Invariant sets of mechanical systems

Abstract

Problems of the existence and stability of invariant sets of mechanical systems are discussed (in particular, steady motions and relative equilibria). The existence problem of stable steady motions (zero-dimensional invariant sets) was first investigated in [1]. The Routh theory [1–15] gives stability conditions of steady motions of conservative mechanical systems with first integrals as well as a construction method for such steady motions. This method was generalized for dissipative systems with first integrals in [12–15]. Recently, the Routh theory was modified for the existence and stability problems of invariant sets of dynamical systems with a nonincreasing energy function and first integrals [7,14] (in particular, for conservative and dissipative mechanical systems with symmetry [16–19]).