I was reading about different integrators that one might use to solve the system of differential equations which governs the $n$-body problem. I read that the Verlet integrator is time-reversible, and thus conserves energy.
I don't understand why time-reversibility implies energy conservation.
Suppose I know the position and velocity of each of $n$ particles in the system at some time $t_0$. I calculate the energy $E_0$ of the system at time $t_0$. I use the Verlet integrator to calculate the approximate positions and velocity of the particles at time $t_0+dt$ (one time step) and I recalculate the energy of the system and find it to be $E_1$. Is it true that $E_0=E_1$? And is there an easy way to prove this?