On Integrable Perturbations of Some Nonholonomic Systems
Integrable perturbations of the nonholonomic Suslov, Veselova, Chaplygin and Heisenberg problems are discussed in the framework of the classical Bertrand-Darboux method. We study the relations between the Bertrand-Darboux type equations, well studied in the holonomic case, with their nonholonomic co...
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irk-123456789-1471512019-02-14T01:25:31Z On Integrable Perturbations of Some Nonholonomic Systems Tsiganov, A.V. Integrable perturbations of the nonholonomic Suslov, Veselova, Chaplygin and Heisenberg problems are discussed in the framework of the classical Bertrand-Darboux method. We study the relations between the Bertrand-Darboux type equations, well studied in the holonomic case, with their nonholonomic counterparts and apply the results to the construction of nonholonomic integrable potentials from the known potentials in the holonomic case. 2015 Article On Integrable Perturbations of Some Nonholonomic Systems / A.V. Tsiganov // Symmetry, Integrability and Geometry: Methods and Applications. — 2015. — Т. 11. — Бібліогр.: 41 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 37J60; 70G45; 70H45 DOI:10.3842/SIGMA.2015.085 http://dspace.nbuv.gov.ua/handle/123456789/147151 en Symmetry, Integrability and Geometry: Methods and Applications Інститут математики НАН України |
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Integrable perturbations of the nonholonomic Suslov, Veselova, Chaplygin and Heisenberg problems are discussed in the framework of the classical Bertrand-Darboux method. We study the relations between the Bertrand-Darboux type equations, well studied in the holonomic case, with their nonholonomic counterparts and apply the results to the construction of nonholonomic integrable potentials from the known potentials in the holonomic case. |
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Tsiganov, A.V. |
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Tsiganov, A.V. On Integrable Perturbations of Some Nonholonomic Systems Symmetry, Integrability and Geometry: Methods and Applications |
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Tsiganov, A.V. |
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Tsiganov, A.V. |
| title |
On Integrable Perturbations of Some Nonholonomic Systems |
| title_short |
On Integrable Perturbations of Some Nonholonomic Systems |
| title_full |
On Integrable Perturbations of Some Nonholonomic Systems |
| title_fullStr |
On Integrable Perturbations of Some Nonholonomic Systems |
| title_full_unstemmed |
On Integrable Perturbations of Some Nonholonomic Systems |
| title_sort |
on integrable perturbations of some nonholonomic systems |
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Інститут математики НАН України |
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2015 |
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http://dspace.nbuv.gov.ua/handle/123456789/147151 |
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On Integrable Perturbations of Some Nonholonomic Systems / A.V. Tsiganov // Symmetry, Integrability and Geometry: Methods and Applications. — 2015. — Т. 11. — Бібліогр.: 41 назв. — англ. |
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Symmetry, Integrability and Geometry: Methods and Applications |
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AT tsiganovav onintegrableperturbationsofsomenonholonomicsystems |
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2025-07-11T01:28:19Z |
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2025-07-11T01:28:19Z |
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1837312048201990144 |
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Symmetry, Integrability and Geometry: Methods and Applications SIGMA 11 (2015), 085, 19 pages
On Integrable Perturbations
of Some Nonholonomic Systems?
Andrey V. TSIGANOV †‡
† St. Petersburg State University, St. Petersburg, Russia
E-mail: andrey.tsiganov@gmail.com
† Udmurt State University, 1 Universitetskaya Str., Izhevsk, Russia
Received May 08, 2015, in final form October 16, 2015; Published online October 20, 2015
http://dx.doi.org/10.3842/SIGMA.2015.085
Abstract. Integrable perturbations of the nonholonomic Suslov, Veselova, Chaplygin and
Heisenberg problems are discussed in the framework of the classical Bertrand–Darboux
method. We study the relations between the Bertrand–Darboux type equations, well stu-
died in the holonomic case, with their nonholonomic counterparts and apply the results to
the construction of nonholonomic integrable potentials from the known potentials in the
holonomic case.
Key words: nonholonomic system; integrable systems
2010 Mathematics Subject Classification: 37J60; 70G45; 70H45
Dedicated to Sergio Benenti on the occasion of his 70th birthday
1 Introduction
In classical mechanics, the Euler–Poisson equations
Iω̇ = Iω × ω + γ × ∂V (γ)
∂γ
, γ̇ = γ × ω (1.1)
describe the rotation of a rigid body with a fixed point using a rotating reference frame with its
axes fixed in the body and parallel to the body’s principal axes of inertia. Here ω = (ω1, ω2, ω3) is
the angular velocity vector of the body, I = diag(I1, I2, I3) is a tensor of inertia, γ = (γ1, γ2, γ3) is
a unit Poisson vector and V (γ) is a potential field. All the vectors are expressed in the so-called
body frame and x × y means the cross product of two vectors in three-dimensional Euclidean
space.
Let us impose a nonholonomic constraint on the angular velocity
f = (ω, a) = 0 or f = (ω, γ) = 0,
where a is a fixed unit vector in the rotating frame for the Suslov problem [33], γ is a fixed unit
vector in the stationary frame for the Veselova problem [40] and (x, y) means the scalar product
of two vectors. In this case the Euler–Poisson equations (1.1) are replaced by equations
Iω̇ = I× ω + γ × ∂V (γ)
∂γ
+ λn, γ̇ = γ × ω, n = a, γ, (1.2)
where λ is a Lagrange multiplier which has to be found from the condition ḟ = 0.
?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
mailto:andrey.tsiganov@gmail.com
http://dx.doi.org/10.3842/SIGMA.2015.085
http://www.emis.de/journals/SIGMA/Benenti.html
2 A.V. Tsiganov
Both systems of differential equations (1.1) and (1.2) are geometrically interpreted in terms
of the vector field X
ẋi = Xi(x1, . . . , x6) (1.3)
in a six-dimensional manifoldM with coordinates x = (ω, γ). The classical Euler–Jacobi theorem
says that the vector field X (1.3) on a six-dimensional manifold M is integrable by quadratures
if it has an invariant volume form (invariant measure) and four functionally independent first
integrals [28, 37].
Equations (1.1) and (1.2) preserve the norm of the unit Poisson vector γ
C1 = (γ, γ) = γ21 + γ22 + γ23 = 1,
and mechanical energy
H1 =
1
2
(Iω, ω) +
1
2
V (γ).
