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ISSN 1560-3547, Regular and Chaotic Dynamics, 2008, Vol. 13, No. 4, pp. 239-249. c Pleiades Publishing, Ltd., 2008.

NONHOLONOMIC MECHANICS

Stability of Steady Rotations in the Nonholonomic Routh Problem
A. V. Borisov* , A. A. Kilin** , and I. S. Mamaev*
Institute of Computer Science, Udmurt State University, ul. Universitetskaya 1, Izhevsk, 426034 Russia
Received Octob er 4, 2007; accepted January 16, 2008
**

Abstract--We have discovered a new first integral in the problem of motion of a dynamically symmetric ball, sub ject to gravity, on the surface of a paraboloid. Using this integral, we have obtained conditions for stability (in the Lyapunov sense) of steady rotations of the ball at the upmost, downmost and saddle point. MSC2000 numbers: 34D20, 70E40, 37J35 DOI: 10.1134/S1560354708040011 Key words: nonholonomic constraint, stationary rotations, stability

1. INTRODUCTION In his fundamental treatise [1] Routh considered the general equations of motion of a symmetric ball rolling without sliding on an arbitrary surface and indicated a series of sp ecial cases in which the equations can b e solved in quadratures. He also considered some questions concerning the stability of some steady motions. In this article we discuss in detail the problem of motion of a ball on the surface of an asymmetric parab oloid in a field of gravity. In that we will assume that the velocity of the ball at the p oint of contact is zero; thus, the nonholonomic model of motion of a ball applies (rolling without sliding). In this case the p oint of contact moves on an equidistant surface. Also we will briefly discuss the question of a ball moving on a surface that is a p erturbation of a parab oloid. As Routh showed [1], the problem of a ball rolling on a surface of revolution in the presence of asymmetric fields is integrable. As we will show b elow, the problem of a ball rolling on an arbitrary analytic surface already b ehaves chaotically and it lacks two additional first integrals to b ecome integrable. In this case one additional first integral may exist under some restrictions on the shap e of the surface and on the p otential of the force field. As shown in the article [2], this integral exists in the case of a ball rolling on a quadric surface in the absence of external forces. That result was generalized [3] to the case of a ball rolling on an arbitrary ellipsoid, sub ject to an external field of a sp ecial typ e. It turns out that one additional first integral exists also in the problem of a ball rolling on an asymmetric parab oloid, sub ject to gravity aligned with one of the principal axes of the surface. This result, as well as the results of [2, 3], is analogous to the problem of motion of a particle on the corresp onding surfaces. It is well-known that the problem of inertial motion of a particle on a quadric surface is integrable (Jacobi's geodesics problem). The problem of motion of a particle on a parab oloid, sub ject to gravity, was considered by PainlevÄ [4], Staude [5] and Chaplygin [13] e (the so-called parab oloidal p endulum problem). Without any further analysis, PainlevÄ only found e an additional first integral, thereby establishing that the system is integrable (this fact is mentioned by Chaplygin [13]). Detailed analysis of the system was p erformed indep endently by Staude (1892)
* ** ***

E-mail: borisov@ics.org.ru E-mail: aka@ics.org.ru E-mail: mamaev@ics.org.ru

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and Chaplygin (1898); using similar methods, they solved the separation of variables problem and qualitatively studied the motion of the system. In some sense the problem of a ball rolling that we consider can b e called the non-holonomic parab oloidal p endulum problem. Below we show that this system needs only one additional first integral to b ecome completely integrable (unlike the general case that needs two first integrals). 2. EQUATIONS OF MOTION OF A BALL ON A SURFACE In contrast to the approach traditional in the rigid b ody dynamics, which uses a b ody-fitted coordinates, for studying motions of a homogeneous ball it is more convenient to write the equations of motion in a fixed coordinate system. In this system the equations that govern the b ehavior of the momentum of the ball and its angular momentum with resp ect to its center, taking into account the reaction force and external forces, have the form mv = N + F , (I)Ç = a ç N + MF , (1)

Fig. 1. A ball rolling on a surface (G is the center of mass, Q is the p oint of contact with the surface).

