Velocity Verlet

Let us show that Velocity Verlet is equivalent to regular Verlet method.

Basic form of Velocity Verlet is as follows:

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

Basic Verlet formulation is as follows:

\[\vec x(t+\Delta t)=2\vec x(t)-\vec x(t-\Delta t)+\vec a(t)\,\Delta t^2\]

Theorem: (3) follows from (1), (2).

\(\Delta\):

Let’s reformulate (1) and (2) for \(t = t - \Delta t\)

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

Subsituting (4) into (1):

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

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