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V. V. KOZLOV
Department of Mechanics and Mathematics Moscow State University, Vorob'ievy Gory 119899, Moscow, Russia

ON THE INTEGRATION THEORY OF EQUATIONS OF NONHOLONOMIC MECHANICS

*

DOI: 10.1070/RD2002v007n02ABEH000203

The paper deals with the problem of integration of equations of motion in nonholonomic systems. By means of well-known theory of the differential equations with an invariant measure the new integrable systems are discovered. Among them there are the generalization of Chaplygin's problem of rolling nonsymmetric ball in the plane and the Suslov problem of rotation of rigid body with a fixed point. The structure of dynamics of systems on the invariant manifold in the integrable problems is shown. Some new ideas in the theory of integration of the equations in nonholonomic mechanics are suggested. The first of them consists in using known integrals as the constraints. The second is the use of resolvable groups of symmetries in nonholonomic systems. The existence conditions of invariant measure with analytical density for the differential equations of nonholonomic mechanics is given.

1. Intro duction
The integration theory of equations of motion for mechanical systems with nonholonomic constraints isn't so complete as in the case of systems with holonomic constraints. This fact has many reasons. First, the equations of nonholonomic mechanics have a more complex structure than the Lagrange equations, which describ e the dynamics of systems with integrable constraints. For example, a nonholonomic system can't b e characterized by the only function of its state and time (cf. [1], ch. XXIV). Second, the equations of nonholonomic mechanics have no invariant measure in the general case (a simple example is given in section 5). The p oint is that nonholonomic constraints may b e realized by action of complementary forces of viscous anisotropic friction with a large viscosity co efficient ([3]). The absence of an invariant measure is a characteristic prop erty of systems with friction. In limit, the anisotropic friction is compatible with the conservation of total energy. But asymptotically stable equilibriums or limit cycles may arise on the manifolds of energy levels (cf. [4]), and this is the reason for nonexistence of additional "regular" integrals of motion. The most p opular metho d to integrate the equations of nonholonomic dynamics is based on the use of the available first integrals or the "conservation laws": if a Lie group that acts on a p osition space preserves the Lagrangian and if vector fields that generate this group are the fields of p ossible velo cities then the equations of motion have the first "vector" integral or the generalized integral of kinetic moment [5,6]. A numb er of problems of nonholonomic dynamics was solved by this metho d, among them, we note esp ecially Chaplygin's problem on an asymmetrical ball rolling over a horizontal plane [5]. Attempts to generalize the Hamilton ­ Jacobi metho d to the systems with nonholonomic constraints were non-effective as well as attempts to present the equations of nonholonomic dynamics in the form of Hamiltonian canonical equations. It turned out that with the help of the Hamilton ­ Jacobi


*

Mathematics Sub ject Classification 37J60, 37J35 Advances in mechanics USSR, V. 8, 3, pp. 85­107, 1985.

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generalized metho d it is p ossible to find at most only some sp ecial solutions of the motion equations. This pap er contains the detailed analysis of these questions. Another general approach to the integration of nonholonomic equations is based on the theory of Chaplygin's reducing factor ([5]): one try to obtain a change of time (different along different tra jectories), such that the equations of motion are presented as Lagrange or Hamilton equations. Though such change exists in exceptional cases only, it allows to solve a numb er of new problems of nonholonomic dynamics (cf. [5]). Let us note that the equations of motion sometimes may b e reduced to the Hamiltonian form by other reasoning (see section 5). The list of exactly solvable problems of nonholonomic dynamics isn't long: almost complete information may b e found in the b o oks [1,5,8]. In this work we present some new integrable problems, consider the characteristic features of b ehavior of nonholonomic systems' tra jectories in the phase space, and prop ose some general theoretical reasonings on metho ds of integrating the equations of nonholonomic dynamics.

2. Differential equations with an integrable measure
Let us consider a differential equation x = f (x), xR
n

(2.1)

and let g t b e its phase flow. Supp ose (2.1) has an integral invariant with some smo oth density M (x), i.e. for any measurable domain D R n the following equation holds for all t M (x) dx =
g t (D ) D

M (x) dx.

(2.2)

Let us recall the well-known Liouville assertion: a smo oth function f : R n R is a density of an invariant M (x) dx if and only if div(M f ) 0. If M (x) > 0 for all x then (2.2) defines a measure in R n invariant with resp ect to the action of f . The existence of an invariant measure simplifies the integration of a differential equation; for example, in the case of n = 2 the equation is always integrable in quadratures. According to Euler, M is also referred to as an integrating factor. Theorem 1. Suppose system (2.1) F1 , . . . , Fn-2 . Suppose the functions F1 , = x Rn : Fs (x) = cs , 1 s n - 2 . Then 1) the solutions of (2.1) that belong to compact component of the level set E c and f 2) Lc is a smooth surface diffeomorphic 3) it is possible to choose angle variables system (2.1) on Lc would have the fol lowing x= with invariant measure (2.2) has n - 2 first integrals . . . , Fn-2 are independent on the invariant set E c =