An additive perturbations (1.2) of the Euler–Poisson vector field (1.1) change the standard
invariant volume form and the second geometric first integral
C2 = (ω, Iγ)→ C2 = (ω, n),
see details in [15]. Thus, according to the Euler–Jacobi theorem equations (1.1) and (1.2) are
integrable by quadratures if there is one more independent first integral H2.
There are several methods to uncover integrals of motion. A one of the simplest method
considers function H2 polynomial in the velocities, with the coefficients being arbitrary functions
of the coordinates, and requires that it is conserved in time
Ḣ2 = 0. (1.4)
This condition yields a system of coupled partial differential equations on the coefficients, which
are some of the most well-studied first-order PDE’s in classical mechanics [26].
In this paper, we want to study what is going on with these well-known PDE’s when we
impose nonholonomic constraints on the rigid body motion. Some partial solutions of these new
PDE’s are discussed in literature, see, e.g., [14, 20, 29] and references within. Our main aim is
to prove that these PDE’s for the nonholonomic Chaplygin, Suslov and Veselova systems can be
easily reduced to the well-studied PDE’s for the Hamiltonian vector field (1.1). Consequently,
we can directly obtain all the possible integrable perturbations of these nonholonomic systems
directly from the well-known integrable potentials of the Hamiltonian mechanics.
The necessary references to the main aspects of nonholonomic mechanics can be found in
several papers of Sergio Benenti dedicated to the analysis of nonholonomic mechanical systems
[3, 5, 6, 7]. Following in the steps of these papers we will consider nonholonomic systems using
only the knowledge of the basic notions of analytical mechanics, i.e., utilizing a ‘user-friendly’
approach to the dynamics of nonholonomic systems proposed by Sergio Benenti [5].
This paper is organized as follows. In Section 2 we will introduce Bertarnd–Darboux equation
for the holonomic particle on the plane and its counterpart for the holonomic particle on the
sphere. Section 3 contains the main result of integrable perturbations for the Suslov system.
We will show that integrable potentials for the nonholonomic Suslov problem satisfy to the
standard Bertarnd–Darboux equation for the holonomic particle in the plane. Sections 4 and 5
are devoted to the review of the known integrable potentials for the Veselova and Chaplygin
systems. We will give the Bertrand–Darboux type equations for these systems and will show
how these equations are reduced to the Bertarnd–Darboux equation for the holonomic particle
on the sphere. Section 6 deals with two nonholonomic systems on the plane and contains the
Bertrand–Darboux type equations for the nonholonomic oscillator and the Heisenberg system
(nonholonomic integrator). Finally, we briefly discuss the quasi-integrable potentials for the
Suslov and Veselova problems, which were introduced by Llibre et al.
On Integrable Perturbations of Some Nonholonomic Systems 3
2 Integrable potentials on the plane and sphere
In [9] Bertrand studied the Newton equations for a particle on the plane
d2q1
dt2
= F1,
d2q2
dt2
= F2, Fk = −∂V (q1, q2)
∂qk
(2.1)
and tried to solve equation (1.4) using linear, quadratic or fractional (linear/linear) anzats in
the velocity for the additional integral of motion.
In particular, according to Bertrand [9], if
H2 =
2∑
i,j=1
Kij(q1, q2)q̇iq̇j + U(q1, q2),
then one equation Ḣ2 = 0 yields two systems of PDE’s. The generic solution of the first system
of equations for the coefficients Kij(q1, q2)
H2 =
(
−α
2
q22 − β2q2 +
γ11
2
)
p21 +
(
−α
2
q21 − β1q1 +
γ22
2
)
p2
+ (αq1q2 + β1q2 + β2q1 + γ12)p1p2 + U(q)
depends on the six constants of integration α, β1, β2, γ11, γ12, γ22.
In order to describe the forces F1,2 that should act on the particle Bertrand also extracted
one linear second-order PDE on potential V from the coupled system of equations on the poten-
tials V and U
(αq1q2 + β1q1 + β2q2 + γ12)(∂22V − ∂11V )
+
(
αq21 − αq22 + 2β1q1 − 2β2q2 + γ11 − γ22
)
∂12V
+ 3(αq1 + β1)∂2V − 3(αq2 + β2)∂1V = 0, (2.2)
where ∂i = ∂/∂qi and ∂ik = ∂2/∂qi∂qk.
In [9] Bertand studied only some partial solutions of this equation, whereas in [17] Darboux
gave a complete solution and, therefore, now equation (2.2) is called the Bertrand–Darboux
equation. Later on Darboux results were included almost verbatim in the classical text of
Whittaker on analytical mechanics, see historical details in [32].
Ideas used by Bertarnd and Darboux to solve the Bertrand–Darboux problem were generali-
zed to study Hamiltonian systems defined in Euclidean spaces of higher dimensions and in other
(pseudo) Riemannian manifolds. According to Eisenhard [21], the first system of PDE’s is the
Killing equation for the Killing tensor of second order with vanishing Haantjes torsion, hereafter
called the characteristic Killing tensor on the Riemannian manifold Q. The second system of
PDE’s on the various Riemannian manifolds was studied by Kalnins, Miller [25], Benenti [2, 4]
etc.
In particular, Benenti formulated the following proposition.
Proposition 1. A natural Hamiltonian H =
∑
gijpipj + V on the cotangent bundle T ∗Q of
a Riemannian manifold Q is separable in orthogonal coordinates iff on Q there exists a Killing K
of second order with simple eigenvalues and normal eigenvectors, so that
d(KdV ) = 0. (2.3)
Separable Hamiltonian flow has a necessary number of first integrals, which can be directly
calculated from the characteristic Killing tensor K, which satisfies to the Killing equation
∇αKβγ +∇βKγα +∇γKαβ = 0,
4 A.V. Tsiganov
where ∇ is the Levi-Civita connection of the Riemannian metric [2, 4]. Tensor K has normal
eigenvectors if and only if its Haantjes torsion is equal to zero. Integrable systems associated with
the Killing tensors of second order with nontrivial Haantjes torsion were found only recently [38].