and the no-slip condition (the velocity at the p oint of contact is zero) is Here m is the mass of the ball, v = x is the velocity of its center of mass, is its angular velocity, I = ÅE is the tensor of inertia of the ball, a is the vector from the center of mass to the p oint of contact, R is the radius of the ball, N is the reaction force at the p oint of contact (see Fig. 1), F and MF are resp ectively the external force and the external torque with resp ect to the p oint of contact. In the case of p otential forces the torque MF can b e expressed in terms of the p otential U (x), which dep ends on the p osition of the center of mass of the ball, as MF = U ç a. x v + ç a = 0. (2)

Eliminating from these equations the reaction force N and adding the kinematic relation equating the velocities of the p oint of contact on the surface and on the ball, we obtain the system of six equations (Å + mR2 ) = mR2 (, ) + MF , x = ç R , (3)

which describ es the dynamics of the angular velocity vector and the coordinates x of the center of mass of the ball. Here is the unit normal vector to the surface at the p oint of contact; it is determined by the relation = (x) , |(x)| (4)

where, following Routh [1], the equation (x) = 0 defines the surface on which the center of mass of the bal l moves. In the article [2] we used the kinetic momentum vector M and for studying the rolling motion of a ball on quadric surfaces. However, the variables and x that we use here are more convenient for generalizing the first integral found in [2] to the case of motion sub ject to gravity that we consider.
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STABILITY OF STEADY ROTATIONS IN THE NONHOLONOMIC ROUTH PROBLEM 241

3. INTEGRALS OF MOTION AND MEASURE Equations (3) in the case of a p otential field with p otential U (x) p ossess the energy integral and the obvious geometric integral 1 H = (M , ) + U (x) = h = const, (x) = 0. (5) 2 Aside from these two integrals, in the case of an arbitrary surface the system has invariant measure with density (Yaroschuk [6]) (, x) = || = x1
2

+

x2

2

+

x3

2

.

(6)

In the general case the system (3) does not p ossess two additional first integrals that it needs to b ecome integrable according to the last multiplier theory (Euler-Jacobi theory), and its b ehavior is chaotic [2]. Examine in greater detail the dynamics of a ball sub ject to gravity under the constraint that the center of mass of the ball moves on an asymmetric parab oloid defined by the equation where B = diag(1/p, 1/q , 0), with p and q b eing the principal radii of curvature of the parab oloid. As shown in [2], in the absence of external forces Eqs. (3) p ossess, taking into account (7), the additional integral F = ( ç , B( ç ))|(x)|2 . (8) (x) = (x, Bx) - 2x3 = 0, (7)

It turns out that when gravity is introduced (U (x) = mgx3 ), the integral (8) admits the generalization 4mg F = ( ç , B( ç ))|(x)|2 - (x, B2 x). (9) Å + mR 2 4. VERTICAL ROTATIONS AND THEIR LINEAR STABILITY Equations (3) have a particular solution of the form x = (0, 0, 0), = (0, 0, ). (10) Dep ending on the signs of p and q this solution describ es the spinning of the ball at the downmost (p > 0 and q > 0) or upmost p oint (p < 0 and q < 0) of an elliptic parab oloid, and at the saddle p oint of a hyp erb olic parab oloid (pq < 0). First we explain the results of Routh [1] concerning the linear stability of this solution, which he stated in his treatise [1] as a problem. That investigation is presented in detail in his Adams Prize Essay in 1877 1) [7]. The linearized equations of motion near the solution (10) have the form D R R + x2 (D2 /p - mg), (Å + D)p (Å + D)q D R R 2 = - 1 - x2 (D2 /q - mg), (Å + D)q (Å + D)p x1 = -R(x2 /q + 2 ), x2 = R(x1 /p + 2 ), x3 = 0, 3 = 0, 1 =
2

(11)

The corresp onding characteristic equation can b e written in the form 2 P4 () = 0, P4 () = 4 +
1)

R

2

mg (Å + D) (q + p) + Å2 pq (Å + D)2

2

2 +

m2 g 2 R 4 . pq (Å + D)2

(12)

Adams Prize comp etition: "The Criterion of Dynamic Stability". Maxwell Judge. Vol. 13 No. 4 2008

