This dissertation explores Hamiltonian variational integrators. Variational integrators are a common class of symplectic integrators, which have primarily been analyzed and constructed by discretizing Hamilton's principle. Hamiltonian variational integrators are derived by discretizing Hamilton's phase space principle and have not been studied as thoroughly. In this dissertation new error analysis theorems and other related results extend the theory of Hamiltonian variational integrators. It is shown that these two formulations of variational integrators are not always numerically equivalent, even when they analytically represent the same map. Numerical simulations show there can be important differences between these two formulations of variational integrators, particularly for averaging methods

Next, a new class of variational integrators is developed based on the Taylor method combined with an augmented shooting method. A symmetric and more computationally efficient version is also developed, as well as a comparison of Lagrangian and Hamiltonian formulations of the integrator. Error analysis results are presented, in addition a proof is given of a sufficient condition for the equivalence of a Hamiltonian and Lagrangian variational integrator. Numerical simulations are presented, as well as a discussion on the role of automatic differentiation in the implementation of Taylor variational integrators.

The last topic focuses on an adaptive framework for symplectic integrators. The Poincare transformation is used to construct an extended Hamiltonian system, which allows for variable step sizes. However, it is shown that the resulting Hamiltonian is often degenerate, and the only plausible framework for variational integrators is to use Hamiltonian variational integrators. Furthermore, the degeneracy of the Hamiltonian is discussed with regards to error analysis and the invertibility of the discrete Legendre transforms. A few monitor functions are considered, and numerical simulations demonstrate the significant gains in efficiency when using adaptive variational integrators.