Variational integrators are numerical integrators

Numerical ordinary differential equations

Numerical ordinary differential equations is the part of numerical analysis which studies the numerical solution of ordinary differential equations...

 for Hamiltonian system

Hamiltonian system

In physics and classical mechanics, a Hamiltonian system is a physical system in which forces are momentum invariant. Hamiltonian systems are studied in Hamiltonian mechanics....

s derived from the Euler-Lagrange equations of a discretized Hamilton's principle

Hamilton's principle

In physics, Hamilton's principle is William Rowan Hamilton's formulation of the principle of stationary action...

. Variational integrators are momentum-preserving and symplectic

Symplectic integrator

In mathematics, a symplectic integrator is a numerical integration scheme for a specific group of differential equations related to classical mechanics and symplectic geometry. Symplectic integrators form the subclass of geometric integrators which, by definition, are canonical transformations...

.

Derivation of a Simple Variational Integrator

Consider a mechanical system with a single particle degree of freedom described by the Lagrangian

,

where is the mass of the particle, and is a potential. To construct a variational integrator for this system, we begin by forming the discrete Lagrangian. The discrete Lagrangian approximates the action for the system over a short time interval:

.

Here we have chosen to approximate the time integral using the trapezoid method, and we use a linear approximation to the trajectory,

between and , resulting in a constant velocity . Different choices for the approximation to the trajectory and the time integral give different variational integrators. The order of accuracy of the integrator is controlled by the accuracy of our approximation to the action; since

,

our integrator will be second-order accurate.

Evolution equations for the discrete system can be derived from a stationary-action principle. The discrete action over an extended time interval is a sum of discrete Lagrangians over many sub-intervals:

.

The principle of stationary action states that the action is stationary with respect to variations of coordinates that leave the endpoints of the trajectory fixed. So, varying the coordinate , we have

.

Given an initial condition , and a sequence of times this provides a relation that can be solved for . The solution is

.

We can write this in a simpler form if we define the discrete momenta,

and

.

Given an initial condition , the stationary action condition is equivalent to solving the first of these equations for , and then determining using the second equation. This evolution scheme gives

and

.

This is a leapfrog integration

Leapfrog integration

Leapfrog integration is a simple method for numerically integrating differential equations of the form\ddot x=F,or equivalently of the form\dot v=F,\;\dot x \equiv v,particularly in the case of a dynamical system of classical mechanics...

scheme for the system; two steps of this evolution are equivalent to the formula above for