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Two nonzero eigenvalues corresp ond to the integrals of motion. The linear approximation to the solution (10) is stable provided that the remaining four eigenvalues have nonnegative real parts. Since P4 () in this case is a biquadratic p olynomial, this means that all its roots are purely imaginary. 1. p > 0 and q > 0 (rotation of the ball at the downmost p oint). In this case all roots of P4 () are certainly purely imaginary; consequently, rotation of the ball at the minimum p oint of the parab oloid is linearly stable at all angular velocities. 2. p < 0 and q < 0 (rotation of the ball at the upmost p oint of the parab oloid). In this case the linear stability condition takes the form [1] Å2 2 > mg(Å + D)( |p| + |q |)2 . (13)

Therefore, for sufficiently large angular velocities rotation of the ball at the upmost p oint of the parab oloid is linearly stable (this effect is called gyroscopic stabilization). 3. pq < 0 (rotation of the ball at the saddle p oint of a hyp erb olic parab oloid). The constant term of P4 () is negative; consequently, rotation of the ball at the saddle p oint of a hyp erb olic parab oloid is linearly unstable. 5. LYAPUNOV STABILITY OF ROTATIONS AT THE DOWNMOST POINT Consider now the Lyapunov stability of solutions (10) of the full nonlinear system (3) assuming that p > 0 and q > 0. (Note that linear instability in the case pq < 0 implies nonlinear instability. Prove first the stability of rotations (10) with resp ect to the variables = (1 , 2 , x1 , x2 ). To this end choose as a Lyapunov function the quadratic part of the expansion of the integral (9) into a series near the solution (10), 8 8 0 0 pq q 8 8 0 p 0 pq (14) F2 = ( , L ), L = 2 8mg 8 0 82 + p2 (Å+D) 0 pq qp 8mg 8 82 0 pq 0 + q2 (Å+D) pq 2 The p ositive definiteness of the function F2 requires the fulfillment of the inequalities q > 0, pq > 0, p3 q > 0, p3 q 3 > 0. (15)

In the case in question p > 0 and q > 0, and so the function (14) is p ositive definite. Therefore, by the Lyapunov theorem the rotations (10) are stable with resp ect to the variables (1 , 2 , x1 , x2 ). Because motions near the solution (10) are b ounded with resp ect to (1 , 2 , x1 , x2 ) and the first integrals of motion H and are invariant, it follows that motions are b ounded with resp ect to all six variables (x, ). Consequently, as the ball rotates at the downmost p oint of the parab oloid the solutions (10) are Lyapunov stable in the full nonlinear setting. 6. LYAPUNOV STABILITY OF ROTATIONS AT THE UPMOST POINT The previous construction with a Lyapunov function is not applicable in the case of rotations at the upmost p oint. However, stability results can b e derived using elements of the KAMtheory (Moser's theorem on invariant curves under mappings of the annulus). We include the main theoretical foundations of this analysis in the App endix. We will apply Theorem 1 from the App endix to prove the Lyapunov stability of p ermanent rotations of the ball at the upmost p oint of the parab oloid in the case p, q < 0 assuming that condition (13) holds (i.e., in the absence of unstability in the linear approximation).
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STABILITY OF STEADY ROTATIONS IN THE NONHOLONOMIC ROUTH PROBLEM 243

Using integrals (5) on the sp ecial energy level E = 1 Å2 we eliminate 3 and x3 from 2 equations (3) and obtain the system of the form = A + w( ) with = (1 , 2 , x1 , x2 ) and the matrix of the linear part determined by equations (11). To analyze the stability of this system we can apply Theorem 1 from App endix. According to this theorem we have to compute the coefficients Å and Åv (see App endix) of the normal form of the system. Parametrize the constants of the system as follows:
2 p = -k1 ,

uv mg = , Å+D k1 k2

Å = , Å+D 2 Å2 2 k2 + k2 = u2 + v 2 + 1 uv , (Å + D)2 k1 k2
2 q = -k2 ,

(16)

where u, v > 0 and the condition (13) certainly holds. Substituting that into (12), we find the eigenvalues of the matrix A; hence u v 1 = , 2 = . k1 k2 k1 k2 Pass to variables Å that reduce the linear part to diagonal form (i.e., to the form (20)); then Å = ( , vÅ ),
Åä k2 (k1 v + k2 u)

Å = 1, 2,
Åä k1 (k1 u + k2 v) ä ä ä

(17)

where vÅ are eigenvectors of AT (i.e., AT vÅ = iÅ vÅ ), for which we find
v1 = v2 = -i -i
43 k1 k2 Åä k1 v + (1 - )k2 u k2 (k1 u + k2 v) 4 k1 k 3 2

k1 u + (1 - )k2 v

,- ,-

34 k1 k2 Åä (1 - )k1 u + k2 v k1 (k1 v + k2 u) 3 k1 k 4 2

(1 - )k1 v + k2 u

,i
ä

k1 (k1 u + k2 v) , 22 k1 k2 k1 (k1 v + k2 u) , 22 k1 k2

k2 (k1 v + k2 u) 22 k1 k2 k2 (k1 u + k2 v) 22 k1 k2

,

,i

.

