Hamilton-Pontryagin Integrators on Lie Groups

In this thesis structure-preserving time integrators for mechanical systems whose configuration space is a Lie group are derived from a Hamilton-Pontryagin (HP) variational principle. In addition to its attractive properties for degenerate mechanical systems, the HP viewpoint also affords a practical way to design discrete Lagrangians, which are the cornerstone of variational integration theory. The HP principle states that a mechanical system traverses a path that extremizes an HP action integral. The integrand of the HP action integral consists of two terms: the Lagrangian and a kinematic constraint paired with a Lagrange multiplier (the momentum). The kinematic constraint relates the velocity of the mechanical system to a curve on the tangent bundle. This form of the action integral makes it amenable to discretization. In particular, our strategy is to implement an s-stage Runge-Kutta-Munthe-Kaas (RKMK) discretization of the kinematic constraint. We are motivated by the fact that the theory, order conditions, and implementation of such methods are mature. In analogy with the continuous system, the discrete HP action sum consists of two parts: a weighted sum of the Lagrangian using the weights from the Butcher tableau of the RKMK scheme, and a pairing between a discrete Lagrange multiplier (the discrete momentum) and the discretized kinematic constraint. In the vector space context, it is shown that this strategy yields a well-known class of symplectic partitioned Runge-Kutta methods including the Lobatto IIIA-IIIB pair which generalize to higher-order accuracy. In the Lie group context, the strategy yields an interesting and novel family of variational partitioned Runge-Kutta methods. Specifically, for mechanical systems on Lie groups we analyze the ideal context of EP systems. For such systems the HP principle can be transformed from the Pontryagin bundle to a reduced space. To set up the discrete theory, a continuous reduced HP principle is also analyzed. It is this reduced HP principle that we apply our discretization strategy to. The resulting integrator describes an update scheme on the reduced space. As in RKMK we parametrize the Lie group using coordinate charts whose model space is the Lie algebra and that approximate the exponential map. Since the Lie group is non abelian, the structure of these integrators is not the same as in the vector space context. We carry out an in-depth study of the simplest integrators within this family that we call variational Euler integrators; specifically we analyze the integrator's efficiency, global error, and geometric properties. Because of their variational character, the variational Euler integrators preserve a discrete momentum map and symplectic form. Moreover, since the update on the configuration space is explicit, the configuration updates exhibit no drift from the Lie group. We also prove that the global error of these methods is second order. Numerical experiments on the free rigid body and the chaotic dynamics of an underwater vehicle reveal that these reduced variational integrators possess structure-preserving properties that methods designed to preserve momentum (using the coadjoint action of the Lie group) and energy (for example, by projection) lack. In addition we discuss how the HP integrators extend to a wider class of mechanical systems with, e.g., configuration dependent potentials and non trivial shape-space dynamics.