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In mathematics, a symplectic integrator (SI) 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. They are widely used in molecular dynamics, discrete element methods, accelerator physics, and celestial mechanics. == Introduction == 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 two-form . A numerical scheme is a symplectic integrator if it also conserves this two-form. Symplectic integrators possess, as a conserved quantity, a Hamiltonian which is slightly perturbed from the original one. 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. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「symplectic integrator」の詳細全文を読む スポンサード リンク
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