Conservation of First Integrals and Methods on Manifolds

Abstract

This chapter deals with the conservation of invariants (first integrals) by numerical methods, and with numerical methods for differential equations on manifolds. Our investigation will follow two directions. We first investigate which of the methods introduced in Chap. II conserve invariants automatically. We shall see that most of them conserve linear invariants, a few of them quadratic invariants, and none of them conserves cubic or general nonlinear invariants. We then construct new classes of methods, which are adapted to known invariants and which force the numerical solution to satisfy them. In particular, we study projection methods and methods based on local coordinates of the manifold defined by the invariants. We discuss in some detail the case where the manifold is a Lie group.