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This paper studies the reduction by symmetry of variational problems on Liegroups and Riemannian homogeneous spaces. We derive the reduced equations ofmotion in the case of Lie groups endowed with a left-invariant metric, and onLie groups that admits a bi-invariant metric. We repeated this analysis forRiemannian homogeneous spaces, where we derive the reduced equations byconsidering an alternative variational problem written in terms of a connectionon the horizontal bundle of the underlying Lie group. We study also the casethat the underlying Lie group admits a bi-invariant metric, and consider thespecial case that the homogeneous space is in fact a Riemannian symmetricspace. These ideas are applied to geodesics for a rigid body on $SO(3)$ toderive geodesic equations on the dual of its Lie algebra (a vector space), theheavy-top in $SE(3)$ to derive reduced equations of motion on the unit sphere$S^2$, geodesics on $S^2$ as a Riemannian symmetric space endowed with abi-invariant metric and optimal control problems for applications to roboticmanipulators.

Reduction by Symmetry and Optimal Control with Broken Symmetries on Riemannian Manifolds