2.1 Bertrand–Darboux type equation on the sphere
Let us consider the standard Hamiltonian vector field describing rotation of a rigid body fixed
at the point
Ṁ = M × ω, γ̇ = γ × ω. (2.4)
Here M is the angular momentum, ω = AM is the angular velocity, γ is the constant unit vector
in a moving frame an A is the diagonal inverse to I matrix
A = I−1 =
a1 0 0
0 a2 0
0 0 a3
, ak =
1
Ik
. (2.5)
According to Euler, there are two first integrals of second order in momenta
H1 =
1
2
(M,AM), H2 = (M,M)
and two geometric integrals
C1 = γ21 + γ22 + γ23 , C2 = γ1M1 + γ2M2 + γ3M3,
which are the Casimir functions of the underlying Poisson structure.
Let us consider perturbation of the free motion (2.4) by adding forces associated with potential
field V1(γ). For the perturbed Hamiltonian vector field
Ṁ = M × ω + γ × ∂V1(γ)
∂γ
, γ̇ = γ × ω, (2.6)
functions C1,2 are also constants of motion and, therefore, we can exclude the third component
of the Poisson vector γ from other calculations
γ3 =
√
1− γ21 − γ22 .
Substituting the polynomials of second order in momenta
H1 =
1
2
(M,AM) + V1(γ), H2 = (M,M) + V2(γ) (2.7)
in the equations Ḣ1,2 = 0 one gets three partial differential equations on V1,2(γ)
∂2(a1V2 − 2V1) = 0, ∂1(a2V2 − 2V1) = 0,
a3(γ2∂1V2 − γ1∂2V2) + 2γ1∂2V1 − 2γ2∂1V1 = 0. (2.8)
Here V1,2(γ) are functions on two independent variables γ1, γ2 and ∂/∂γk = ∂k.
The generic solution of these equations (2.8)
V1 = −1
2
(
a2a3γ
2
1 + a1a3γ
2
2 + a1a2γ
2
3
)
, V2 = a1γ
2
1 + a2γ
2
2 + a3γ
2
3
is associated with the Clebsh system [26].
On Integrable Perturbations of Some Nonholonomic Systems 5
At C2 = (γ,M) = 0 the phase space is equivalent to the cotangent bundle of the two-
dimensional sphere [12]. In this case equation Ḣ2 = 0 yields only two equations(
γ22(a2 − a3)− a2
)
∂1V2 − γ1γ2(a1 − a3)∂2V2 + 2∂1V1 = 0,(
γ21(a1 − a3)− a1
)
∂2V2 − γ1γ2(a2 − a3)∂1V2 + 2∂2V1 = 0, (2.9)
from which we can easily get the equation on one potential. Namely, when equations are
differentiated by γ1,2 and subtracted from each other, the result is
∂1
((
γ22(a2 − a3)− a2
)
∂1V2 − γ1γ2(a1 − a3)∂2V2
)
− ∂2
((
γ21(a1 − a3)− a1
)
∂2V2 − γ1γ2(a2 − a3)∂1V2
)
= 0. (2.10)
It is well-known that the characteristic equation (2.3), (2.10) has a continuum of solutions
labelled by two arbitrary functions G1,2
V1 =
u2G1(u1)− u1G2(u2)
2(u2 − u1)
, V2 =
G1(u1)−G2(u2)
u2 − u1
. (2.11)
Here u1, u2 are sphero-conical coordinates on the sphere
γi =
√
(u1 − ai)(u2 − ai)
(aj − ai)(am − ai)
, i 6= j 6= m.
If pu1,2 are the corresponding momenta defined by relations
Mi =
2εijmγjγm(aj − am)
u1 − u2
(
(ai − u1)pu1 − (ai − u2)pu2
)
,
then first integrals H1,2 (2.7) satisfy to the following separation relations
4(ui − a1)(ui − a2)(ui − a3)p2ui +Gi(ui)− uiH2 + 2H1 = 0, i = 1, 2. (2.12)
If we take homogeneous polynomials of N -th order G1,2 = uN , the corresponding poten-
tials V
(N)
1,2 satisfy to the well-known recurrence relations
V
(1)
1 = 0, V
(1)
2 = 1, 2V
(N)
1 = ρV
(N−1)
2 , V
(N)
2 = σV
(N−1)
2 − 2V
(N−1)
1 ,
where
σ = (u1 + u2) = (a1 + a2 + a3)− a1γ21 − a2γ22 − a3γ23 ,
ρ = u1u2 = a2a3γ
2
1 + a1a3γ
2
2 + a1a2γ
2
3 . (2.13)
According to Bogoyavlenskii [12] these potentials are equal to
V
(N)
2 =
[N/2]∑
k=0
(−1)k
(
N − k
k
)
ρkσN−2k,
2V
(N)
1 =
[(N−1)/2]∑
k=0
(−1)k
(
N − k − 1
k
)
ρk+1σN−2k−1. (2.14)
Here [z] is an integer part of the rational number z. If G1,2(u) = u−K , one gets rational potentials
V
(−K)
2 =
1/uK1 − 1/uK2
u2 − u1
=
1
(u1u2)K
uK2 − uK1
u2 − u1
= −V
(K)
2
ρK
6 A.V. Tsiganov
and
2V
(−K)
1 =
u2/u
K
1 − u1/uK2
u2 − u1
=
1
(u1u2)K
uK+1
2 − uK+1
1
u2 − u1
= −V
(K+1)
2
ρK
Of course, any linear combination of these polynomial and rational potentials also satisfies (2.9).
For instance, at N = 2 one gets the Neumann system
H
(2)
1 =
1
2
(M,AM)− a2a3γ21 − a1a3γ22 − a1a2γ23 ,
H
(2)
2 = (M,M) + a1γ
2
1 + a2γ
2
2 + a3γ
2
3 ,
and the Braden system at K = 1
H
(−1)
1 =
1
2
(M,AM) +
a1γ
2
1 + a2γ
2
2 + a3γ
2
3 −
(
γ21 + γ22 + γ23
)
(a1 + a2 + a3)
a2a3γ21 + a1a3γ22 + a1a2γ23
,
H
(−1)
2 = (M,M) +
1
a2a3γ21 + a1a3γ22 + a1a2γ23
.
In [41] Wojciechowski presented another family of potentials associated with other symmetric
functions G1,2 (2.11) on variables u1,2 and parameters a1, a2, a3. For instance, if
F
(2)
1,2 (u) = u
(
u2 − (a1 + a2 + a3)u+ a1a2 + a1a3 + a2a3
)
,
F
(3)
1,2 (u) = u(u− a1)(u− a2)(u− a3)
then the second integrals of motion read as
H
(2)
2 = (M,M) +
∑
a2i γ
2
i −
(∑
aiγ
2
i
)2
,
H
(3)
2 = (M,M) +
∑
a3i γ
2
i − 2
(∑
aiγ
2
i
)(∑
a2kγ
2
k
)
+
(∑
aiγ
2
i
)3
.