E c may be found by quadratures. If Lc is a connected = 0 on Lc then to a two-dimensional torus, x, y mo d 2 on L c so that, after the change of variables, form y= µ , (x, y ) (2.3)

, (x, y )

where , µ = const, || + |µ| = 0 and is a smooth positive function 2 -periodical with respect to x and y .
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Let us mention the main p oints of the pro of. Since the vector field f is tangent to E c , differential equation (2.1) is b ounded on Ec . This equation on Ec has an integral invariant M d , Vn-2 where d is the element of area of Ec considered as a surface emb edded into R n , Vn-2 is the (n - 2)-dimentional volume of the parallelepip ed in R n , the gradients of F1 , . . . , Fn-2 b eing its sides. Now the integrability by quadratures on Ec follows from Euler's remark. The first conclusion of theorem 1 (which was firstly mentioned by Jacobi) is proved by this reasoning. The second conclusion is the well-known top ological fact: any connected, compact, orientable, two-dimensional manifold that admits a tangent field without singular p oints is diffeomorphic to a two-dimensional torus. The third conclusion is, in fact, the Kolmogorov theorem on reduction of differential equations on a torus with a smo oth invariant measure [9]. Equations (2.3) have invariant measure (x, y ) dx dy . By averaging the right-hand sides of (2.3) with resp ect to this measure we get the differential equations u = , µ v = ;
2 2

= 1 4

2 0 0

dx dy .

(2.4)

Prop osition 1. Let : T 2 R be a smooth (analytical ) function. Then for almost al l pairs (, µ) R there exists a smooth (analytical ) change of angle variables x, y u, v that reduces (2.3) to (2.4). The pro of is presented in [9,10]. Note that if (2.3) can(not) b e reduced to (2.4) for a pair (, µ) then the same is true for all pairs ( , µ), = 0. So, the prop erty of reducibility dep ends on
arithmetical prop erties of µ that is referred to as the rotation numb er of the tangent vector field on T 2 = x, y mo d 2 .

Prop osition 2. Let (x, y ) = m,n exp i(mx + ny ), m,n = -m,-n . If (2.3) may be reduced to (2.4) by a differentiable change of angle variables u = u(x, y ), v = v (x, y ) then m,n m + nµ
2

|m|+|n|=0

< .

(2.5)

If the ratio µ is rational then the torus T 2 is stratified into a family of closed tra jectories. In this case the reducibility condition is equivalent to the equality of p erio ds of rotation for different closed tra jectories. In the general case (the Fourier decomp osition of contains harmonics) the p oints (, µ) R n with rationally indep endent (, µ), for which series (2.5) diverges, are everywhere dense in R n . The questions of reducibility of (2.3) are discussed in [9]. The general prop erties of solutions of (2.3) see in [10].

3. S. A. Chaplygin's problem
Let us consider as an example the problem of rolling of a balanced, dynamically non-symmetric ball on a horizontal rough plane (see [5]). The motion of the ball is describ ed by the following system of
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equations in R6 = R3 { } â R3 { }: k + â k = 0, k = I + ma â ( â ).
2

+ â = 0;

(3.1)

Let b e the vector of the angular rotation velo city of the ball, the unit vertical vector, I the tensor of inertia of the ball with resp ect to its center, m the mass of the ball, and a its radius. These equations have the invariant measure with density M=
2 )-1

(ma

1 - , (I + ma2 E)

-1



,

E=

ij

.

(3.2)

Taking into account the existence of four indep endent integrals F 1 = k , , F2 = k , , F3 = = , , F4 = k , k , we see that (3.1) is integrated by quadratures. Note that system of equa tions (3.1) has no equilibriums on the non-critical level sets E c . Indeed, if = 0, then and are linearly dep endent. This fact implies the linear dep endence of dF 1 and dF2 . The simplest case of integration by quadratures of equations (3.1) is the case of zero value of the constant in the "area" integral F2 . In elliptic co ordinates , on the Poisson sphere , = 1 the equations of motion on the level Ec are reduced to the following form = P5 ( ) (
-1

-

-1

) ( , ) =

;

=

P5 ( ) (
-1

-

-1

) ( , )

;

(a - )(a - ).

Co efficients of the p olynomial P5 of the fifth order and the constant a dep end on parameters of the problem and on the constants of the first integrals (see [5] for details). The variables , ranges a 2 , b1 b2 where P5 is nonnegative. The uniformizing over different closed intervals a1 substitution
a
2

x=
a
1

z dz , P5 (z ) z dz , P5 (z )



-1

1 =
a
1

z dz , P5 (z )
2



b

(3.3) z dz , P5 (z )

y=µ
b
1

µ

-1

1 =
b
1

intro duces the angle variables x, y mo d 2 on E c , and the equations of motion take form (2.3) x= =
-1

, (x, y )
-1

y=

µ , (x, y )

(3.4)

(x) -

(y )

a - (x) a - (y ) .

