# Lagrangian Reduction by Stages

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*Hernán Cendra; Jerrold E. Marsden; Tudor S. Ratiu*

This booklet studies the geometry of the reduction of Lagrangian systems with
symmetry in a way that allows the reduction process to be repeated; that is, it
develops a context for Lagrangian reduction by stages}. The Lagrangian
reduction procedure focuses on the geometry of variational structures and how
to reduce them to quotient spaces under group actions. This philosophy is well
known for the classical cases, such as Routh reduction for systems with cyclic
variables (where the symmetry group is Abelian) and Euler-Poincaré
reduction (for the case in which the configuration space is a Lie group) as
well as Euler-Poincaré reduction for semidirect products.

The context established for this theory is a Lagrangian analogue
of the bundle picture on the Hamiltonian side. In this picture, we
develop a category that includes, as a special case, the
realization of the quotient of a tangent bundle as the Whitney sum
of the tangent of the quotient bundle with the associated adjoint
bundle. The elements of this new category, called the
Lagrange–Poincaré category, have enough geometric
structure so that the category is stable under the procedure of
Lagrangian reduction. Thus, reduction may be repeated, giving the
desired context for reduction by stages. Our category may be viewed as a
Lagrangian analog of the category of Poisson manifolds in Hamiltonian
theory.

We also give an intrinsic and geometric way of writing the reduced
equations, called the Lagrange–Poincaré equations, using
covariant derivatives and connections. In addition, the context includes
the interpretation of cocycles as curvatures of connections and is general
enough to encompass interesting situations involving both semidirect products
and central extensions. Examples are given to illustrate the general
theory.

In classical Routh reduction one usually sets the conserved quantities
conjugate to the cyclic variables equal to a constant. In our development, we
do not require the imposition of this constraint. For the general theory along
these lines, we refer to the complementary work of [2000], which studies the
Lagrange-Routh equations.

#### Table of Contents

# Table of Contents

## Lagrangian Reduction by Stages

- Contents vii8 free
- Chapter 1. Introduction 112 free
- Chapter 2. Preliminary Constructions 718
- Chapter 3. The Lagrange–Poincaré Equations 1728
- Chapter 4. Wong's Equations and Coordinate Formulas 3748
- Chapter 5. The Lie Algebra Structure on Sections of the Reduced Bundle 4354
- Chapter 6. Reduced Tangent Bundles 5162
- Chapter 7. Further Examples 8596
- Chapter 8. The Category LB* and Poisson Geometry 95106
- Bibliography 103114