More complicated functions G1,2 (2.11) yield more complicated potentials, for example, rational
functions
G1,2(u) =
3∑
i=1
bibju+
(a1 − a3)(a1 − a2)b21
u− a1
+
(a2 − a3)(a2 − a1)b22
u− a2
+
(a3 − a1)(a3 − a2)b23
u− a3
give rise to the Rosochatius potentials [31]
H
(Ros)
1 =
1
2
(M,AM) +
b21
(
a2γ
2
3 + a3γ
2
2
)
γ21
+
b22
(
a1γ
2
3 + a3γ
2
1
)
γ22
+
b23(a1γ
2
2 + a2γ
2
1)
γ23
,
H
(Ros)
2 = (M,M) +
b21
γ21
+
b22
γ22
+
b23
γ23
up to the constant terms, see also [24].
In [19] Dragović considered functions G1,2 (2.11), which are the Laurent polynomials, and
ingeniously coupled the corresponding potentials V1,2 together with the Appell hypergeometric
function.
These familiar well-studied potentials are permanently rediscovered both in holonomic [24, 39]
and nonholonomic mechanics [22, 29].
On Integrable Perturbations of Some Nonholonomic Systems 7
3 Suslov problem
One of the most widely known mechanical nonholonomic systems is the Suslov problem descri-
bing motion of a rigid body under the following constraint on its angular velocity
(ω, a) = 0, (3.1)
where a is a fixed unit vector in the body frame [33]. It means that there is no twisting around
this vector a.
Imposing this constraint we have to add some terms with the Lagrangian multiplier to the
initial Hamiltonian vector field
Iω̇ = I× ω + γ × ∂V1(γ)
∂γ
+ λa, γ̇ = γ × ω. (3.2)
Differentiating the constraint (3.1) by time and using the equation of motion we obtain
λ =
1(
I−1a, a
) (I−1a, I× ω +
1
2
γ × ∂V1(γ)
∂γ
)
.
Vector field (3.2) preserves the mechanical energy
H1 = (M,AM) + V1(γ) (3.3)
and the geometric constants of motion
C1 = (γ, γ) = 1, C2 = (ω, a) = 0,
which allows us to remove the redundant variable from the calculations
γ3 =
√
1− γ21 − γ22 .
If we assume that a is an eigenvector of the tensor of inertia [14, 20, 29], i.e., that tensor of
inertia is diagonal
I =
I1 0 0
0 I2 0
0 0 I3
,
and vector a is equal to a = (0, 0, 1) in some coordinate frame, the constraint is trivial
ω3 = 0.
Substituting the standard anzats for the second integral of motion
H2 = f1(γ)ω2
1 + f2(γ)ω2
2 + f3(γ)ω1ω2 + V2(γ)
with unknown functions fk and V2 on the components γ1,2 of the Poisson vector in the equation
Ḣ2 = 0 we obtain the well-known Bertrand–Darboux equation on potential [9, 17].
Proposition 2. For vector field (3.2) the following statements are equivalent:
1. There is an additional independent first integral of second order in velocities
H2 =
(
α
√
I1I2
2
γ21 + β1
√
I1γ1 +
γ11
√
I1
2
√
I2
)
ω2
1
+
(
α
√
I1I2
2
γ22 + β2
√
I2γ2 +
γ22
√
I2
2
√
I1
)
ω2
2
+
(
α
√
I1I2γ1γ2 + β1
√
I1γ2 + β2
√
I2γ1 + γ12
)
ω1ω2 + V2(γ).
8 A.V. Tsiganov
2. Potential V1 satisfies the Bertrand–Darboux equation (2.2) with
q1 = γ1
√
I2, q2 = γ2
√
I1. (3.4)
3. Potential V1 is separable. A characteristic coordinate system for the Bertrand–Darboux
equation provides separation for V1 and can be taken as one of the following four orthogonal
coordinate systems on the q1,2-plane: elliptic, parabolic, polar or Cartesian.
The proof is completely similar to the one for the original Bertrand–Darboux theorem [17, 32].
This result allows us to suppose that the nonholonomic Suslov system is equivalent to the
holonomic motion on the plane with coordinates q1, q2 after some singular change of time, but
its study is out of the framework of the present note.
Another reduction to the Bertrand–Darboux equation was proposed in [29]. Integrable vector
field remains integrable for any fixed value of mechanical energy H1, for instance on the zero-
energy hypersurface
H1 = 0.
In the Hamiltonian case the separation of variables of this null Hamilton–Jacobi equation is
equivalent to the ordinary separation of the image of the original Hamiltonian under a generalized
Jacobi–Maupertuis transformation [8].
If we substitute potential
V1 = −1
2
(
mu21
I2
+
mu22
I1
)
and velocities
ω1 =
mu2
I1
, ω2 = −µ1
I2
, ω3 = 0
from Theorem 1 in [29] into the first integral H1 (3.3), one gets H1 = 0. Here µ1,2 are functions
on the components of the Poisson vector γ1,2, which satisfy equations (3.2)
γ̇1 = −
√
1− γ21 − γ22
µ1
I2
, γ̇2 = −
√
1− γ21 − γ22
µ2
I1
.
Differentiating these equations with respect to time we obtain the Newton equations (2.1)
on q1,2 (3.4) with forces labelled by two functions µ1,2(q1, q2). Thus, on the zero-energy hy-
persurface of the initial Hamiltonian one gets an initial Bertrand problem with the well-known
solution. Of course, on this zero-energy hypersurface we can find other solutions associated
with nonitegrable potentials on the whole phase space. In [29] such potentials were called quasi-
implicitly integrable or locally integrable potentials.
4 Veselova system
Let us consider the nonholonomic Veselova system describing the motion of a rigid body under
the following constraint
(ω, γ) = 0, (4.1)
where γ is a unit Poisson vector fixed in space [40]. It means that there is no twisting around
vector γ.
