Euler–Poincaré Reduction by a Subgroup of Symmetries as an Optimal Control Problem

Abstract

Given a Lagrangian density Ldt defined in the 1-jet bundle $J^{1}P$ of a principal G-bundle $P\rightarrow \mathbb {R}$, invariant with respect to the action of a closed subgroup $H\subset G$, its Euler–Poincaré reduction in $(J^{1}P)/H=C(P)\times _{\mathbb {R}}(P/H)$ (C(P): the bundle of connections, P / H: the bundle of H-structures) induces an optimal control problem. The control variables of this problem are connections $\sigma$, the dynamical variables $\bar{s}$ are H-structures, the Lagrangian density $l(t,\sigma ,\bar{s})dt$ is the reduction of Ldt and the dynamical equations are $\nabla ^{\sigma }\bar{s}=0$. We prove that the solution of this problem are solutions of the original reduction problem. We study the Hamilton–Cartan–Pontryagin formulation of the problem under an appropriate regularity condition. Finally, the theory is illustrated with the example of the heavy top, for which the symplectic structure of the set of solutions with zero vertical component of the angular momentum is also provided.

Dedicated to our friend and colleague Jaime Muñoz Masqué on the occasion of his 65th birthday