Symmetric Reduction of a Regular Controlled Lagrangian System with a Momentum Map

Abstract

In this paper, the author first defines a regular controlled Lagrangian (RCL for short) system on a symplectic fiber bundle, establishing a good expression of the dynamical vector field of an RCL system. This dynamical vector field synthesizes the Euler-Lagrange vector field and its changes under the actions of the external force and the control. Moreover, the author describes the RCL-equivalence, the RpCL-equivalence, and the RoCL-equivalence, proving regular point and regular orbit reduction theorems for the RCL system and the regular Lagrangian system with symmetry and a momentum map. Finally, as an application the author considers the regular point reducible RCL systems on a generalization of Lie group.

Keywords

Regular controlled Lagrangian system / Legendre transformation / RCL-equivalence / Momentum map / Regular point reduction / Regular orbit reduction / 70H33 / 53D20 / 70Q05

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