Finally we obtain the coefficients Åv in the form 11 = - 12 =
2 2 2 2 1 v 2 (1 - )(k1 - k2 )2 v u + k1 k2 (k1 + k2 )(u2 - v 2 ) , 4 (u2 - v 2 )3

2 2 22 4 4 2 2 1 v -uk1 k2 (k1 + k2 )(u2 - v 2 )(2 - ) + 4k2 k1 v 3 + 2 -k2 - k1 + (k1 - k2 )2 v u2 , 4 (u2 - v 2 )3 2 2 22 4 4 2 2 1 u (2 - )k2 k1 v (k1 + k2 )(u2 - v 2 ) + 4k2 k1 u3 - 2 k1 + k2 - (k1 - k2 )2 uv 4 (u2 - v 2 )3 2

21 = -

,

22 =

2 2 2 2 1 u2 (1 - )(k1 - k2 )2 v u - k1 k2 (k1 + k2 )(u2 - v 2 ) . 4 (u2 - v 2 )3 66 k1 k2 u k6 k6 v + 21 2 2 2 2 + . . . 2 - v2 1 1 u u -v

For the integral (9) we have the expansion F =-

Remark 1. The vectors vÅ are determined up to rescaling; corresp ondingly, Åv and the integral are also determined up to some multiplicative constants. Finally for the twist value we obtain D1 =
55 4 4 2 2 22 1 uv k2 k1 3u2 v 2 -k2 - k1 + (k1 - k2 )2 + 2k2 k1 (u2 + v 2 )2 - v 2 u2 2 (u2 - v 2 )4

.

Relying on Theorem 1, we can prove the Lyapunov stability of rotations of the ball at the upmost p oint of the parab oloid assuming the fulfillment of (13) with the exception of the cases in which the parameters satisfy one of the equalities D1 = 0, m1 u + m2 v = 0, or |m1 | + |m2 | < 4.
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7. NUMERICAL RESULTS Here we present some results of numerical modelling using constructions of PoincarÄ mappings. e Figure 2 shows the three-dimensional PoincarÄ mapping for the values of the energy integral e E = 50, of the additional integral F = 20, and of the parameters p = 1, q = 10, m = 1, g = 1, D = 1, R = 0.1. For the plane of cross-section we chose the plane 1 = 0, and the cross-section is constructed in the space (x1 , x2 , 2 ). The results of modelling show that no other additional first integral can exist and the problem is not integrable. However, the existence of the first integral F causes the chaotic layers and invariant tori to lie on two-dimensional level surfaces of this integral in the space (x1 , x2 , 2 ).

Fig. 2. The PoincarÄ cross-section of the unp erturb ed problem at the level sets E = 50 of the energy integral e and F = 20 of the additional first integral.

