CHAPTER 1

Introduction

Reduction theory for mechanical systems with symmetry has its origins in the

classical works of Euler, Lagrange, Hamilton, Routh, Jacobi, Liouville and Poincare,

who studied the extent to which one can reduce the dimension of the phase space of

the system by making use of any available symmetries and associated conservation

laws. Corresponding to the main two views of mechanics, namely Hamiltonian and

Lagrangian mechanics, one can also adopt two views of reduction theory.

In symplectic and Poisson reduction, which are now well developed and much

studied subjects, one focuses on how to pass the symplectic two form and the

Poisson bracket structure as well as any associated Hamiltonian dynamics to a

quotient space for the action of a symmetry group (see, for example, Meyer [1973],

Marsden and Weinstein [1974], Marsden and Ratiu [1986] and the expositions in

Abraham and Marsden [1978], Arnold [1989], Libermann and Marie [1987] and

Marsden [1992]).

In Lagrangian reduction theory, which proceeds in a logically independent way,

one emphasizes how the variational structure passes to a quotient space (see, for

example, Cendra and Marsden [1987], Cendra, Ibort and Marsden [1987], Mars-

den and Scheurle [1993a, 1993b], Bloch, Krishnaprasad, Marsden and Ratiu [1996],

Cendra, Holm, Marsden and Ratiu [1998], Holm, Marsden and Ratiu [1998a], Jal-

napurkar and Marsden [2000] and Marsden, Ratiu and Scheurle [2000]). Of course,

the two methodologies are related by the Legendre transform, although not always

in a straightforward way.

The main purpose of this work is to further the development of Lagrangian

reduction theory. There are several aspects to this program. First, we provide

a context that allows for repeated Lagrangian reduction by the action of a sym-

metry group. Second, we provide the geometry that is useful for the expression

of the reduced equations, called the Lagrange-Poincare equations. Further details

concerning the main results of this work are given shortly.

1.1. Background

As we have mentioned, in the last few years there has been considerable ac-

tivity in the area of Lagrangian reduction in which one focuses on the reduction

of variational principles. We shall review the background for this theory briefly,

starting with the best known classical results.

Classical Cases. Several classical instances of Lagrangian reduction are well known,

such as Routh reduction which was developed by Routh [1877] in connection with

his studies of the stability of relative equilibria. Routh began the development of

what we would call today Lagrangian reduction for Abelian groups. One thinks of

this case as treating mechanical systems with cyclic variables.