Verlet integration

Verlet integration (French pronunciation: ​) 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 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 Cowell and 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. Verlet integration (French pronunciation: ​) 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 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 Cowell and 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 x → ¨ ( t ) = A → ( x → ( t ) ) {displaystyle {ddot {vec {x}}}(t)={vec {A}}{ig (}{vec {x}}(t){ig )}} with initial conditions x → ( t 0 ) = x → 0 {displaystyle {vec {x}}(t_{0})={vec {x}}_{0}} and x → ˙ ( t 0 ) = v → 0 {displaystyle {dot {vec {x}}}(t_{0})={vec {v}}_{0}} , an approximate numerical solution x → n ≈ x → ( t n ) {displaystyle {vec {x}}_{n}approx {vec {x}}(t_{n})} at the times t n = t 0 + n Δ t {displaystyle t_{n}=t_{0}+n,Delta t} with step size Δ t > 0 {displaystyle Delta t>0} can be obtained by the following method: