2)}$.

First of all, we note that the manifold

$\displaystyle\mathcal{M}_{0}^{4}=\\{(\omega,v_{1},x,y,\varphi)\,|\,\omega=0\\}$

defines a Poisson submanifold with the Casimir functions

$\displaystyle C_{1}=\varphi,\qquad C_{2}=x\sin\varphi-y\cos\varphi.$

Thus, the entire phase space of the system $\mathcal{M}^{5}$ is a union of
three nonintersecting Poisson submanifolds

$\displaystyle\mathcal{M}^{5}=\mathcal{M}^{5}_{+}\cup\mathcal{M}_{0}^{4}\cup\mathcal{M}^{5}_{-},$
$\displaystyle\mathcal{M}^{5}_{+}=\\{(\omega,v_{1},x,y,\varphi)\,|\,\omega>0\\},\qquad\mathcal{M}^{5}_{-}=\\{(\omega,v_{1},x,y,\varphi)\,|\,\omega<0\\}.$

Let us consider $\mathcal{M}^{5}_{+}$ in more detail and pass from
$(v_{1},\omega)$ to (polar) coordinates $(h,\psi)$:

$\displaystyle v_{1}=Aah\cos\psi,\qquad\omega=h\sin\psi,\qquad
A^{2}=1+\frac{I}{ma^{2}}>1,\qquad E=\frac{1}{2}mA^{2}a^{2}h^{2},$

where $h\in(0,\infty)$ and $\psi\in(0,\pi)$. In the new variables
$(\psi,x,y,\varphi,h)$ the Poisson bracket becomes

$\displaystyle{\bf J}^{(2)}=\begin{pmatrix}0&0&0&0&-hA^{-1}\sin\psi\\\
0&0&0&0&hAa\cos\psi\cos\varphi\\\ 0&0&0&0&hAa\cos\psi\sin\varphi\\\
0&0&0&0&h\sin\psi\\\ *&*&*&*&0\end{pmatrix}.$ (18)

As a result, the Casimir functions (18) have the form

$\displaystyle C_{1}=A\psi+\varphi,\qquad
C_{2}=x+A^{2}a\int_{\psi^{*}}^{\psi}\tan(\xi)\cos(A\xi-C_{1})d\xi,$
$\displaystyle C_{3}=y-A^{2}a\int_{\psi^{*}}^{\psi}\tan(\xi)\sin(A\xi-
C_{1})d\xi,$ (19)

where $\psi^{*}\in(0,\pi)$.

The level surface of the Casimir function $C_{1}={\rm const}$ foliates the
domain $\mathcal{M}^{5}_{+}$ into four-dimensional surfaces. The projection of
each such surface into the three-dimensional space

$\displaystyle
S^{1}\times\mathbb{R}^{2}_{+}=\\{(\varphi,\omega,v_{1})\,|\,\omega>0\\}$

is a ruled surface (similar to a helicoid). Its equation can be represented in
the following parametric form

$\displaystyle
v_{1}=Aah\cos\left(\frac{\varphi+C_{1}}{A}\right),\qquad\omega=h\sin\left(\frac{\varphi+C_{1}}{A}\right),\qquad\varphi=\varphi.$
(20)

In the definition (20) the periodicity in $\varphi$ should, of course, be
taken into account (see Fig. 6).

Figure 6: Characteristic surface defined by the Casimir function $C_{1}$ in
the domain $\mathcal{M}^{5}_{+}$ for $C_{1}=2$ and $A=3$.

Proceeding in a similar way for $\mathcal{M}^{5}_{-}$, it is easy to check
that relations (19) also define the Casimir functions with the only difference
that $\psi^{*}\in(\pi,2\pi)$.

