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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). \(\not\Delta\)