Variational Integrators for Noncanonical Hamiltonian Systems

We propose a new approach to the derivation of variational integrators for noncanonical Hamiltonian systems of the form \(\dot{q} = \bar{\Omega}^{-1} (q) \, \nabla H (q)\). Such systems originate from degenerate Lagrangians of the form \(L (q, \dot{q}) = \left< \vartheta(q) , \dot{q} \right> - H(q)\), where \(\vartheta\) is a general, possibly nonlinear function of \(q\) and \(\bar{\Omega} = \nabla \vartheta^{T} - \nabla \vartheta\).
We do not directly discretise the degenerate Lagrangian as usually done with variational integrators, but instead we construct a formal Lagrangian and derive discrete Euler-Lagrange equations for the corresponding extended system. After a simple reduction procedure, we obtain variational Runge-Kutta integrators of the dynamics of the original system. We discuss gauge invariance and symplecticity and show how a generalised discrete Noether theorem can be used in order to prove conservation of momentum maps of the discrete system.