With leapfrog you can use x + v dt/2 for the positions to calculate the energy. Or use this update rule: x += v*dt/2 v += a(x)*dt x += v*dt/2 This

With leapfrog you can use x + v dt/2 for the positions to calculate the energy. Or use this update rule: x += v*dt/2 v += a(x)*dt x += v*dt/2 This gives you the x,v at the same time, yet still doesn't require storing the accelerations. The acceleration calculation can imperatively update v :) In fact, if you call that update leapfrog(dt), then you can get a fourth order symplectic method like so: leapfrog(c1*dt) leapfrog(c2*dt) leapfrog(c1*dt) Where c1,c2 are particular coefficients which you have to ask Yoshida.