Consider a general set of Hamilton's equations $$ \begin{align} \dot{q}(q, p) &= \partial_p H(q, p) \\ \dot{p}(q, p) &= -\partial_q H(q, p) \end{align} $$ A first-order integrator one could use is the sympletic-Euler. There are two second-order methods I know of that can be used to numerically integrate these equations: Leapfrog and position-Verlet.

Is there any difference at all between using Leapfrog or position-Verlet?

Leapfrog (second order, sympletic, reversible)

This is like a sympletic Euler but sandwiched between a half-momentum step at the beginning and at the end. $$ \begin{align} p_{t+\delta/2} &= p_t + \frac{\delta}{2}\dot{p}(q_t, p_t) \\ q_{t+\delta} &= q_t + \delta \dot{q}(q_t, p_{t+\delta/2}) \\ p_{t+\delta} &= p_{t+\delta/2} + \frac{\delta}{2}\dot{p}(q_{t+\delta}, p_{t+\delta/2}) \end{align} $$

Position-Verlet (second order, sympletic)

Similar to the Leapfrog method but starts with a half-position update rather than half-momentum update. \begin{align} p_{t+\delta/2} &= p_t + \frac{\delta}{2}\dot{p}(q_t, p_t) \\ q_{t+\delta} &= q_t + \delta \dot{q}(q_t, p_{t+\delta/2}) \\ p_{t+\delta} &= p_{t+\delta/2} + \frac{\delta}{2}\dot{p}(q_{t+\delta}, p_{t+\delta/2}) \end{align}

Is there any difference between using Leapfrog or position-Verlet? Either numerically or theoretically.