2 Example : The Nonlinear Pendulum

We provide new existence and uniqueness results for the discrete-time Hamilton (DTH) equations of a symplectic-energy-momentum (SEM) integrator. In particular, we identify points in extended-phase space where the DTH equations of SEM integration have no solution for arbitrarily small time steps. We use the nonlinear pendulum to illustrate the main ideas. 1 Background Is symplectic-energy-momentum integration well-posed? Loosely speaking, the answer is no. Points exist in the extended phase-space of a Hamiltonian system where the equations of a symplectic-energy-momentum (SEM) inte-grator have no solution for arbitrarily small time steps. Before considering this question in more detail, we provide a brief review of SEM integration. Hamiltonian dynamics is at the heart of modern physics and arises naturally in applications such as optimal control theory and geometric optics. Hamiltonian dynamics is also the inspiration for the relatively new field of symplectic geometry. A symplectic-energy-momentum (SEM) integrator is a numerical integrator that preserves the following key properties associated * Dedicated to the memory of my father Shibberu Wolde Mariam.