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Concept# Symplectic integrator

Summary

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 denotes the position coordinates, the momentum coordinates, and is the Hamiltonian.
The set of position and momentum coordinates are called canonical coordinates.
(See Hamiltonian mechanics for more background.)
The time evolution of Hamilton's equations is a symplectomorphism, meaning that it conserves the symplectic 2-form . A numerical scheme is a symplectic integrator if it also conserves this 2-form.
Symplectic integrators also might possess, as a conserved quantity, a Hamiltonian which is slightly perturbed from the original one (only true for a small class of simple cases). By virtue of these advantages, the SI scheme has been widely applied to the calculations of long-term evolution of chaotic Hamiltonian systems ranging from the Kepler problem to the classical and semi-classical simulations in molecular dynamics.
Most of the usual numerical methods, like the primitive Euler scheme and the classical Runge–Kutta scheme, are not symplectic integrators.
A widely used class of symplectic integrators is formed by the splitting methods.
Assume that the Hamiltonian is separable, meaning that it can be written in the form
This happens frequently in Hamiltonian mechanics, with T being the kinetic energy and V the potential energy.
For the notational simplicity, let us introduce the symbol to denote the canonical coordinates
including both the position and momentum coordinates. Then, the set of the Hamilton's equations given in the introduction can be expressed in a single expression as
where is a Poisson bracket.

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