Invariant higher-order variational problems: Reduction, geometry and applications

Abstract

This thesis is centred around higher-order invariant variational problems defined on Lie groups. We are mainly motivated by applications in computational anatomy and quantum control, but the general framework is relevant in many other contexts as well. We first develop a higher-order analog of Euler–Poincare reduction theory for variational problems with symmetry and discuss the important examples of Riemannian cubics and their higher-order generalisations. The theory is then applied to higher-order template matching and the optimal curves on the Lie group of transformations are shown to satisfy higher-order Euler–Poincare equations. Motivated by questions of model selection in interpolation problems of computational anatomy, we then study the relationship between Riemannian cubics on manifolds with a group action (‘object manifolds’) and Riemannian cubics on the corresponding group itself. It is shown, for example, that in Type I symmetric spaces only those Riemannian cubics can be lifted horizontally that lie in flat, totally geodesic submanifolds. We then return to higher-order template matching and provide an alternative derivation of the Euler–Lagrange equations using Lagrange multipliers, which leads to a geometric interpretation of the equations in terms of higher-order Legendre–Ostrogradsky momenta. Building on this approach, we develop a variational integrator that respects the geometric properties of continuous-time solution curves. We also derive the corresponding adjoint equations. The remainder of the thesis is concerned with an application to quantum control, namely, to the problem of experimentally steering a quantum system through a series of target states at prescribed times. We show that the Euler–Lagrange equations lead to Riemannian cubic splines on the special unitary group, under whose action the system evolves optimally. Finally, we perform numerical experiments for two-level quantum systems and extend the formalism to the control of coherent states in bosonic multi-particle systems.