Here (x) and (y ) are 2 -p erio dic functions of x and y arising as the inversions of Ab elian integrals (3.3). These equations imply Prop osition 3. The rotation number of a tangent vector field on two-dimensional invariant tori in Chaplygin's problem is equal to the ratio of real periods of the Abelian integral z dz . P5 (z )
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Remark. This assertion is true for integrable problems of dynamics of a heavy body with a fixed point defined by the system of Euler-Poisson equations (see [10]). Since the Euler-Poisson equations are Hamiltonian, by the Liouville theorem, in integrable cases they always can be reduced to form (2.4) on two-dimensional invariant tori. It seems that equations (3.4) have no such property; inequality (2.5) is not fulfilled on all invariant non-resonant tori.

Let us make a change of time t by the formula dt = (a - )(a - ) d . (3.5)

Equations (3.4) preserve their form but the variables x, y in function are separated = Prop osition 4.
-1

(x) -

-1

(y ).

Suppose that µ and in (2.3) are nonzero and = (x) + (y ).

Then equations (2.3) can be reduced to form (2.4) by an invertible change of angle variables on T 2 . The pro of is presented in [10]. Note that if = (x) + (y ), then series (2.5) n n
2

n + nµ

2

n=0

converges for all , µ = 0. So, taking into account change of time (3.5) one may reduce (3.4) to the form du = U, d dv = V , d (3.6)

where U and V dep end on the constants of the first integrals only, and U, V = 0. This result can cause the temptation to use Chaplygin's reducing multiplier theorem: if we can reduce (3.1) using change of time (3.5) to the Euler-Lagrange equations of some variational problem (they are written, as well as the classical Euler-Poisson equations, in redundant variables) then according to the Liouville theorem, the equations of motion on the two-dimensional invariant tori in some angle variables u, v mo d 2 have form (3.6). It is p ossible to show that this metho d do es not lead to the goal. In conclusion, note that S. A. Chaplygin himself never considered the problem of the ball's rolling in connection with the reducing multiplier theory.

4. A generalization of S.A.Chaplygin's problem
We are going to show that the problem of rolling of a balanced, dynamically non-symmetric ball on a rough plane is still integrable (in the sense of section 1), if the particles of the ball are attracted by this plane prop ortionally to the distance. Since the center of mass of the ball coincides with its geometrical center, we can calculate the p otential by the formula V ( ) = 2 r,
2

dm = I , , 2

(4.1)

where is the unit vertical vector, r is the radius vector of particles of the ball, I is the tensor of inertia of the ball with resp ect to its center. The attraction forces generate the rotational moment - r â r , dm = - r , (r â ) dm = â V
¡



= â I .
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In order to obtain the moment of forces with resp ect to the contact p oint, it is necessary to add the moment of the combined force r , dm = r dm, ,

which is equal to zero, since the center of mass of the ball coincides with its geometrical center. According to the theorem ab out the change of kinetic moment b ehavior with resp ect to the contact p oint (see [5], [6]), the equations of rolling of the ball can b e presented in the following form k + â k = â I , Theorem 2. Indeed, they have four indep endent integrals F1 = k , + I , , F2 = k , , F3 = , = 1, F4 = k , k - A , , where elements Ai of a diagonal matrix A are expressed through the principal moments of inertia I by the formulae A1 = (I2 + ma2 )(I3 + ma2 ), . . .
i

+ â = 0.

(4.2)

Differential equations (4.2) are integrable by quadratures.

Since equations (4.2) have the invariant measure with density (3.2), they are integrable by theorem 1. It would b e interesting to integrate this equation explicitly and test if prop osition 3 remains true for equations (4.2). Note that the problem of rotation of a b o dy ab out a fixed p oint in an axisymmetric force field with p otential (4.1) is also integrable ([1]). In addition to the classical integrals F 1 , F2 , F3 , there is the integral F4 , where one must put a = 0. This integral was found indep endently by Clebsh in the problem on motion of a b o dy in an ideal fluid and by Tisseran, who investigated rotational motion of heavenly b o dies.

5. G. K. Suslov's problem and its generalization
Following G. K. Suslov with the nonintegrable of reference. Supp ose Following the metho d ([11], ch. 53), we consider the problem of rotation ab out a fixed p oint of a b o dy constraint a , = 0, where a is a vector that is constant in the moving frame that the b o dy rotates in an axisymmetric force field with the p otential V ( ). of Lagrange multipliers, we write down the equations of motion ([11], ch. 46): + â = 0, a , = 0. (5.1)

I + â I = â V + a ,

Using the constraint equation a , = 0, the Lagrange factor can b e expressed as the function of and = - a , I -1 (I â ) + I -1 ( â V ) a , I -1 a . Equations (5.1) always have three indep endent integrals: F1 = I , 2 + V ( ), F2 = , , F3 = a , .

