Periodic Solutions of Symmetric Hamiltonian Systems

Abstract

Until now, the dynamical systems under consideration have been dissipative ones — that is, we have not required energy to be conserved (and indeed the concept of energy has not been given any emphasis). For many models arising from mechanics, however, it is important to build in the constraint of energy-conservation. Historically this development led to a general formulation of mechanics in terms of ‘Hamiltonian systems’. In this final chapter we consider effects of symmetry on Hamiltonian dynamics. This is a very broad area, and we shall discuss only certain topics within it that link to the ideas developed earlier. Our approach is closely related to the technique of ‘reduction’ in symmetric Hamiltonian systems — essentially the passage to the orbit space of the symmetry group — which has been extensively developed and applied: see for example Abraham and Marsden [1], Lewis [356, 357], Lewis and Ratiu [358], Lewis and Simo [359], Lewiset al.[360], Marsden [374], Marsden and Weinstein [376], Ortega and Ratiu [419], Patrick [426, 427], Roberts and Dias [451].