  This paper defines a class of variational problems on Lie groups that admit
involutive automorphisms. The maximum Principle of optimal control then
identifies the appropriate left invariant Hamiltonians on the Lie algebra of
the group. The corresponding Hamiltonian equations admit a spectral
representation with important applications to the theory of integrable systems.
Particular cases make a link to Moser's work relating the solutions of Kepler's
problem to the geodesics on spaces of constant curvature, Newmann's mechanical
problem on the sphere and related geodesic problem of Jacobi on an ellipsoid.