For real motions, F2 = 1, F3 = 0. In this case, we can reduce the problem of integration of equations (5.1) to the problem of existence of an invariant measure (the existence isn't evident) and the fourth indep endent integral. Prop osition 5. If a is an eigenvector of operator I , then the phase flow of system (5.1) preserves the "standard" measure in R 6 = R3 { } â R3 { }.

To prove the prop osition we have to verify the following fact: the divergence of the right-hand side of (5.1) is equal to zero as Ia = µa .
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G. K. Suslov has considered a particular case of the problem, when the b o dy is not under action of exterior forces: V 0. In this case the first equation of (5.1) is closed relatively to . We can show that it is integrable by quadratures (see [11], ch. 53). The analysis of these quadratures shows that if a isn't an eigenvector of the inertia op erator, then all tra jectories (t) approach asymptotically as t ± to some fixed straight line on the plane a , = = 0 (see Fig. 1). Consequently, the equation with resp ect to and complete system (5.1) have no invariant measure with continuous density. In this case theorem 1 isn't applicable, so, the question ab out the p ossibility to find the vector (t) by quadratures remains op en. But if Ia = µa then equations (5.1) have the additional integral: the value of the kinetic moment is preserved F4 = I , I .

Fig. 1

Equations (5.1) are integrable by theorem 1. However, this p ossibility may b e easily realized directly. It seems that in the most general case, the existence of an invariant measure is connected with the hyp othesis of prop osition 5: Ia = µa . From now on, we supp ose that this equality is fulfilled. Now supp ose that the b o dy rotates in the homogeneous force field V = b , . If a , b = 0, then equations (5.1) have the integral F4 = I , b consequently, her work [12]. generality we equation 3 = they are integrable by quadratures. This case was indicated by E. I. Kharlamova in We are going to consider an "opp osite" case, when b = a , = 0. Without loss of can assume that the vector a has the comp onents (0, 0, 1). Taking into account the 0, we obtain that two first equations (5.1) have the following form I1 1 = 2 , I2 2 = -1 ; = (1 , 2 , 3 ).

Therefore I1 1 = 2 , I2 2 = -1 . ¨ ¨ Using the Poisson equations 1 = -2 3 , 3 = 1 3 we get I1 1 = 3 1 , ¨ The energy integral
2 2 (I1 1 + I2 2 ) 2 + 3 = h

I2 2 = 2 2 . ¨

(5.2)

makes it p ossible to express 3 through 1 and 2 . After that, equations (5.2) may b e rewritten as the Lagrange equations Ii2 i = V d L = L ¨ i dt i i L = TV , T=
22 2 I1 1 + I 2 2 , 2

(i = 1, 2),
2 2 2

2 I1 1 + I 2 V = 1 h- 2 2

.

These equations have the energy integral T + V . For real motions its value is equal to 2 /2. Let us emphasize that unlike the reducing multiplier theory our reduction of equation (5.1) to Lagrange (or Hamilton) equations do esn't require the change of time (cf. [11]). The change Ii i = ki corresp onding to the transition from the angular velo city to the kinetic moment reduces the considered problem of rotation of a b o dy to the problem of motion of a material p oint in the p otential force field ¨ ki = - V ki (i = 1, 2),
2 1 h - k1 I V= 2 -1 1 2 + k2 I 2 -1 2 2

.

(5.3)

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At I1 = I2 we have the motion of p oint in a central field. This motion corresp onds to the well-known integrable "Lagrange case" of the generalized Suslov problem. As well as in Lagrange's classical problem of a heavy symmetric top, the equations of motion are integrable in this case in elliptical functions of time. If I1 = I2 , then the equations apparently have no additional analytical integral indep endent of the energy integral. The following observation confirms this supp osition. Put formally I 1 = -I2 = 1. Then at h = 0 equations (5.3) practically coincide with the equations of the Young-Mills homogeneous two-comp onent mo del, non-integrability of which is established in [14]. If the value of h is fixed, the p oint moves in the area defined by the inequality V 2 /2. For different h, these areas are shown in Fig 2. The tra jectories of vibrational motions, when one of the comp onents k1 or k2 b ecomes zero, are esp ecially interesting. These motions are expressed through elliptical functions of time. One more case of integrability of (5.1) is given by Theorem 3. Suppose Ia and the potential V ( ) is defined by formula (4.1). Then equations (5.1) are integrable by quadratures. Proof. Let us show that equations (5.1) have the Clebsch ­ Tisseran integral F4 = 1 I , I - 1 A , , 2 2 Indeed, F4 = I , â I + a , I + A , â = A = I
-1

Fig. 2

det I .

= , I ( â I ) + µ a , + , â A = I -1 and can I2 .