On Integrable Perturbations of Some Nonholonomic Systems 9
According to [40] this constraint shifts the initial Hamiltonian vector field (2.4)
Ṁ = M × ω + λγ, γ̇ = γ × ω, (4.2)
where the Lagrangian multiplier λ is chosen so that the constraint (4.1) is satisfied at any time
λ =
(AM ×M,Aγ)
(Aγ, γ)
.
There are integrals of motion of second order in momenta
H1 =
1
2
(M,AM), H2 = (M,M)− (γ, γ)−1(γ,M)2
and two geometric constants of motion
C1 = (γ, γ), C2 = (γ, ω) = 0.
In the presence of the potential field equations of motion (4.2) become
Ṁ = M × ω + λγ + γ × ∂W1(γ)
∂γ
, γ̇ = γ × ω, (4.3)
where
λ =
(AM ×M + γ × ∂W1(γ)/∂γ,Aγ)
(Aγ, γ)
.
As usual, functions C1,2 remain constants of motion, and we can exclude the redundant variable
γ3 =
√
1− γ21 − γ22 .
Vector field (4.3) is a conformally Hamiltonian field, see, for instance, [15, 36]. Substituting the
following anzats for integrals of motion
H1 =
1
2
(M,AM) +W1(γ), H2 = (M,M)− (γ, γ)−1(γ,M)2 +W2(γ) (4.4)
in Ḣ1,2 = 0 one gets two first-order equations on potentials W1,2
2
(
γ22
(
a−12 − a
−1
3
)
− a−12
)
∂1W1 − 2γ1γ2
(
a−11 − a
−1
3
)
∂2W1 + ∂1W2 = 0,
2
(
γ21
(
a−11 − a
−1
3
)
− a−11
)
∂2W1 − 2γ1γ2
(
a−12 − a
−1
3
)
∂1W1 + ∂2W2 = 0. (4.5)
Proposition 3. After the inversion of parameters ak → a−1k and substitution
W2 = 2V1, W1 =
V2
2
equations (4.5) coincide with equations (2.9) for potentials on the two-dimensional sphere.
Thus, all the integrable potentials for the nonholomic Veselova system are easily expressed via
well-known integrable potentials V1,2 for the holonomic system on the two-dimensional sphere.
For functions G1,2 (2.11) which are the Laurent polynomials in u1,2 these expressions were found
in [20]. We only want to note that it is true for any integrable potentials.
Following Theorem 2 in [29] let us fix the values of velocities by equations
I1ω1γ2 − I2ω2γ1 −Ψ2 = 0, pω3 −Ψ1 = 0, ω1γ1 + ω2γ2 + ω3γ3 = 0,
10 A.V. Tsiganov
where p =
√
I1I2I3
(
γ21
I1
+
γ22
I2
+
γ23
I3
)
and Ψ1,2(γ) are functions on γ. Substituting these velocities
and potential
W1 = − Ψ2
1 + Ψ2
2
2(I1γ22 + I2γ21)
into the mechanical energy H1 (4.4) one gets H1 = 0.
The remaining three equations of motion of the components of the Poisson vector γ are easily
reduced to equations of motion for the holonomic particle on the sphere with forces labelled by
two functions Ψ1,2(γ). Thus, on the zero-energy hypersurface of the initial Hamiltonian one gets
standard characteristic equation (2.10) with the well-known solutions.
5 Chaplygin ball
As in [16] we consider the rolling of a dynamically balanced ball on a horizontal absolutely rough
table without slipping or sliding. ‘Dynamically balanced’ means that the geometric center
coincides with the center of mass, but mass distribution is not assumed to be homogeneous.
Because of the roughness of the table this ball cannot slip, but it can turn about the vertical
axis without violating the constraints.
After reduction [16] motion of the Chaplygin ball is defined by the following vector field
Ṁ = M × ω, γ̇ = γ × ω. (5.1)
Here M is the angular momentum of the ball with respect to the contact point, ω is the angular
velocity vector of the rolling ball. Its mass, inertia tensor and radius will be denoted by m,
I = diag(I1, I2, I3) and b respectively. All the vectors are expressed in the so-called body frame,
which is firmly attached to the ball, and its axes coincide with the principal inertia axes of the
ball.
The angular velocity vector is equal to ω = AgM , here matrix
Ag = A + dg(γ)Aγ ⊗ γA
is defined by the nondegenerate matrix A (2.5) and function
g(γ) =
1
1− d
(
a1γ21 + a2γ22 + a3γ23
) , d = mb2. (5.2)
It is easy to prove that vector field (5.1) preserves two polynomials of second order in momenta
H1 =
1
2
(M,AgM), H2 = (M,M)
and two geometric constants of motion
C1 = γ21 + γ22 + γ23 = 1, C2 = γ1M1 + γ2M2 + γ3M3,
see details in [13, 34, 37].
Indeed, equations of motion of the ball in the potential field
Ṁ = M × ω + γ × ∂U1(γ)
∂γ
, γ̇ = γ × ω (5.3)
have the same form as the equations (2.6) in rigid body dynamics. In fact, the principal difference
between holonomic and nonholonomic systems is hidden within the relation of the angular
velocity to the angular momentum.
On Integrable Perturbations of Some Nonholonomic Systems 11
According to [35, 36], integrals of motion for the Veselova system are expressed via the
integrals of motion for the Chaplygin ball. So, we can easily express integrable potentials for
the Chaplygin system via integrable potentials for the Veselova system W1,2 (4.5) and then via
integrable potentials V1,2 (2.9) for the holonomic system on the two-dimensional sphere. Of
course, we can get the same result by directly substituting standard ansatz
H1 =
1
2
(M,AgM) + U1(γ), H2 = (M,M) + U2(γ) (5.4)
in Ḣ1,2 = 0 we obtain
a1(a2 − a3)γ1γ2∂1U2 + a1
(
(a2 − a3)γ22 + a3 − 1
)
∂2U2 + 2g−1∂2U1 = 0,
a2
(
(a1 − a3)γ21 + a3 − 1
)
∂1U2 + a2(a1 − a3)γ1γ2∂2U2 − 2g−1∂1U1 = 0,
a3γ2
(
(a1 − a2)γ1 + a2 − 1
)
∂1U2 + a3γ1
(
(a1 − a2)γ22 − a1 + 1
)
∂2U2
+ 2g−1
(
γ1∂2U1 − γ2∂1U1
)
= 0.