The second Poisson bracket. In this case it turns out that when $E\neq 0$, by
Proposition 2.6, the system (17) can be represented in conformally Hamiltonian
form

$\displaystyle\dot{\boldsymbol{x}}=(2E)^{(-1)}{\bf J}^{(4)}\frac{\partial
E}{\partial\boldsymbol{x}},$

with a Poisson bracket of rank 4 using the symmetry fields as follows:
$\widehat{\boldsymbol{u}}_{x}$ and $\widehat{\boldsymbol{u}}_{y}$

$\displaystyle{\bf
J}^{(4)}=\boldsymbol{v}\wedge\boldsymbol{u}+\boldsymbol{u}_{x}\wedge\boldsymbol{u}_{y}=\\!\\!\begin{pmatrix}0&a\omega^{3}+\frac{ma}{I+ma^{2}}\omega
v_{1}^{2}&-v_{1}^{2}\cos\varphi&-v_{1}^{2}\sin\varphi&-\omega v_{1}\\\
*&0&-v_{1}\omega\cos\varphi&-v_{1}\omega\sin\varphi&-\omega^{2}\\\
*&*&0&1&0\\\ *&*&*&0&0\\\ *&*&*&*&0\end{pmatrix}.$

######  Remark 5.1.

We note that the Jacobi identity for the matrix $\frac{{\bf J}^{(4)}}{E}$ does
not hold.

In this case, $\omega=0$ defines the Poisson submanifold ${\bf J}^{(4)}$ on
which ${\operatorname{rank}{\bf J}^{(4)}=2}$, and the Casimir functions are

$\displaystyle C_{1}=\varphi,\qquad
C_{2}=Aa(x\sin\varphi-y\cos\varphi)-h^{-1}.$

In the remaining region of the phase space $\overline{{\bf J}}$ possesses the
only Casimir function

$\displaystyle C_{1}=A\psi+\varphi,\qquad\tan\varphi=\frac{v_{2}}{Aa\omega}.$

The bracket ${\bf J}^{(4)}$ was obtained previously in the initial coordinates
$(v_{1},\omega,x,y,\varphi)$ in [13].

###  5.3 The Chaplygin sleigh on an inclined plane

Below we consider the motion of the Chaplygin sleigh on an inclined plane. We
direct the axis $Ox$ along the line of maximum slope, then

$\displaystyle U=m\mu(x+a\cos\varphi),\qquad\mu=g\sin\chi,$

where $\chi$ is the angle of inclination of the plane to the horizon. In this
case, the energy integral has the form

$\displaystyle
E=\frac{I+ma^{2}}{2}\omega^{2}+\frac{mv_{1}^{2}}{2}+m\mu(x+a\cos\varphi).$

Moreover, two symmetry fields are preserved in the system (17)

$\displaystyle\boldsymbol{u}_{x}=\frac{\partial}{\partial
x},\qquad\boldsymbol{u}_{y}=\frac{\partial}{\partial y}.$

Since $\boldsymbol{u}_{x}(E)=m\mu$, Theorem 1 applies to this system.

As a result, its equations of motion are represented in Hamiltonian form with
the Poisson bracket ${\bf J}^{(2)}$ of rank 2:

$\displaystyle{\bf
J}^{(2)}=\boldsymbol{v}\wedge\boldsymbol{u}_{x}=\begin{pmatrix}0&\omega&0&0&0\\\
*&0&-v_{1}\sin\varphi&\mu\cos\varphi-a\omega^{2}&\frac{ma}{I+ma^{2}}(\omega
v_{1}-\mu\sin\varphi)\\\ *&*&0&0&0\\\ *&*&*&0&0\\\ *&*&*&*&0\end{pmatrix},$

where $\boldsymbol{v}$ is the initial vector field defined by the system (17).

In [13] it is shown that the dynamics of this system, which is noncompact, has
a complicated, seemingly random behavior, leading to the absence of an
invariant measure and global analytic Casimir functions in this system.

##  6 Conclusion

Thus, in this paper it has been shown that the Hojman construction (using
conformal symmetry fields) allows one to obtain Poisson brackets for various
nonholonomic systems. As a rule, the resulting Poisson brackets do not possess
a maximal rank, and in the general case a smooth invariant measure and global
Casimir functions may be absent for these brackets. Nevertheless, as shown in
[3], the above-mentioned Poisson brackets can be useful to investigate
stability problems.

### Acknowledgments

Section 3 was prepared by A.V. Borisov under the RSF grant No. 15-12-20035.
Section 2 was written by I.S. Mamaev within the framework of the state
assignment for institutions of higher education. The work of I.A. Bizyaev
(Sections 4 and 5) was supported by RFBR grant No. 15-31-50172. The authors
thank A.V. Tsiganov and A.V. Bolsinov for useful discussions and the referees
for numerous comments, which have contributed to the improvement of this
paper.

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