Variational principles are at the core of the formulation of mechanical problems. What happens in the presence of symmetry when variables can be eliminated? I will discuss the geometry underlying this reduction process and present the induced constrained variational principle and the associated Euler-Lagrange equations. The rigid body and the Euler equations for ideal fluids are examples of such reduced Euler-Lagrange equations in convective and spatial representations, respectively. This geometric structure permits the introduction of a new class of optimal control problems that have the remarkable property that the control satisfies precisely these reduced Euler-Lagrange equations. As an example, it is shown that geodesic motion for the normal metric can be controlled by geodesics on the symmetry group. In the case of fluids, these optimal control problems yield the classical Clebsch variables and singular solutions for the Camassa-Holm equation. Relaxing the constraint to a quadratic penalty yields associated optimization problems. Time permitting, the equations of metamorphosis dynamics in imaging will be deduced from this optimization problem.