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Abstract

In this review we discuss methods of investigation of steady motions of nonholonomic mechanical systems. General conclusions are illustrated by examples from the rigid bodies dynamics on a absolutely rough plane.Problems existence, stability and bifurcation of steady motions of mechanical systems with first integrals was first investigated by E. J. Routh [1, 2] and H. Poincare [3]. Their results were developed in . In spite of the fact that general theorems of Routh method are valid for arbitrary dynamical systems with first integrals, the most part of results [1-26] concern to holonomic systems. Indeed, for holonomic systems the existence of first integrals and steady motions is determined by the same symmetry groups. This fact is not correct for nonholonomic systems: usually, the existence of first integrals and steady motions of these systems is determined by different symmetry groups. Moreover, first integrals of nonholonomic systems (excluding the energy integral in the conservative case) are less common than steady motions and usually its explicit form is unknown.This research was supported by the Russian Foundation for Basic Research (00-15-96150, 01-01-00141). General statements of Routh theory1.1. Introduction. Consider a general dynamical system (in particular, mechanical) defined by the system of ordinary differential equations ẋ = f (x).(1.1)a scalar product of two vectors a and b by (a • b) and their vector product by [a × b].Let x(t; x 0 ) denote a solution of equations (1.1) with the initial conditions x(0; x 0 )= x 0 ∈ X (t = 0 can be chosen as the initial time without loss of generality, because equations (1.1) do not depend on time explicitly). This solution corresponds to some motion of the dynamical system, so we will call it "motion x(t; x 0 )" of the system. Definition 1.1. A set X 0 ⊂ X is an invariant set of the system if x(t; x 0 ) ∈ X 0 for all t 0 and arbitrary x 0 ∈ X 0 .Remark 1.1. If dim X 0 = 0 then X 0 = {x 0 }, where x 0 ∈ X is an invariant point of equations (1.1) (the point, where f (x 0 ) = 0), so x(t; x 0 ) ≡ x 0 is a steady motion of the system.

Tippe Top Equations and Equations for the Related Mechanical Systems

“…3 we have taken the values of the parameters m = 0.02, R = 0.02, α = 0.3, I 1 = 235 625 mR 2 and I 3 = 2 5 mR 2 . A similar surface showing the stationary points θ min (λ, D) for values of λ and D has also been discussed in [12]. Points on and above this surface correspond to solutions to equation (1.20) for the rolling TT.…”

“…The effective potential for a rolling axisymmetric sphere was derived as early as [3]. In Karapetyan and Kuleshov [12] the same potential is found as a special case of a general method for calculating the potential of a conservative system with n degrees of freedom and k < n integrals of motion. The potential for the rolling TT in the above form was derived and analysed in [8].…”

“…1). This model has been used in many works, for example by Hugenholtz [9], Cohen [5] and Karapetyan [11,12]. An alternative model of the TT could be to consider the TT as two spherical segments joined by a rigid rod, as proposed in [23].…”