Analysis of adaptive higher order exponential variational integrators

Variational integrators are powerful tools for advanced numerical solutions of mechanical problems
appeared in mathematics and physical sciences. Compared to standard schemes they may be applied to complex
systems where the computational cost is very high. In the present paper, we make an attempt to explore whether
their methodology may be effective in adaptive time step variational integrators with the use of the space-time
geodesic approach of classical mechanics while being combined with a simultaneous decrease, as much as
possible, of the corresponding cost. Following the advantages of our previously deduced variational
integrators, we now formulate a derivation of time adaptive high order exponential variational integrators. As a
first step, this is successfully achieved for systems of which their Lagrangian is of separable form. Towards this
end, we start from unfolding the standard Euler-Lagrange system to its space-time manifold and then we rewrite
it as a geodesic problem with zero potential energy. Simulation results, without the need to optimise the step
size, show that one can employ the space-time geodesic formulation to generate an adaptive scheme that still
preserves general underlying geometric structure properties of the system