First step in Velocity Verlet

\[\vec{x}(\Delta t) = \vec{x}(0) + \vec{v}(0) \Delta t + \frac{1}{2} \vec{a}(0) \Delta t^2\] \[\vec{v}(\Delta t) = \vec{v}(0) + \frac{\vec{a}(0) + \vec{a}(\Delta t)}{2} \Delta t\]

Now then,

\[\vec{x}(2 \Delta t) = \vec{x}(\Delta t) + \vec{v}(\Delta t) \Delta t + \frac{1}{2} \vec{a}(\Delta t) \Delta t^2\]

\[\frac{2}{\Delta t^2} (\vec{x}(2 \Delta t) - \vec{x}(\Delta t) - \vec{v}(\Delta t) \Delta t) = \vec{a}(\Delta t)\]

So

\[\vec{v}(\Delta t) = \vec{v}(0) + \frac{1}{2}\vec{a}(0)\Delta t + \frac{1}{\Delta t} (\vec{x}(2 \Delta t) - \vec{x}(\Delta t) - \vec{v}(\Delta t) \Delta t)\]

\[2 \vec{v}(\Delta t) = \vec{v}(0) + \frac{1}{2}\vec{a}(0)\Delta t + \frac{1}{\Delta t} (\vec{x}(2 \Delta t) - \vec{x}(\Delta t))\]

Now

\[\frac{1}{\Delta t} (\vec{x}(2 \Delta t) - \vec{x}(\Delta t))\]

is a forward finite-difference approximation of \(\vec{v}(\Delta t)\), i.e.

\[\vec{x}(2 \Delta t) = \vec{x}(\Delta t) + \vec{\hat{v}}(\Delta t) \Delta t + \frac{1}{2} \vec{a} (\Delta t) \Delta t^2 + O(\Delta t^3)\]

\[\frac{\vec{x}(2 \Delta t) - \vec{x}(\Delta t)}{\Delta t} = \vec{\hat{v}}(\Delta t) + \frac{1}{2} \vec{a} (\Delta t) \Delta t + O(\Delta t^2)\]

\[\frac{1}{\Delta t} (\vec{x}(2 \Delta t) - \vec{x}(\Delta t)) = \vec{\hat{v}}(\Delta t) + \frac{1}{2} \vec{a} (\Delta t) \Delta t + O(\Delta t^2)\]

So we get

\[2 \vec{v}(\Delta t) = \vec{v}(0) + \frac{1}{2}\vec{a}(0)\Delta t + \vec{\hat{v}}(\Delta t) + \frac{1}{2} \vec{a} (\Delta t) \Delta t + O(\Delta t^2)\]

\[2 \vec{v}(\Delta t) = \vec{v}(\Delta t) + \vec{\hat{v}}(\Delta t) + O(\Delta t^2)\]

\[\vec{v}(\Delta t) = \vec{\hat{v}}(\Delta t) + O(\Delta t^2)\]

where \(\vec{\hat{v}}\) is a precise value.