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.