Symplectic integrator

In mathematics, a symplectic integrator (SI) is a numerical integration scheme for Hamiltonian systems. Symplectic integrators form the subclass of geometric integrators which, by definition, are canonical transformations. They are widely used in nonlinear dynamics, molecular dynamics, discrete element methods, accelerator physics, plasma physics, quantum physics, and celestial mechanics.

Symplectic integrators are designed for the numerical solution of Hamilton's equations, which read

where ${\displaystyle q}$ denotes the position coordinates, ${\displaystyle p}$ the momentum coordinates, and ${\displaystyle H}$ is the Hamiltonian. The set of position and momentum coordinates ${\displaystyle (q,p)}$ are called canonical coordinates. (See Hamiltonian mechanics for more background.)

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