VelocityVerletVerfahren

VelocityVerletVerfahren, also known as the Velocity Verlet algorithm, is a time integration scheme used to numerically solve Newton’s equations of motion. It belongs to the family of symplectic integrators and is widely employed in molecular dynamics, astrophysics, and other fields that require long‑term stability of phase‑space trajectories.

The method starts from Taylor expansions of position and velocity. By inserting the acceleration (force divided by mass) into a second‑order approximation for the position, one obtains a predictor step that yields the position at the next time step. A subsequent evaluation of the acceleration at the new position allows a corrector step that updates the velocity. The algorithm achieves second‑order accuracy in both position and velocity while preserving the symplectic structure, which is essential for accurate energy conservation over many time steps.

Algorithmically the procedure is:

1. Compute the half‑step velocity:

v(t+½Δt) = v(t) + (½ Δt) a(t).

2. Update position to the next instant:

r(t+Δt) = r(t) + Δt v(t+½Δt).

3. Evaluate the new acceleration a(t+Δt) from forces at r(t+Δt).

4. Complete the velocity update:

v(t+Δt) = v(t+½Δt) + (½ Δt) a(t+Δt).

This simple sequence avoids the need for higher‑order force evaluations and maintains time reversibility.

Compared with the basic Verlet scheme, Velocity Verlet stores velocities directly, which facilitates coupling to thermostat algorithms and integration of velocity‑dependent forces. It is computationally efficient, requiring only one force evaluation per time step, and is robust for stiff systems when combined with adaptive time‑stepping or constraint solvers such as SHAKE or RATTLE.