= , I ( â I ) + â A = 0,

since I ( â I ) = -( â conclusion of prop osition 5 Let us show, how one and > 0, I3 I1 , I3 differential equations:

) det I . To complete the pro of, we have to take into account the to use theorem 1. explicitly integrate equation (5.1). For definiteness, let a = (0, 0, 1) Then (5.1) may b e presented as the following closed system of four I2 2 = (I3 - I1 )1 3 ,
2 2 2 3 = 1 - 1 - 2 .

I1 1 = (I1 - I2 )2 3 , 1 = -2 3 ,

2 = 1 3 ,

Let us intro duce the new time by formula d = 3 dt and denote the differentiating with resp ect to by prime. Then equations of motion take form of a linear system with constant co efficients I1 I2
1 2

= (I2 - I3 )2 , = (I3 - I1 )1 ,



2 1

= 1 , = -2 .

They can b e presented in the equivalent form
1

+ 2 1 = 0, 1



2 = (I3 - I1 ) I2 , 1 Put 1 1 1 = - arctg , 2
168

2 2 2

+ 2 2 = 0, 2 = (I3 - I2 ) I1 .

2 2 2 = arctg . 1
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These variables are angle variables on two-dimensional invariant tori with 1 = 1 , Consequently, 1 = 1 /, 2 = 2 /; = (1 - c2 sin2 1 - c2 sin2 2 ) 1 2
-1/2

1 = 2 .

.

The constants c1 and c2 (c2 + c2 1) can b e expressed as functions of constant values of the energy 1 2 integral and Clebsch ­ Tisseran integral. The remarkable prop erty of this problem is the fact that the ratio of frequencies 1 /2 is indep endent of initial data and dep ends only on the constants of parameters of the problem. Consequently, if the numb er (I3 - I1 )I (I3 - I2 )I
1 2

is rational, then all solutions are p erio dic; otherwise practically all tra jectories aren't closed (except degenerated motions, when 1 0 or 2 0). Let s (0) = as . Then


t=
0 2 1 2

dx 1 - c sin (1 x + a1 ) - c sin (2 x + a2 )
2 2 2

.

If c1 = 0 (or c2 = 0) then 1 and 2 (and consequently, 1 , 2 , 3 ) are elliptical functions of time. This conclusion is true in the case 1 = 2 (i. e., when I1 = I2 or I3 = I1 + I2 ) for all c1 , c2 . In the most general case the analytical character of the solutions is essentially more complex. In conclusion, note that series (2.5) diverges in this problem if the irrational ratio 1 /2 is approximated by rational numb ers anomalously fast.
Remark. Equations (5.1) are also integrable for potentials of the general form
2 2 2 V ( ) = 1 (c11 1 + c22 2 + c33 3 + 2c12 1 2 ). 2

Using the change of time d = 3 dt, the equations of motion are reduced to the linear system I2 1 = - V , 1 I1 2 = - V ; 2 V =V
2 2 3 =1-1 - 2 2

.

In the general case, potential V does not have a simple physical interpretation.

6. The first integrals used as constraints
Let L(x, x, t) b e a Lagrangian of a nonholonomic system that satisfies the following "regularity" condition: the quadratic form 2L , x is p ositively definite. In particular, det L xx = 0. The constraints (not necessarily linear) are given by the equations f1 (x, x, t) = . . . = fm (x, x, t) = 0 (6.1) with indep endent co-vectors f1 fm , ..., . x x
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The equations of motion can b e presented as the Lagrange equations with the multipliers d L - L = dt x x
m


s=1

s

fs , x

f1 = . . . = f

m

= 0.

(6.2)

Prop osition 6. If the functions L, f1 , . . . , fm satisfy the above conditions, then a unique solution of (6.2) corresponds to every initial state that is permissible by constraints (6.1). Indeed, under these supp ositions the multipliers 1 , . . . , m are smo oth functions of x, x, t by the explicit function theorem. Now supp ose equations (6.2) have a first integral F (x, x, t). We get the following Prop osition 7.
f1 f If the co-vectors F , , ..., x x
m

x

are independent then x(t) is a solution

may turn out to b e completely integrable. In this case the study of motions that b elong to the level set F = c is reduced to the investigation of some holonomic system. We do not have to integrate here the constraint equations, since the variables can b e considered as redundant one's, and the equations of motion may b e written as Hamilton equations in redundant variables (see [11], [15]). Let us consider as an example Suslov's problem in a homogeneous force field in the Kharlamova integrable case. The equations a , = 0 and I , b = 0 form an integrable field of directions on the manifold of the rigid b o dy p ositions (on the group SO(3)). Thus, Suslov's problem is reduced in this case to a system with one degree of freedom. Though the one-dimensional manifold of states of such system isn't closed in SO(3) in the general case. If the constraints are non-linear with resp ect to velo cities, it is natural to use the energy integral H (x, x) = L x - L x as the first integral1 . For example, let us consider the App ell ­ Hamel system with the Lagrangian L = 1 (x2 + y 2 + z 2 ) + g z , 2 and the non-linear constraint x2 + y 2 = k 2 z 2 , (see [16] and [17]). By the energy integral 1 (x2 + y 2 + z 2 ) - g z = h 2 k = const = 0 g = const

