Relative Equilibria in Mechanical Systems

Abstract

This chapter presents some of the ideas in Smale's papers on topology and mechanics. It reviews some of the proofs and generalizes some of the theorems. The chapter illustrates Lagrange's equation for a mechanical system. It reviews some generalities of G-manifolds and defines the momentum and show that it is an invariant of the motion. Relative equilibria correspond to critical points of a certain function on state space. Under a nondegeneracy hypothesis, the relative equilibria, considered as points in phase space, are precisely the critical points of the energy–momentum function. This result is essentially found for the special case of the planar n-body problem. The chapter discusses to the n-body problem and then to the planar n-body problem. The computations show where the difficulties in generalizing the results to the three-dimensional case lies.