Consider a p erturb ed situation, in which the surface the ball is rolling on differs from the parab oloid (we will refer the parab oloidal case as unp erturb ed). Define a p erturbation of the surface in the form x3 = 1 (x, B x) + 1 (111 x3 + 112 x2 x2 + 122 x1 x2 + 222 x3 ), where ij k , i, j, k = 1, 2, 1 1 2 2 2 3 are some coefficients that determine the deviation of the surface from the parab oloidal one. All crosssections given b elow are constructed for the values of the parameters p = 1, q = 10, m = 1, g = 1, D = 1, R = 0.1. If the condition Å = 0 holds (and it implies that RÅ = O(3 ), see the App endix), which imp oses certain constraints on ij k , then as the energy grows, the invariant tori of the unp erturb ed problem are destroyed very slowly. For small values of the energy (Fig. 3) the crosssection is almost indistinguishable from the unp erturb ed problem. At that, the two-dimensional tori very slowly "dissipate", forming nested solid tori. As the energy grows (Fig. 4), some of the tori are destroyed and form a three-dimensional chaotic layer. Large domains of regularity are also observed. The articles [8, 9], devoted to a generalization of the KAM-theory to reversible systems with first integrals, show that for a symmetric p erturbation of the surface (for which the system is reversible) there exist invariant manifolds that prevent the "dissipation" of two-dimensional tori near the p eriodic solution corresp onding to rotation of the ball at the downmost p oint of the surface. As computer exp eriments show, such b ehavior in the case in question is also observed far from the said p eriodic solution. Domains of regular motion filled with two-dimensional tori are clearly seen in Fig. 5. Chaotic layers observed on its PoincarÄ cross-section may also b e related to the existence e of invariant manifolds. For an asymmetric p erturbation Å = 0, RÅ = O(2 ) (violating the reversibility condition) twodimensional tori dissipate much faster than in the case Å = 0. For small energy three-dimensional tori also app ear, but they have much more complicated structure (Figs. 6 and 7). As the energy grows, dynamics of the system develops highly dissipative character (Fig. 8).
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STABILITY OF STEADY ROTATIONS IN THE NONHOLONOMIC ROUTH PROBLEM 245

Fig. 3. The PoincarÄ cross-section of the p erturb ed e problem on the level set E = 0.5 of the energy integral for the p erturbation parameters 111 = 0.1, 112 = 0.05, 122 = 0, 222 = 0.

Fig. 4. The PoincarÄ cross-section of the p erturb ed e problem on the level set E = 12 of the energy integral for the p erturbation parameters 111 = 0.1, 112 = 0.05, 122 = 0, 222 = 0.

Fig. 5. The PoincarÄ cross-section of the p erturb ed problem on the level set E = 4 of the energy integral for e the p erturbation parameters 111 = 0.2, 112 = 0, 122 = 0.1, 222 = 0.

The corresp onding two-dimensional phase p ortrait of the normal Note in conclusion that an application of the methods of [8, 9] to homogeneous ellipsoid on an absolutely rough plane is included in that article the domains of stability of p ermanent rotations numerically.

Finally, we include the phase p ortrait of the normal form for the parameter values corresp onding to Figs. 6 and 7. It has the form R1 = (-0.0007232806807R1 + 0.00006530012756R2 )R1 + O(R3 ), (18) R2 = (0.001446561361R1 - 0.0003265006378R2 )R2 + O(R3 ). form is depicted in Fig. 9. the problem of motion of a heavy in the article [10]. In particular, of the ellipsoid are constructed

APPENDIX Consider a system of ordinary differential equations in R4 of the form = v ( ) = A + w( ),
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(19)


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Fig. 6. The PoincarÄ cross-section of the p erturb ed e problem on the level set E = 0.5 of the energy integral for the p erturbation parameters 111 = 0.1, 112 = 0.05, 122 = 0, 222 = 0.2.

Fig. 7. Close-up of a part of the PoincarÄ crosse section in Fig. 6.

Fig. 8. The PoincarÄ cross-section of the p erturb ed problem on the level set E = 3 of the energy integral for e the p erturbation parameters 111 = 0.1, 112 = 0.05, 122 = 0, 222 = 0.2.

where A is some constant real matrix and w( ) b egins with second order terms; i.e., = 0 is a fixed p oint and A is the matrix of the linearization. Assume that 1. The system (19) preserves a measure (i.e., ?( )v ( ) = 0) with density ( ) analytic in a neighb orhood of = 0. 2. There exists a first integral nondegenerate in a neighb orhood of = 0 (its expansion has the form P ( ) = F0 + fÅv m v + . . . ). 3. The eigenvalues of A are purely imaginary and pairwise distinct (1 = -2 = i1 , 3 = -4 = i2 , 1 = 2 ). In order to state stability criteria for an equilibrium write the system in a basis consisting of eigenvectors of A. It is known that in this case the coordinates are complex conjugate numb ers = (1 , 1 , 2 , 2 ). Also, select terms of a sp ecial typ e from the cubic part of the resulting system. We have 1 = i(1 + 11 1 1 + 12 2 2 )1 + . . . , (20) 2 = i(2 + 21 1 1 + 22 2 2 )2 + . . . , where the dots denote second and higher order terms (excluding the terms made explicit in (20)).
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STABILITY OF STEADY ROTATIONS IN THE NONHOLONOMIC ROUTH PROBLEM 247

Fig. 9. Phase p ortrait of the system (18).

