Rolling on a Plane. A. V. Borisov and I. S. Mamaev. Received November 1, 2004. Key words: invariant measure, rolling ellipsoid, Liouville equation, Celtic stone.
Mathematical Notes, vol. 77, no. 6, 2005, pp. 855–857. Translated from Matematicheskie Zametki, vol. 77, no. 6, 2005, pp. 930–932. c Original Russian Text Copyright 2005 by A. V. Borisov, I. S. Mamaev.
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The Nonexistence of an Invariant Measure for an Inhomogeneous Ellipsoid Rolling on a Plane A. V. Borisov and I. S. Mamaev Received November 1, 2004
Key words: invariant measure, rolling ellipsoid, Liouville equation, Celtic stone.
In this paper, we obtain new conditions for the nonexistence of an invariant measure for an inhomogeneous ellipsoid with special distribution of mass rolling on an absolutely rough plane. They supplement earlier results on the absence of a measure for a rolling Celtic stone. 1. INTRODUCTION General conditions for the existence of a smooth invariant measure for a dynamical system were obtained in [1, 2]. Suppose that Rn is a phase space with coordinates x = (x1 , . . . , xn ) and v(x) is a smooth vector field on it. The corresponding phase flow is determined by the system x ∈ Rn ,
x˙ = v(x),
v = (v1 , . . . , vn ).
(1)
Following [1], we investigate the existence of an integral invariant
f (x) dn x,
m(D) = D
D ⊂ Rn ,
(2)
with smooth positive density f (x) for the system (1). This problem makes sense, e.g., in a neighborhood of an equilibrium point. The function f satisfies the Liouville equation div(f v) = 0 ; making the change ω = − ln f , we can rewrite it in the form ω˙ = div v,
(3)
where the dot denotes differentiation along the field v . The solvability of system (3) near the singular point (1) was analyzed in [1, 2]. Assuming that the singular point coincides with the origin ( x = 0), we expand the vector field in the series n λrs xr + · · · , s = 1, . . . , n. (4) vs (x) = r=1
For smooth ω , we have ω = ω0 + (a, x) + · · · , 0001-4346/2005/7756-0855
ω0 ∈ R,
a = (a1 , . . . , an ).
c 2005 Springer Science+Business Media, Inc.
(5) 855
856
A. V. BORISOV, I. S. MAMAEV
Substituting (4) and (5) into Eq. (3), we easily obtain the following solvability condition for the Liouville equation: (6) tr Λ = tr λrs = 0. In [1, 2], it was shown that this condition does not hold for, e.g., the dynamics of a Celtic stone. This is the dynamical system describing rolling without slipping on the horizontal plane of a nonsymmetric solid body for which the principal geometric and dynamical axes are different at the contact points. It was proved that there is no invariant measure near permanent vertical rotations, which are equilibrium positions for this system. The absence of an invariant measure in the dynamics of Celtic stones determines their strange and mysterious behavior described by many authors (see [3, 4]). In particular, in [3], it was shown that, in the phase spaces of such systems, strange attractors typical of dissipative systems arise for certain energies. 2. EQUATIONS OF PLANE MOTION OF AN ELLIPSOID: VERTICAL ROTATIONS Consider a triaxial ellipsoid rolling without slipping on a horizontal plane in the gravity field. It is convenient to write its equations of motion in the moving coordinate system rigidly attached to the ellipsoid and such that the coordinate axes pass through the center of the ellipsoid and are directed along its principal axes. In these coordinates (x1 , x2 , x3 ) , the equation of the ellipsoid can be written as (r, B−1 r) = 1,
B = diag(b1 , b2 , b3 ),
bi = const,
r = (x1 , x2 , x3 ),
(7)
γ˙ = γ × ω ,
(8)
and the equations of motion have the form −1 ω˙ = IQ (M × ω − mr × (ω × r˙ ) + mgr × γ),
where ω is the vector of angular velocity, γ is the unit vertical vector perpendicular to the horizontal plane, (γ , γ) = 1 , and mg is the weight of the body. In Eqs. (8), IQ = I + (mr2 − mr ⊗ r),
r = −
Bγ (γ , Bγ)
M = Iω + mr × (ω × r) = IQ ω ;
,
(9)
i.e., IQ and M are the inertia tensor and the kinetic momentum with respect to the contact point Q . We also assume that the geometric and dynamical axes of the ellipsoid coincide, i.e., I = diag(I1 , I2 , I3 ) . In this situation, the absence of an invariant measure is most difficult to prove, because such a measure exists in some special cases [5]. It turns out that, under the additional condition I1 (b2 − b3 ) + I2 (b3 − b1 ) + I3 (b1 − b2 ) = 0, i.e . Ii = α + βbi ,
(10)
α, β = const,
Eqs. (8) have a two-parameter family of special solutions, namely, vertical rotations [4] of the form γ = ξ = const,
ω = kξ,
where
k2 =
mg , (β − m) (ξ, Bξ)
(ξ, ξ) = 1.
(11)
Cumbersome calculations give the following explicit trace formula for the matrix of the linear part of system (8) near solution (11): 2 mg(β − m) i,j ,k bi (bj + bk )ξi , (12) βmξ1 ξ2 ξ3 (b1 − b2 )(b2 − b3 )(b3 − b1 ) tr Λ = A (ξ, Bξ) MATHEMATICAL NOTES
Vol. 77
No. 6
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INVARIANT MEASURE FOR AN INHOMOGENEOUS ROLLING ELLIPSOID
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where A = (ξ , IBξ)(ξ , BB2 ξ) − (β − m)
bi bj (bi − bj )2 (α(β − m) + β 2 bk )ξi2 ξj2 ,
i
