Verlet Integration

The Verlet algorithm and its velocity form are the standard integrators of molecular dynamics, prized because they are time-reversible, symplectic and conserve energy well over the millions of steps a simulation requires.

Definition

Verlet integration is a time-reversible, symplectic method for integrating Newton's equations of motion that updates particle positions using current and previous positions and the acceleration, yielding stable trajectories for molecular dynamics.

Scope

This topic covers the Verlet family of integrators: the original position Verlet scheme, the equivalent leapfrog and velocity Verlet formulations, their time-reversibility and symplectic structure, and the resulting long-term energy conservation. It situates these methods within the broader theory of symplectic integration of Hamiltonian systems.

Core questions

• How does the Verlet scheme advance positions and velocities from the forces?
• Why is the Verlet algorithm time-reversible and symplectic?
• Why does Verlet integration conserve energy well over very long simulations?
• How do the position, leapfrog and velocity Verlet formulations relate to one another?

Key theories

Symplectic and time-reversible structure

Verlet integration preserves the symplectic geometry of phase space and is invariant under time reversal, which together prevent the systematic energy drift that plagues non-symplectic integrators of conservative systems.

Shadow Hamiltonian conservation

Although the discrete Verlet trajectory does not exactly conserve the true energy, it nearly conserves a closely related shadow Hamiltonian, keeping the energy error bounded and oscillatory rather than growing.

Equivalent formulations

The position Verlet, leapfrog and velocity Verlet schemes produce the same trajectory but differ in how and when velocities are available, with velocity Verlet being preferred when synchronized positions and velocities are needed.

Clinical relevance

Verlet integration is the default time-stepping engine in essentially all molecular dynamics codes, from simple Lennard-Jones fluids to large biomolecular simulations, and the same symplectic principle is used in long-term orbital integration in astronomy.

History

The scheme was used by astronomer Carl Stormer early in the twentieth century and popularized for molecular simulation by Loup Verlet in his 1967 study of Lennard-Jones fluids; later analysis showed it to be a symplectic integrator, explaining its excellent long-term stability.

Key figures

• Loup Verlet
• Carl Stormer
• Ernst Hairer

Seminal works

• verlet1967
• hairer1993

Frequently asked questions

Why is Verlet preferred over a higher-order Runge-Kutta method in molecular dynamics?

Although Runge-Kutta can be more accurate per step, it is not symplectic and slowly drifts in energy over long runs. Verlet's symplectic, time-reversible structure keeps the energy bounded over millions of steps, which matters far more than per-step accuracy for equilibrium simulations.

Does Verlet conserve energy exactly?

No. It conserves a nearby shadow Hamiltonian rather than the exact energy, so the measured energy oscillates within a bounded band instead of drifting away, which is sufficient for computing stable thermodynamic averages.