The sufficiency is obvious for x(t) satisfies (6.2) if we put m+1 = 0. On the contrary, let x(t) b e a solution of a system of form (6.2), where s ranges from 1 to m + 1. Let y (t) b e the unique motion of system (6.2) with the initial data y (0) = x(0), y (0) = x(0). Evidently, F y(t) = c. The function y (t) as well as x(t) satisfies the equations of motion of the extended system with m+1 = 0. To complete the pro of, it remains to use the conclusion of prop osition 6. Let us discuss one of p ossible applications of prop osition 7. Supp ose f i are linear with resp ect to velo cities and constraints (6.1) are non-integrable. If equations of motion have the linear integral F , then equations f1 = . . . = fm = fm+1 = 0 (fm+1 = F - c)

of (6.2) with the constant value of the integral F = c if and only if this function is a motion of mechanical system with the Lagrangian L and constraints f 1 = . . . = fm = fm+1 = c, where fm+1 = = F - c.

(6.3)

The equation of motion has the integral of energy, if the constraints are homogeneous with respect to velocities and the Lagrangian does not depend on time explicitly.

1

¡

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and (6.3) we get the equation of the "integrable" constraint z (1 + k 2 ) - g z = h. 2 (6.4)

Consequently, the co ordinate z changes with the constant acceleration g /(1 + k 2 ). Excluding nonlinear integrable constraint (6.4) (i. e., considering z as a known function of time) we get a more simple system with two degrees of freedom, the Lagrangian L = 1 (x2 + y 2 ) 2 and the constraint x2 + y 2 = f (t), where f = k 2 z 2 is a known quadratic function of time. The further integration may b e easily fulfilled.

7. Symmetries of nonholonomic systems
We supp ose that the vector field v (x) = 0 is a symmetry field of a nonholonomic system with Lagrangian L(x, x) and constraints f1 (x, x) = . . . = fm (x, x) = 0,
s if the phase flow gv of the differential equation

dx = v (x) dt preserves L and f1 , . . . , fm . Prop osition 8. A phase flow of a symmetry field converts solutions of a nonholonomic system to solutions of the same system. Proof. By the theorem on rectification of tra jectories, the phase flow g nates x1 , . . . , xn is the following one-parameter group x1 x1 + s; x 2 x2 , . . . , x n xn .
s v

in some lo cal co ordi-

With resp ect to these variables, L and f i do not dep end on x1 , consequently, the equations of motion do not contain this variable, to o. This fact implies prop osition 8. In the case of integrable constraints, the symmetry field corresp onds to a linear with resp ect to velo cities first integral of the equations of motion. It is not so in the case of nonholonomic systems. Prop osition 9.
s If gv preserves the Lagrangian and v is the field of possible velocities, i. e.

f1 f v = . . . = m v = 0, x x then the equations of motion have the first integral L = const. x This assertion ("the Noether theorem") is discussed in [6], for example.
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Theorem 4. Suppose the equations of motion (6.2) have n - m first integrals f m+1 , . . . , fn . If 1) at points of the set Ec = {f1 = . . . = fm = 0, fm+1 = cm+1 , . . . , fn = cn } the Jacobian (f1 , . . . , fn ) , x1 , . . . , x n is nonzero, 2) there exist fields v1 , . . . , vn-1 that are linearly independent at al l points E C and generate a s solvable Lie algebra with respect to the commutation operation, while their phase flows g vi preserve L and f1 , . . . , fn , 3) there are no vectors x = s vs (x), s R among solutions of the system of equations f1 = . . . = f
m

= 0,

f

m+1

=c

m+1

, ...,

f n = cn ,

(7.1)

then solutions of (6.2) that belong to E c are found by quadratures.
Remark. In some cases the existence of first integrals of nonholonomic systems can be established by the following observation. Let F (x, x) be the first integral of a "free" holonomic system with Lagrangian L. This function is an integral of a nonholonomic system with the same Lagrangian L and constraints f 1 = . . . = fm = 0 in the case of 2 L -1 fs · F = 0, 1 s m, x x x2 if f1 = . . . = fm = 0. This condition of invariancy is fulfilled for the Clebsch-Tisseran integral in Suslov's problem (theorem 3). Besides, it is fulfilled for the energy integral in the case of homogeneous constraints and for the Noether integral L · v , if the field v is the field of possible velocities (proposition 9). x

Proof of theorem 4. By the explicit function theorem we obtain from (7.1) that x = vc (x). By conditions 2 and 3, the vectors vc , v1 , . . . , vn-1 are linearly indep endent at all p oints s flows gvi convert solutions of (7.2) to solutions of the same equation (prop osition 8). To pro of, it remains to apply the well-known Lie theorem on integrability by quadratures differential equations (see, for example, [18]). Let us consider as an illustrating example the problem of sliding of a balanced skate ice. One can cho ose units of length, time and mass so that the Lagrangian would take form: L = 1 (x2 + y 2 + z 2 ). 2 (7.2) x. The phase complete the of systems of on horizontal the following (7.3)

