Editing Verlet integration (section)

===Computing velocities – Størmer–Verlet method===
The velocities are not explicitly given in the basic Størmer equation, but often they are necessary for the calculation of certain physical quantities like the [[kinetic energy]]. This can create technical challenges in [[molecular dynamics]] simulations, because kinetic energy and instantaneous temperatures at time <math>t</math> cannot be calculated for a system until the positions are known at time <math>t + \Delta t</math>. This deficiency can either be dealt with using the [[#Velocity Verlet|velocity Verlet]] algorithm or by estimating the velocity using the position terms and the [[mean value theorem]]:
:<math>
\mathbf{v}(t) =
\frac{\mathbf{x}(t + \Delta t) - \mathbf{x}(t - \Delta t)}{2\Delta t}
+ \mathcal{O}\left(\Delta t^2\right).
</math>

Note that this velocity term is a step behind the position term, since this is for the velocity at time <math>t</math>, not <math>t + \Delta t</math>, meaning that <math>\mathbf v_n = \tfrac{\mathbf x_{n+1} - \mathbf x_{n-1}}{2\Delta t}</math> is a second-order approximation to <math>\mathbf{v}(t_n)</math>. With the same argument, but halving the time step, <math>\mathbf v_{n+\frac12} = \tfrac{\mathbf x_{n+1} - \mathbf x_n}{\Delta t}</math> is a second-order approximation to <math>\mathbf{v}\left(t_{n+\frac12}\right)</math>, with <math>t_{n+\frac12} = t_n + \tfrac12 \Delta t</math>.

One can shorten the interval to approximate the velocity at time <math>t + \Delta t</math> at the cost of accuracy:
:<math>\mathbf{v}(t + \Delta t) = \frac{\mathbf{x}(t + \Delta t) - \mathbf{x}(t)}{\Delta t} + \mathcal{O}(\Delta t).</math>