Verlet integration

Verlet integration (vɛʁˈlɛ) is a numerical method used to integrate Newton's equations of motion. It is frequently used to calculate trajectories of particles in molecular dynamics simulations and computer graphics. The algorithm was first used in 1791 by Jean Baptiste Delambre and has been rediscovered many times since then, most recently by Loup Verlet in the 1960s for use in molecular dynamics. It was also used by P. H. Cowell and A. C. C. Crommelin in 1909 to compute the orbit of Halley's Comet, and by Carl Størmer in 1907 to study the trajectories of electrical particles in a magnetic field (hence it is also called Störmer's method). The Verlet integrator provides good numerical stability, as well as other properties that are important in physical systems such as time reversibility and preservation of the symplectic form on phase space, at no significant additional computational cost over the simple Euler method. For a second-order differential equation of the type with initial conditions and , an approximate numerical solution at the times with step size can be obtained by the following method: set , for n = 1, 2, ... iterate Newton's equation of motion for conservative physical systems is or individually where is the time, is the ensemble of the position vector of objects, is the scalar potential function, is the negative gradient of the potential, giving the ensemble of forces on the particles, is the mass matrix, typically diagonal with blocks with mass for every particle. This equation, for various choices of the potential function , can be used to describe the evolution of diverse physical systems, from the motion of interacting molecules to the orbit of the planets. After a transformation to bring the mass to the right side and forgetting the structure of multiple particles, the equation may be simplified to with some suitable vector-valued function representing the position-dependent acceleration. Typically, an initial position and an initial velocity are also given.