Here x, y are the co ordinates of the p oint of contact, z is the angle of rotation of the skate. The constraint equation is f = x sin z - y cos z = 0. (7.4) The equations of motion have two first integrals x2 + y 2 + z 2 = h, z = c. (7.5)

The second one is obtained by using prop osition 9 with the help of the vector field v 3 = (0, 0, 1). By (7.4) and (7.5) we obtain the field v h,c = h - c2 cos z , h - c2 sin z , c . The fields v1 = (1, 0, 0) and v2 = (0, 1, 0) are commuting symmetry fields. If c = 0 then vectors v h,c , v1 and v2 are linearly indep endent, consequently, in this case we can apply theorem 4. Let us emphasize that v 1 and v2 do not generate the conservation laws.
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Theorem 4 imp ose strict restrictions on the nonholonomic system. These restrictions can b e weakened if we replace condition 2 by the condition 2) for the fixed c = (cm+1 , . . . , ch ) there exist n - 1 linearly indep endent fields v i (x, c) that generate a solvable Lie algebra and commute with v c (x). Let us add to Lagrangian (7.3) the term -x/2. Thus, we have placed the skate onto an inclined plane. Equations (7.4)­(7.5) hold if we replace x by h + x. Then the field v h,c is equal to h - c2 + x cos z , If h and c = 0 are fixed, then the fields v1 = 2 h - c2 + x, -(cos z ) c, 0 , v2 = (0, 1, 0) h - c2 + x sin z , c .

and vh,c are indep endent, and all their commutators vanish. In the same way one can solve the problem of rolling of a homogeneous disk on a rough plane, the problem of rolling of a ball in a vertical pip e and a series of other problems of nonholonomic mechanics.

8. Existence of an invariant measure
The existence of an integral invariant with a p ositive density is interesting not only from the standp oint of integration of differential equations. It is interesting by itself, from the standp oint of p ossible applications, for example, in ergo dic theory. We are going to consider the problem of existence of an invariant measure for systems of differential equations. We are esp ecially interested in its applications to nonholonomic mechanics. By the theorem on rectification of tra jectories, in a sufficiently small neighb orho o d of an ordinary p oint there always exists an invariant measure with a smo oth stationary density. Therefore, the problem of existence of an invariant measure is esp ecially interesting near equilibriums as well as in sufficiently big domains of the phase space, where tra jectories have the prop erty of returning. Let us consider the first p ossibility. Let the p oint x = 0 b e an equilibrium of an analytical system of differential equations x = x + . . . (8.1) We say that a set of (complex) eigenvalues 1 , . . . , n of the matrix is resonant, if mi i = 0 for some natural mi . Note that a weaker resonance condition: mi i = 0 for some integer mi 0 and |mi | = 0 is usually used for investigation of system (8.1) (for example, in the theory of normal forms). Prop osition 10. If a set 1 , . . . , n is not resonant then in a smal l neighborhood of the point x = 0 equation (8.1) has no integral invariant with an analytical density. The non-resonance condition is fulfil led, for example, in the case of Re i 0 ( 0) and Re i > 0 (< 0). Proof. Let us expand the density M (x) in a convergent series with resp ect to homogeneous forms: M = Ms + M
s+1

+...,

s

0.

Evidently, Ms is the density of the integral invariant for the linear system x = x. One can assume that as already reduced to the canonical Jordan form. Let us arrange the monomials of the form M s in some lexicographical order: Ms =
mi 0 m1 +...+mn =s


a

m1 ...m

n

x

m 1

1

...x

m n

n

.

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It is evident that div Ms (x) is some form of the same degree. By equating its co efficients to zero, we get a linear homogeneous system of equations with resp ect to a m1 ...mn . The determinant of this system is equal to the pro duct (m1 + 1)1 + . . . + (mn + 1)n .
mi 0 m1 +...+mn =s


This pro duct is nonzero by supp osition. Consequently, all a

m1 ...m

n

= 0.

Remark. If a more strict condition of absence of resonant ratios in the traditional sense is fulfilled then equation (8.1) has no first integrals analytical in a neighborhood of the point x = 0.

