Even a low-order symplectic integrator will conserve the phase-space volume of a problem, and as such, never introduce or remove energy systematically. Below is a demo of how a second order symplectic integrator can outperform a fourth order non-symplectic integrator. Beware, of course, that if your Hamiltonian is not formulated to conserve a quantity, a symplectic integrator will not guarantee that quantity is conserved (in this case, the phase angle of our orbit).

### A Simple Orbital Problem

A particle orbiting another in a central inverse-square potential has the following Hamiltonian, for the position vector $p$ and momentum vector $q$. $$H = -\frac{1}{q}+\frac{p^2}{2}$$ Solving Hamilton's equations gives these equations of motion: $$\dot p = {|q|}^{-2}$$ $$\dot q = p$$ These are the equations of motion for a planet orbiting a star, for example. Let's take a look at what happens when we throw an elliptical orbit set of initial conditions at two different integrators, and see how they conserve energy and angular momentum.