Symmetry reduction of discrete Lagrangian mechanics on Lie groups

2000/04/04 by Marsden, Jerrold E., Pekarsky, Sergey, Shkoller, Steve
#FOS: Mathematics #Numerical Analysis (math.NA) #Symplectic Geometry (math.SG)

paper · doi:10.48550/arxiv.math/0004018

Abstract

For a discrete mechanical system on a Lie group G determined by a (reduced) Lagrangian ℓ we define a Poisson structure via the pull-back of the Lie-Poisson structure on the dual of the Lie algebra \mathfrak g^* by the corresponding Legendre transform. The main result shown in this paper is that this structure coincides with the reduction under the symmetry group G of the canonical discrete Lagrange 2-form ω_\mathbbL on G × G. Its symplectic leaves then become dynamically invariant manifolds for the reduced discrete system. Links between our approach and that of groupoids and algebroids as well as the reduced Hamilton-Jacobi equation are made. The rigid body is discussed as an example.

Related

• Discrete Lagrangian reduction, discrete Euler-Poincare equations, and semidirect products
• Discrete Euler-Poincaré and Lie-Poisson Equations
• The local description of discrete Mechanics
• Reduction by symmetries of contact mechanical systems on Lie groups
• Tulczyjew's Triplet for Lie Groups II: Dynamics