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Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. 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.The time evolution of Hamilton''s equations is a symplectomorphism, meaning that it conserves the symplectic two-form dp wedge dq. A numerical scheme is a symplectic integrator if it also conserves this two-form.