This system of equations has only one solution
U1 = −1
2
(
a2a3γ
2
1 + a1a3γ
2
2 + a1a2γ
2
3
)
, U2 = a1γ
2
1 + a2γ
2
2 + a3γ
2
3 ,
which coincides with the single solution of the initial system (2.8) associated with the Clebsch
model. This integrable potential has been found in [27].
At C2 = 0 conditions Ḣ1,2 = 0 are thus
gγ1γ2
(
a2(a1 − a3)γ21 + a1(a2 − a3)γ22 − a1a2 + (a1 + a2 − 1)a3
)
∂1U2
− g
(
a1
(
x21 + x22 − 1
)(
a2x
2
2 − 1
)
− a3
(
a1
(
x42 − 1
)
+ x21
(
a2x
2
2 − 1
)))
∂2U2
− 2γ1γ2∂1U1 − 2(γ22 − 1)∂2U1 = 0,
g
(
a2
(
x21 + x22 − 1
)(
a1x
2
1 − 1
)
− a3
(
a2
(
x41 − 1
)
+ x22
(
a1x
2
1 − 1
)))
∂1U2
− gγ1γ2
(
a2(a1 − a3)γ21 + a1(a2 − a3)γ22 − a1a2 + (a1 + a2 − 1)a3
)
∂2U2
+ 2(x21 − 1)∂1U1 + 2γ1γ2∂2U1 = 0. (5.5)
Here g ≡ g(γ) is the function defined by (5.2). If we change the parameters
e1 =
a1
1− a1
, e2 =
a2
1− a2
, e3 =
a3
1− a3
and substitute in (5.5)
2U1 =
(
e2e3γ
2
1 + e1e3γ
2
2 + e1e2γ
2
3
)
V2 + 2V1,
U2 = d
(
1 + (e1 + e2 + e3)− e1γ21 − e2γ22 − e3γ23 + e2e3γ
2
1 + e1e3γ
2
2 + e1e2γ
2
3
)
V2, (5.6)
then the equations (5.5) become(
γ22(e2 − e3)− e2
)
∂1V1 − γ1γ2(e1 − e3)∂2V2 + 2∂1V1 = 0,(
γ21(e1 − e3)− e1
)
∂2V1 − γ1γ2(e2 − e3)∂1V2 + 2∂2V1 = 0.
It is easy to see that this system coincides with the initial system of equations (2.9) defining
integrable potentials on the sphere up to ak → ek.
Thus, for the Chaplygin ball imposition of the nonholonomic constraint leads to deformation
of potentials (5.6) and to replacement of parameters ak → ek.
12 A.V. Tsiganov
Proposition 4. At C2 = 0 conformally Hamiltonian vector field (5.3) has two integrals of
motion (5.4) with potentials
2U1 = ρV2 + 2V1, U2 = d(ρ+ σ + 1)V2.
Here V1,2 are integrable potentials on the sphere (2.11) after replacement of parameters ak → ek,
and
σ =
(
γ21 + γ22 + γ23
)
(e1 + e2 + e3)− e1γ21 + e2γ
2
2 + e3γ
2
3 ,
ρ = e2e3γ
2
1 + e1e3γ
2
2 + e1e2γ
2
3
are the same polynomials of second order in variables γ as above (2.13).
Following to S.A. Chaplygin [16] we can introduce the sphero-conical coordinates u1, u2
γi =
√
(u1 − ei)(u2 − ei)
(ej − ei)(em − ei)
, i 6= j 6= m,
and explicitly present some solutions of the Bertrand–Darboux type equations (5.5). Namely,
integrals of motion H1,2 (5.4) satisfy the separation relations
4(e1 − ui)(e2 − ui)(e3 − ui)
d(e1 + 1)(e2 + 1)(e3 + 1)
p2ui +Gi(ui)−
uiH2
d(ui + 1)
+ 2H1 = 0, i = 1, 2,
which can be considered as gentle deformation of the initial relations (2.12). Thus, separable
potentials in this case read as
U2 =
d(u1 + 1)(u2 + 1)(G1(u1)−G2(u2))
u2 − u1
,
U1 =
u1u2(G1(u1)−G2(u2)) + u2G1(u1)− u1G2(u2)
2(u2 − u1)
.
The passage to limit d → 0 reduces equations of motion for the Chaplygin ball (5.3) to the
standard Euler–Poisson equations. However, at d → 0 we have to simultaneously change the
definition of the second potential U2 in (5.6) and, therefore, we present another family of solu-
tions for equations (5.5).
Let us introduce variables v1,2
γi =
√
(1− daj)(1− dam)
(1− dv1)(1− dv2)
·
√
(v1 − ai)(v2 − ai)
(aj − ai)(am − ai)
, i 6= j 6= m,
and the conjugated momenta pv1,2 , see [34] for details. In this variables the separated relations
have the following form
4(1− dvi)(vi − a1)(vi − a2)(vi − a3)p2vi + Ui(vi) + viH2 − 2H1 = 0, i = 1, 2,
and integrable potentials
U2 =
G1(v1)−G2(v2)
v2 − v1
, U1 =
v2G1(v1)− v1G2(v2)
2(v2 − v1)
are the same functions on variables v1,2 as the integrable potentials on the sphere (2.11).
On Integrable Perturbations of Some Nonholonomic Systems 13
Proposition 5. At C2 = 0 vector field (5.3) has integrals of motion of second order in veloci-
ties (5.4) if potentials U1,2 have the same form as integrable potentials on the sphere (2.14)
U
(N)
2 =
[N/2]∑
k=0
(−1)k
(
N − k
k
)
%kςN−2k,
2U
(N)
1 =
[(N−1)/2]∑
k=0
(−1)k
(
N − k − 1
k
)
%k+1ςN−2k−1,
and
U
(−K)
2 = −U
(K)
1
%K
, 2U
(−K)
1 = −U
(K+1)
1
%K
.
Of course, any linear combination of these polynomial and rational potentials also satisfies the
equations (5.5). These potentials differ from (2.14) by replacement of polynomials σ and ρ for
the following functions
ς = g(γ)
(
σ + d
(
a1(a2 + a3)γ
2
1 + a2(a1 + a3)γ
2
2 + a3(a1 + a2)γ
2
3
))
,
% = g(γ)
(
ρ+ da1a2a3
(
γ21 + γ22 + γ23
))
,
which at d = 0 become initial polynomials (2.13).
