this is for holding javascript data
Ghlen Livid added First_step_in_Velocity_Verlet__.tex
almost 8 years ago
Commit id: 488e594f0ff025492a12f1c68080006cd601b815
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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
$$\vec{v}(\Delta t) = \vec{v}(0) + \frac{1}{2}\vec{a}(0)\Delta t + \frac{1}{\Delta t} (\vec{x}(2 \Delta t) - \vec{x}(\Delta t) - \vec{v}(\Delta t) \Delta t)$$
$$2 \vec{v}(\Delta t) = \vec{v}(0) + \frac{1}{2}\vec{a}(0)\Delta t + \frac{1}{\Delta t} \vec{x}(2 \Delta t) - \frac{1}{\Delta t} \vec{x}(\Delta t)$$
