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# 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):

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

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

From (5), we can see that

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

Then (6) turns into

$\vec{x}(t + \Delta t) = \vec{x}(t) + \vec{x}(t) - \vec{x}(t - \Delta t) + \vec{a}(t) \Delta t^2$

$\vec{x}(t + \Delta t) = 2 \vec{x}(t) - \vec{x}(t - \Delta t) + \vec{a}(t) \Delta t^2$

which is exactly (3). $$\not\Delta$$