For instance, at N = 2 we have the following analogue of the Neumann system
H
(2)
1 =
1
2
(M,AM) + %, H
(2)
2 = (M,M) + ς,
and at K = 1 the following counterpart of the Braden system
H
(−1)
1 =
1
2
(M,AM)− ς
%
, H
(−1)
2 = (M,M)− 1
%
.
Of course, we can also single out other families of solutions of the equations (5.5), for instance,
see [20].
6 Nonholonomic oscillator and Heisenberg system
Let us consider the Lagrangian of the particle in Euclidean space R3
L =
m
2
(
ẋ2 + ẏ2 + ż2
)
− V (x, y, z), (6.1)
where m is the mass of the particle. When this system is subject to a nonholonomic constraint,
the resulting mechanical system may or may not preserve energy and the phase space volume,
and their integrability and reduction theories are completely different from the Hamiltonian
case [1, 18, 23]. In this Section we consider two first-order nonholonomic constraints, which
displays all the basic properties of first-order nonholonomic systems in the control theory [11].
The first nonholomic constraint and potential in (6.1) for the so-called nonholonomic oscil-
lator have the following form
f = ż − kyẋ = 0, V =
y2
2
k ∈ R,
14 A.V. Tsiganov
whereas second constraint and potential in (6.1) for the so-called Heisenberg system read as
f = ż − (yẋ− xẏ) = 0, V = 0.
The Heisenberg system (nonholonomic integrator) can be pointed out as a benchmark example
of nonholonomic system with a first-order nonintegrable constraint, which mimics the kinematic
model of a wheeled mobile robot of the unicycle type.
In generic case at V (x, y, z) = V (x, y) in (6.1) the third degree of freedom also decouples from
the rest of the system and after nonholonomic reduction we obtain a two-degrees of freedom
system of the Chaplygin type [11, 23, 30]. For the such generalized nonholonomic oscillator the
reduced equations of motion are
ẋ =
px
m
, ẏ =
py
m
, ṗx = − 1
1 + k2y2
(
k2ypxpy + ∂xV
)
, ṗy = −∂yV. (6.2)
For the generalized Heisenberg system the reduced equations of motion read as
ẋ =
px
m
, ẏ =
py
m
,
ṗx = −(x2 + 1)∂xV + xy∂yV
m(1 + x2 + y2)
, ṗy = −(y2 + 1)∂yV + xy∂xV
m(1 + x2 + y2)
. (6.3)
Below we will study potentials V (x, y) in (6.2) and (6.3), so that the corresponding four-
dimensional vector fields X have an additional first integral and possess an invariant volume
form. Of course, equations on these potentials have the form of the characteristic equation (2.3)
and can be considered as an analogue of the Bertrand–Darboux equation (2.2).
6.1 The generalised nonholonomic oscillator
The vector field for the reduced nonholonomic oscillator (6.2) after the following change of
variables
p1 =
√
k2y2 + 1px, p2 =
py√
k2y2 + 1
, q1 = x, q2 = y
becomes the conformally Hamiltonian vector field
X = −µPdH1, P =
(
0 I
−I 0
)
with respect to the canonical Poisson bivector P and reduced Hamiltonian
H1 =
2∑
i,j=1
gijpipj + V (q1, q2) =
p21
2m
+
p22(k
2q22 + 1)
2m
+ V (q1, q2).
Conformal factor
µ =
1√
k2q22 + 1
is a nowhere vanishing smooth function on an open dense subset of the plane q2 6= ∞, which
defines an invariant volume form Ω̂ = µdq ∧ dp.
Substituting a linear function in velocities
H2 = g1(q1, q2)p1 + g2(q1, q2)p2
On Integrable Perturbations of Some Nonholonomic Systems 15
into the equation Ḣ2 = 0 one gets
g1 = c1 + c2 ln
(
kq2 +
√
k2q22 + 1
)
, g2 = −
√
k2q22 + 1(c2kq1 − c3)
and
V (q1, q2) = G
(
−
c1 ln
(
kq2 +
√
k2q22 + 1
)
k
− c2
(
kq21
2
+
∫
ln
(
kq2 +
√
k2q22 + 1
)√
k2q22 + 1
)
− c3q1
)
.
If we want to consider a single valued integral H2, we have to put c2 = 0 and
c3 = 0, V = G(q1) or c1 = 0, V = G(q2).
Substituting polynomials of second order in velocities
H2 =
2∑
i,j=1
Kij(q1, q2)pipj + U(q1, q2),
where Kij and U are single valued functions on q1, q2 in the equation Ḣ2 = 0, we obtain the
following expression for the second integral of motion
H2 =
c1
(
k2q22 + kq2
√
k2q22 + 1 + 1
)
kq2 +
√
k2q22 + 1
p1p2 + c2
(
k2q22 + 1
)
p22 + U(q1, q2),
and the following counterpart of the Bertrand–Darboux equation
c1√
k2q22 + 1
((
k2q22 + 1
)
∂22V + k2q2∂2V − ∂11V
)
− 2c2∂12V = 0. (6.4)
This equation has one physical and one formal solution
c1 = 0, V = G1(q1) +G2(q2) and c2 = 0, V = G1(q+) +G2(q−),
where
q± = q1 ±
ln
(
kq2 +
√
k2q22 + 1
)
k
.
Thus, for the nonholonomic oscillator we obtain only trivial perturbations in the framework of
the Bertrand–Darboux method.
6.2 The generalized Heisenberg system
The vector field for the reduced Heisenberg system (6.3) after the following change of variables
p1 =
m
(
(1 + y2)px − xypy
)
1 + x2 + y2
, p2 =
m
(
(1 + x2)py − xypx
)
1 + x2 + y2
, q1 = x, q2 = y
is conformally Hamiltonian vector field
X = −µPdH1, P =
(
0 I
−I 0
)
(6.5)
with respect to canonical Poisson bivector P and reduced Hamiltonian
H1 =
2∑
i,j=1
gijpipj + V (q1, q2) =
q21 + q22 + 1
2m
(
p21 + p22 + (q1p1 + q2p2)
2
)
+ V (q1, q2).
16 A.V. Tsiganov
Conformal factor
µ =
(
1 + q21 + q22
)−1
is a nowhere vanishing smooth function on an open dense subset of the plane q1,2 6= ∞, which
defines an invariant volume form Ω̂ = µdq ∧ dp.
