Energy conservation in systems with discrete time

Question

Numerical simulations of classical particle dynamics usually break energy conservation due to discretization of time. Is there any explicit numerical scheme that does not break energy conservation (without global rescaling of velocities)? Is it even possible for a particle systems with discretized time to have something like energy conservation law?

Answer 1

Yes. Symplectic integrators are designed to do exactly that. They are intended for solving Hamiltonian systems where the Hamiltonian is conserved.

https://en.wikipedia.org/wiki/Symplectic_integrator