Variational reduction of Lagrangian systems with general constraints

Sergio Grillo; Marcela Zuccalli

doi:10.3934/jgm.2012.4.49

Article Contents

2012, Volume 4, Issue 1: 49-88. Doi: 10.3934/jgm.2012.4.49

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Variational reduction of Lagrangian systems with general constraints

Sergio Grillo^1, and
Marcela Zuccalli^2,

1.
Instituto Balseiro, U.N. de Cuyo - C.N.E.A., San Carlos de Bariloche, R8402AGP
2.
Departamento de Matemtica, Facultad de Ciencias Exactas, U.N.L.P., La Plata, Buenos Aires

Received: October 2011

Revised: April 2012

Published: March 2012

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Abstract

In this paper we present an alternative procedure for reducing, in the Lagrangian formalism, the equations of motion of first order constrained mechanical systems with symmetry. The procedure involves two principal connections: one of them is used to define the reduced degrees of freedom and the other one to decompose variations into horizontal and vertical components. On the one hand, we show that this new procedure is particularly useful when the configuration space is a trivial principal bundle over the symmetry group, which is the case of many interesting examples. On the other hand, based on that procedure, we extend in a natural way the variational reduction methods to the Lagrangian systems with higher order constraints. Examples are discussed in order to illustrate the involved theorethical constructions.

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Mathematics Subject Classification: Primary: 37J05, 70H30; Secondary: 70H50.

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