  Current molecular dynamic simulations of biomolecules using multiple time
steps to update the slowingly changing force are hampered by an instability
occuring at time step equal to half the period of the fastest vibrating mode.
This has became a critical barrier preventing the long time simulation of
biomolecular dynamics. Attemps to tame this instability by altering the slowly
changing force and efforts to damp out this instability by Langevin dynamics do
not address the fundamental cause of this instability. In this work, we trace
the instability to the non-analytic character of the underlying spectrum and
show that a correct splitting of the Hamiltonian, which render the spectrum
analytic, restores stability. The resulting Hamiltonian dictates that in
additional to updating the momentum due to the slowly changing force, one must
also update the position with a modified mass. Thus multiple-timestepping must
be done dynamically.
