Second Order Lagrangian Dynamics On Double Cross Product Groups

Abstract

We observe that the iterated tangent group of a Lie group may be realized as a double cross product of the 2nd order tangent group, with the Lie algebra of the base Lie group. Based on this observation, we derive the 2nd order Euler-Lagrange equations on the 2nd order tangent group from the 1st order Euler-Lagrange equations on the iterated tangent group. We also present in detail the 2nd order Lagrangian dynamics on the 2nd order tangent group of a double cross product group.