Accordingly, the first integral in the new variables has the form 1 F = F0 + (a1 1 1 + a2 2 2 ). 2 The following theorem holds. Theorem 1. The equilibrium = 0 of the system (19) is Lyapunov stable if either a1 Ç a2 > 0, or a1 Ç a2 < 0, m1 1 + m2 2 = 0 for |m1 | + |m2 | < 4 and D = a1 (1 22 - 2 12 ) - a2 (1 21 - 2 11 ) = 0. Proof. The first case is obvious: the first integral is a Lyapunov function of the system. Indeed, it is of fixed sign in a neighb orhood of = 0 and F = 0. Before dealing with the second case we prove the following useful lemma, which constrains the coefficients in the normal form (20) under the assumption that an invariant measure and a first integral exist. Lemma. Suppose that a system (19) possesses a first integral and an invariant measure both analytic in a neighborhood of = 0, and that nonresonance conditions of the form |1 | = |2 |, |1 | = 3|2 |, 3|1 | = |2 |, hold. Then in the normal form al l coefficients Åv are real. Proof. Write Åv = Åv , 1 = X1 + iY1 , 2 = X2 + iY2 , where XÅ and Yv are real, and expand the integral into a p ower series in XÅ and Yv : F=
k =0

Fk (X, Y ),

where Fk is a homogeneous p olynomial of degree k. Using the method of undertermined coefficients, we find that for the system (20) the first integral admits an expression of the form F = F0 + F2 + F4 + O( 5 ), 1 2 2 F2 = a1 (X1 + Y12 ) + a2 (X2 + Y22 ) , 2 F4 = and the conditions 11 = 22 = 0, a1 12 + a2 21 = 0
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(21)

1 2 2 2 2 b11 (X1 + Y12 )2 + b12 (X1 + Y12 )(X2 + Y22 ) + b22 (X2 + Y22 )2 , 4

(22)


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must b e satisfied. Expanding similarly the invariant measure density in a neighb orhood of the origin while using (22), we find that 1 2 2 c1 (X1 + Y12 ) + c2 (X2 + Y22 ) + O( 3 ) 2 under the additional assumptions that = 0 + 12 = 21 = 0.

We will prove the second case by applying Moser's theorem on invariant curves under mappings of the annulus [11]. In order to construct a mapping necessary for such application we reduce the system (20) to normal form up to third order terms. By the arguments used in the proof of the PoincarÄ-Dulac theorem [12], a p olynomial change of variables Å = Å + PÅ ( ), Å = 1, 2, can e reduce the system (20) to a form Å = i(Å + Å1 1 1 + Å2 2 2 )Å + O( 5 ); the p olynomial PÅ ( ) b egins with second order terms and all its coefficients are real. Following [11], make the change of variables Å = RÅ eiÅ , RÅ = O(3 ), Å = 1, 2, (23)

where RÅ > 0 and is a small parameter. The system and the first integral (23) take the form = Å + 2 (Å1 R1 + Å2 R2 ) + O(3 ), F 1 F = 2 = (a1 R1 + a2 R2 ) + O(2 ). 2 a1 2c R1 - + O(2 ) a2 a1 (24) (25)

On the level surface F = c of the first integral eliminate R2 by the formula R2 = -

and pass to a new indep endent variable 2 . We will obtain a 2 -p eriodic nonautonomous system of the form dR1 = O(3 ), d2 1 + 11 R1 - d1 = d2 2 + 21 R2 -
|a1 | 2 12 a1 a2 21 a1 a2

R1 - R1 -

2c a1 2c a1

+ O(3 ).