Substituting linear function in velocities
H2 = g1(q1, q2)p1 + g2(q1, q2)p2
into the equation Ḣ2 = 0 one gets the following first integral
H2 =
(
p1q
2
1 + p2q1q2 + p1
)
c1 +
(
p1q1q2 + p2q
2
2 + p2
)
c2 + (p1q2 − p2q1)c3, ck ∈ R,
and potential
V = G
(
(c1q1 + c2q2)
2 + 2c3(c1q2 − c2q1) + c21 + c22 − c23
(c1q2 − c2q1 − c3)2
)
depending on the arbitrary function G.
Substituting polynomials of second order in velocities
H2 =
2∑
i,j=1
Kij(q1, q2)pipj + U(q1, q2)
into the equation Ḣ2 = 0 one gets the following expression for the second integral of motion
H2 = (p1q2 − p2q1)(p1q21 + p2q1q2 + p1)c1 +
(
p1q
2
1 + p2q1q2 + p1
)(
p1q1q2 + p2q
2
2 + p2
)
c2
+ (p1q2 − p2q1)
(
p1q1q2 + p2q
2
2 + p2
)
c3 − (p1q2 − p2q1)2c4
+
(
q22
(
q21 + 1
)
p21 + 2q32q1p1p2 +
(
q42 + q21 + 2q22 + 1
)
p22
)
c5
+
((
q41 + 2q21 + q22 + 1
)
p21 + 2q31q2p1p2 + q21
(
q22 + 1
)
p22
)
c6 + U(q1, q2).
In this case equation (2.3) looks like
A∂11V + 2B∂12V + C∂22V +
1
1 + q21 + q22
(
a∂1V + b∂2V
)
= 0,
where A, B, C are the polynomials of second order in q1,2
A =
(
q21 + 1
)
(q1c1 − c2) + q2
(
q21 − 1
)
c3 + 2q1q2(c6 − c4),
B = q2
(
q21 + 1
)
c1 + q1
(
q22 + 1
)
c3 +
(
q21 − q22
)
c4 −
(
q21 + 1
)
c5 +
(
q22 + 1
)
c6,
C =
(
q22 + 1
)
(q2c3 + c2) + q1
(
q22 − 1
)
c1 + 2q1q2(c4 − c5),
and a, b are the polynomials of fourth order
a =
(
2q41 + 2q21q
2
2 + 5q21 − q22 + 3
)
c1 + q1
(
q21 − 3q22 + 1
)
c2 + 2q1q2
(
q21 + q22 + 3
)
c3
− 2q2
(
q21 + q22 + 3
)
c4 + 4q2
(
q21 + 1
)
c5 − 2q2
(
q21 − q22 − 1
)
c6,
b = 2q1q2
(
q21 + q22 + 3
)
c1 + q2
(
3q21 − q22 − 1
)
c2 +
(
2q21q
2
2 + 2q42 − q21 + 5q22 + 3
)
c3
+ 2q1
(
q21 + q22 + 3
)
c4 − 2q1
(
q21 − q22 + 1
)
c5 − 4q1
(
q22 + 1
)
c6.
Following Darboux [17] we can find the canonical form of the corresponding Killing tensor
and a few families of solutions to this equation. For instance, if c6 = 1 and other constants of
On Integrable Perturbations of Some Nonholonomic Systems 17
integration are equal to zero, then solutions of the equation Ḣ2 = 0 are labelled by two arbitrary
functions G1,2
V (q1, q2) =
q21 + q22 + 1
2m
G1(q2) +
1
2m
G2
(
q22 + 1
q21
)
,
U(q1, q2) = q21G1(q2) +G2
(
q22 + 1
q21
)
.
If c4 = 1 and other constants of integration are equal to zero, then we have solution
V (q1, q2) = G1(r)−
r2 + 1
2mr2
G2(ϕ), U(q1, q2) = G2(ϕ)
associated with polar coordinates on the plane
q1 = r cosϕ, q2 = r sinϕ.
Here G1,2 are arbitrary functions. In similar manner we can get solutions associated with
parabolic and elliptic coordinates on the plane, but they are bulky and, therefore, we do not
present these solution explicitly.
In order to get these solutions we can also use the Birkhoff method. Namely, let us consider
a general natural system of two degrees of freedom described in certain generalized coordinates
by the following Lagrangian
L =
2∑
i,j=1
gij(q)
dqi
dt
dqj
dt
− V (q1, q2).
According to Birkhoff [10] using change of time t → τ and coordinates (q1, q2) → (x, y) this
Lagrangian can always be reduced to the form of
L =
(
dx
dτ
)2
+
(
dy
dτ
)2
− U(x, y).
Thus, taking the well-known solutions U(x, y) of the classical Bertrand–Darboux equation (2.2)
and applying the inverse Birkhoff transformation we are able to obtain integrable potentials
V (q1, q2) for the two-dimensional holonomic system with nonstandard metric.
We can apply this method for the given nonholonomic case because the corresponding vec-
tor field X (6.5) is a conformally Hamiltonian vector field [30, 36], i.e., it can be reduced to
a Hamiltonian vector field by changing of time. Recall, that equation Ḣ2 = 0 (1.4) is invariant
with respect to change of time, which we have to use both in the Birkhoff method and in the
reduction of the conformally Hamiltonian vector field to the Hamiltonian one.
7 Conclusion
In this paper, we consider perturbations of the five well-known two-dimensional nonholonomic
systems, which are integrable by the Euler–Jacobi theorem. We show that the Bertrand–
Darboux method is applicable to these systems and that all the obtained Bertrand–Darboux
type equations in nonholonomic case can be reduced to the Bertrand–Darboux type equations
in holonomic case. Consequently, we can directly obtain all the possible integrable potentials for
these nonholonomic systems directly from the well-known integrable potentials of the Hamilto-
nian mechanics.
18 A.V. Tsiganov
Acknowledgements
We are greatly indebted B. Jovanović and the anonymous referees for a relevant contribution
to improve the paper. The work on the revised, final version of this paper was supported by
Russian Science Foundation (project No 15-12-20035).
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1 Introduction
2 Integrable potentials on the plane and sphere
2.1 Bertrand-Darboux type equation on the sphere
3 Suslov problem
4 Veselova system
5 Chaplygin ball
6 Nonholonomic oscillator and Heisenberg system
6.1 The generalised nonholonomic oscillator
6.2 The generalized Heisenberg system
7 Conclusion
References
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