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}
    q_{t+\delta/2} &= q_t + \frac{\delta}{2}\dot{q}(q_t, p_t) \\ 
    p_{t + \delta} &= p_t + \delta \dot{p}(q_{t+\delta}, p_t) \\
    q_{t+\delta} &= q_{+\delta/2} + \frac{\delta}{2}\dot{q}(q_{t+\delta/2}, p_{t+\delta})
\end{align}

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