Current molecular dynamics simulations of biomolecules using multiple time steps to update the slowly changing force are hampered by instabilities beginning at time steps near the half period of the fastest vibrating mode. These “resonance” instabilities have became a critical barrier preventing the long time simulation of biomolecular dynamics. Attempts to tame these instabilities by altering the slowly changing force and efforts to damp them out by Langevin dynamics do not address the fundamental cause of these instabilities. In this work, we trace the instability to the nonanalytic character of the underlying spectrum and show that a correct splitting of the Hamiltonian, which renders the spectrum analytic, restores stability. The resulting Hamiltonian dictates that in addition to updating the momentum due to the slowly changing force, one must also update the position with a modified mass. Thus multiple-time stepping must be done dynamically.

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