(26)

If we consider only the values |c| < lies within the limits

of the integral then for sufficiently small the value of R1 1 R1 2. (27)

At that, R2 > 0, and in terms of the original variables 2 < 1 1 < 22 . Consider now the PoincarÄ cross-section of the system (26) by the plane 2 = 0 (mod 2 ). Up e to 3 it is given by the relations R(2 ) = R(0) + O(3 ), (2 ) = (0) + D0 + 2 D1 R1 + O(3 ), 1 c(12 2 - 22 1 ) D0 = 2 1+ , 2 1 2 a2 2 D1 = (1 22 - 2 12 )a1 - (1 21 - 2 11 )a2 . a2 2 2

(28)

Because the original system preserves the invariant measure, so does the mapping (28); thus, each closed curve around the origin in the plane 1 2 intersects its own image under the mapping (28). Therefore, the invariant curves theorem [11] can b e applied to the mapping (28), and if D1 = 0
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STABILITY OF STEADY ROTATIONS IN THE NONHOLONOMIC ROUTH PROBLEM 249

then in the annulus (27) there is an invariant curve. This implies that each solution lying on a given energy level c and starting inside that curve will never go outside it. Adjusting the small parameter , we can make the domain enclosed by an invariant curve arbitrarily small, which guarantees the stability of the equilibrium. ACKNOWLEDGMENTS Research 08-01-00651 Presidential 5239.2008.1 was partially supp orted by the Russian Foundation of Basic Research (pro jects Nos. and 07-01-92210). A.A.K. and I.S.M also acknowledge the supp ort from the RF Program for Supp ort of Young Scientists (pro jects Nos. MK-6376.2008.1 and MDresp ectively). REFERENCES
1. Routh, E.J., Advanced Dynamics of a System of Rigid Bodies, 6th ed., London: MacMillan Company, 1905. Reprinted by New York: Dover Publications, 1955. 2. Borisov A.V., Mamaev, I.S., and Kilin, A.A., A New Integral in the Problem of Rolling a Ball on an Arbitrary Ellipsoid Dokl. Phys., 2002, vol. 47, no. 7, pp. 544-547 [Translated from Dokl. Akad. Nauk, 2002, vol. 385, no. 3, 2002, pp. 338-341]. 3. Mamaev, I.S., New Cases when the Invariant Measure and First Integrals Exist in the Problem of a Body Rolling on a Surface, Regul. Chaotic Dyn., 2003, vol. 8, no. 3, pp. 331-335. 4. PainlevÄ, P., LeÈons sur l'intÄgration des equations diffÄrentiel les de la MÄcanique et Applications, Paris, e c e e e A. Hermann, 1895, pp. 291. 5. Staude, O., Ein Beitrag zur Discussion der Bewegungsleichungen eines Punktes, Math. Ann., vol. 41, no. 2, 1892, pp. 219-259. 6. Yaroshchuk, V.A., New Cases of the Existence of an Integral Invariant in a Problem on the Rolling of a Rigid Body, Without Slippage, on a Fixed Surface, Vestnik Moskov. Univ. Ser. I Mat. Mekh., 1992, no. 6, pp. 26-30 (in Russian). 7. Routh, E.J., A Treatise on the Stability of a Given State of Motion: Particularly Steady Motion (Adams Price Essay), London: MacMillan Company, 1877. 8. Matveev, M.V., Lyapunov Stability of Equilibrium Positions of Reversible Systems, Math. Notes, 1995, vol. 57, no. 1-2, pp. 63-72 [Translated from Mat. Zametki, 1995, vol. 57, no. 1, pp. 90-104]. 9. Matveyev, M.V., Reversible Systems with First Integrals, Physica D, 1998, vol. 112, pp. 148-157. 0. Glukhikh, Yu.D., Tkhai, V.N., Chevallier, D.P., On the Stability of Permanent Rotations of a Heavy Homogeneous Ellips oid on an Ideally Rough Plane, in Problems in the Investigation of the Stability and Stabilization of Motion, Part I, Ross. Akad. Nauk, Vychisl. Tsentr im. A. A. Dorodnitsyna, Moscow, 2000, pp. 87-104 (in Russian). 1. Moser, J.K., Lectutes on Hamiltonian Systems, Memoirs Am. Math. Soc., 1968, vol. 81, pp. 1-60. 2. Arnold, V.I., Geometrical Methods in the Theory of Ordinary Differential Equations, New York: SpringerVerlag, 1988. 3. Chaplygin, S.A., On a Paraboloid Pendulum, 1898; reprinted in: Polnoe sobranie sochinenii (Collected Works), Leningrad: Akad. Nauk SSSR, 1933, vol. 1, pp. 194-199.

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REGULAR AND CHAOTIC DYNAMICS

Vol. 13

No. 4

2008