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.
Previous Page Next Page