Symplectic and Reversible Integrators

In an Hamiltonian problem, the symplectic condition and microscopic reversibility are inherent properties of the true time trajectories which, in turn, are the exact solution of Hamilton's equation. A step-wise integration defines a $t$-flow mapping which may or may not retain these properties. Non symplectic and/or non reversible integrators are generally believed [68,69,70,71] to be less stable in the long-time integration of Hamiltonian systems. In this section we shall illustrate the concept of reversible and symplectic mapping in relation to the numerical integration of the equations of motion.