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Discrete classical molecular dynamics using the Verlet algorithm (VA) possesses a hidden energy invariance. This finding questions the formulation of classical dynamics and its relation to Newtonian dynamics.

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Area of Science:

  • Computational Physics
  • Classical Mechanics
  • Molecular Dynamics

Background:

  • Discrete classical molecular dynamics simulations often employ numerical algorithms like the Verlet algorithm (VA).
  • The VA introduces time discretization, potentially leading to deviations from true Hamiltonian dynamics.
  • Previous work suggested the existence of a 'shadow Hamiltonian' for VA, where discrete trajectories approximate analytic ones.

Purpose of the Study:

  • To rigorously prove the existence of an exact hidden energy invariance for Verlet algorithm dynamics.
  • To investigate the implications of this invariance for the formulation of classical dynamics.
  • To discuss the relationship between discrete VA dynamics and other discrete dynamics models.

Main Methods:

  • Mathematical proof to demonstrate the existence of a conserved quantity (hidden energy invariance) in VA.
  • Analysis of the discrete dynamics generated by the Verlet algorithm.
  • Comparison of VA dynamics with continuous Newtonian dynamics and other discrete dynamics formulations.

Main Results:

  • An exact hidden energy invariance, denoted E(*), is proven to exist for discrete classical molecular dynamics simulated with the Verlet algorithm.
  • This invariance is independent of the existence of a 'shadow Hamiltonian' or analytic approximations.
  • The discrete VA dynamics exhibit the same fundamental invariances as continuous Newtonian dynamics.

Conclusions:

  • The Verlet algorithm in molecular dynamics possesses an inherent exact energy invariance.
  • This finding challenges the interpretation of discrete dynamics and raises questions about the most appropriate formulation of classical mechanics.
  • The study clarifies the connection between VA and other discrete dynamics models, such as Lee's work.