While great advances have been made in the field of intermolecular potentials for molecular modeling and material science, the choice in constant energy integration dynamics have been often relegated to the St{\o}rmer-Verlet. In the past 30 years the symplecticity of this method and relatively low local error in initial conditions have made this integrator a hallmark of stability. However, the need to perform N-body particle simulations to reach greater time domains has encouraged practitioners to approach higher timesteps and the stability limit.

This article explores implicit classes of adaptive Verlet algorithms for potential use in the two-stage coupled equations of motion. At the heart of these methods are state dependant, time re-scaling functions that offer bounded variable timesteps. There is not many demonstrations of their overall energy conservation for a choice in timestep value, beyond global error demonstrations in discrete time. Presented here is the derived linear stability for arbitrary time transformation functions with the implicit Adaptive Verlet scheme. Considerations for selecting a stable range of timesteps in fulfilling stable harmonic trajectory are also outlined.

Lastly we observe evidence that the implicit Adaptive scheme paired with a bounded timestep model may contain a family of solutions that contribute additional spectral elements. Regular modes of spectral peaks shown in our monochromatic system point towards additional solutions in the Hamiltonian system. Before moving towards variable timestep methods we demonstrate the need to develop a transformation functional that meets symmetric criteria outlined by previous authors.