Let us consider as an example the problem on p ermanent rotations of a convex rigid b o dy with an analytical convex b ound on a horizontal absolutely rough plane (see [4]). The motion of such b o dy is describ ed by a system of six differential equations that have the integral of energy and the geometrical integral. In a particular case, when one of principal central axes of inertia of the b o dy is orthogonal to its surface, we have a one-parameter family of stationary rotations ab out the vertical axis of inertia. Singular p oints of equations of motion corresp ond to the stationary motions. The characteristic equation has the following form: 2 (a4 4 + a3 3 + a2 2 + a1 1 + a0 ) = 0. The dep endence of the co efficients as on numerous parameters of the problem is rather complicated; practically, they are arbitrary. The existence of the double zero ro ot is connected with the existence of two indep endent integrals, since the differentials of the energy integral and of the geometrical integral are indep endent in the general case at p oints that corresp ond to p ermanent rotations. Fixing the levels of the first integrals we get differential equations on four-dimensional manifolds that have no invariant measure with an analytical density in the general case. Consequently, the initial equations also have no invariant measure in a neighb orho o d of stationary motions. Now let us consider the problem of existence of invariant measure for systems of differential equations that are similar to integrable systems that satisfy the conditions of theorem 1. It is natural to take the constants of the first integrals I1 , . . . , In-2 as indep endent variables in a neighb orho o d of invariant tori of a non-disturb ed integral system and to take angle variables x, y mo d 2 on the invariant tori. In these variables, the p erturb ed system has the following form: Is = fs (I , x, y ) + . . . , x= (I ) + X (I , x, y ) + . . . , (I , x, y ) s = 1, . . . , n - 2,

µ(I ) + Y (I , x, y ) + . . . (I , x, y ) Fig. 3 (8.2) We assume that all functions in the right-hand sides of these differential equations are analytical in the direct pro duct D â T 2 , where D is a domain in Rn-2 = {I1 , . . . , In-2 }, T 2 = {x, y mo d 2 }; is a small parameter. For system (8.2), it is natural to consider the problem of existence of an invariant measure, the density of which is analytical with resp ect to I , x, y , 2 -p erio dic with resp ect to x, y and analytically dep ends on : M = M0 + M1 + 2 M2 + . . . (8.3) y= The unp erturb ed problem has an invariant measure with the density M 0 . According to the well-known averaging principle, we average the right-hand sides of (8.2) with resp ect to the measure
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dm = dx dy . As a result, we get the closed system of equations for changing of slow variables I in the domain D Is = Fs (I ), 1 s n - 2,
2 2 2 2

Fs = 1
0 0

fs dx dy ,

=
0 0

dx dy .

(8.4)

Prop osition 11. Suppose m(I ) + nµ(i) 0 in domain D for al not equal to zero simultaneously. If averaged system (8.4) has no invariant density, then initial system (8.2) also has no invariant measure with density System (8.4) is simpler than (8.2); the sufficient condition of nonexistence for (8.4) is given by proposition 10. Proof of proposition 11. Co efficients M0 and M1 of (8.3) satisfy equations M0 M0 +µ = 0, x y

l integer m, n that are measure with analytical (8.3). of an invariant measure

(8.5)

s

(M f ) + M X + M Y = - M1 + µ M1 . 0s 0 0 Is x y x y

(8.6)

Since /µ is irrational for almost all I D , equation (8.5) implies M 0 = (I ). Substituting this relation into (8.6) and averaging with resp ect to x, y we get the following equation: F = 0. Is s (8.7)

s

Remark. One can show that (under conditions of proposition 11) if averaged system (8.4) has no analytical first integral in D then initial system (8.2) has no integral that can be expressed as a series g 0 + g1 + . . . with coefficients gs analytical in D â T 2 .

Consequently, is the density of the integral invariant of (8.4). It remains to show that 0. If it is not true then M0 = 0. But in this case the function M1 + M2 + . . . is the density of an invariant measure for (8.2) If M1 0, this op eration may b e rep eated once more. The prop osition is proved.

Let us consider in more details the particular case, when n = 3. The index s may b e omitted. Let F (I ) 0. If F (I ) = 0 at some p oint of the interval D then (8.4) evidently has no invariant measure. Therefore, we assume that F (I ) = 0 in D . Consider the following Fourier expansions: X = F Y = F f = F X Y f
mn

(I ) exp i(mx + ny ),

mn

(I ) exp i(mx + ny ),

mn

(I ) exp i(mx + ny ).

The resonant set is the set of p oints I D , such that
|m|+|n|=0

amn m + nµ

2

= ,

a

mn

=

dfmn + i(mX dI

mn

+ nY

mn

).

Prop osition 12. Suppose 1) (I ) µ(I ) const, 2) the intersection D is not empty. Then (8.2) has no integral invariant with density (8.3).
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Indeed, correlation (8.7) implies = c/F , where c = const. Let M1 = b
mn

(I ) exp i(mx + ny ).

Equation (8.6) gives us the set of correlations -(m + nµ)b Let I . Then the condition implies c = 0. The author is grateful to professor V. F. Zhuravlev who has read the pap er and made a numb er of remarks. |b
mn mn

= ca

mn

.

|2 <

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[8] Yu. I. Neimark, N. A. Fufaev. Dynamics of nonholonomic systems. M.: Nauka. 1967. P. 519. [9] A. N. Kolmogorov. On dynamical systems with integral invariant on torus. Dokl. USSR Acad. of Sci. V. 93. 5. P. 763­766 . 1953.


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