LeapfrogVerlet

The leapfrog Verlet algorithm is a time integration method used to solve Newton’s equations of motion in systems such as molecular dynamics and celestial mechanics. It is a variant of Verlet integration that employs a staggered (half-step) update of the velocity, causing positions and velocities to “leapfrog” over each other. The method is explicit, second-order accurate in time, and is valued for its simplicity and good long-term stability.

A common formulation keeps positions x at full time steps and velocity (or velocity at half-steps) offset by half a time step. A typical sequence is:

- v at the half-step: v(t+Δt/2) = v(t) + (a(t) Δt)/2, where a(t) is the acceleration from the forces at x(t).

- Position update: x(t+Δt) = x(t) + v(t+Δt/2) Δt

- Compute new acceleration a(t+Δt) from x(t+Δt)

- Velocity update to the next half-step: v(t+Δt/2) = v(t+Δt/2) + (a(t+Δt) Δt)/2

This yields a symmetric, time-reversible integration scheme and is often described as symplectic, meaning it preserves the structure of Hamiltonian dynamics over long simulations.

Advantages include its explicitness, low computational overhead, and favorable energy behavior in conservative systems. It is widely used in molecular dynamics, N-body simulations, and other problems where long-term qualitative accuracy is important.

Limitations include reduced suitability for velocity-dependent or non-conservative forces without modification, sensitivity to stiff or highly coupled motions requiring smaller time steps, and the need for additional schemes (such as SHAKE or RATTLE) when applying holonomic constraints. The method was introduced in variations of the Verlet family in the 1960s and 1967 by Loup Verlet, with the leapfrog variant gaining prominence in computational physics and chemistry.