Editing Verlet integration (section)

===Verlet integration (without velocities)===
To discretize and numerically solve this [[initial value problem]], a time step <math>\Delta t > 0</math> is chosen, and the sampling-point sequence <math>t_n = n\,\Delta t</math> considered. The task is to construct a sequence of points <math>\mathbf x_n</math> that closely follow the points <math>\mathbf x(t_n)</math> on the trajectory of the exact solution.

Where [[Euler's method]] uses the [[forward difference]] approximation to the first derivative in differential equations of order one, Verlet integration can be seen as using the [[central difference]] approximation to the second derivative:
:<math>\begin{align}
\frac{\Delta^2\mathbf x_n}{\Delta t^2}
&= \frac{\frac{\mathbf x_{n+1} - \mathbf x_n}{\Delta t} - \frac{\mathbf x_n - \mathbf x_{n-1}}{\Delta t}}{\Delta t}\\[6pt]
&= \frac{\mathbf x_{n+1} - 2 \mathbf x_n + \mathbf x_{n-1}}{\Delta t^2} = \mathbf a_n = \mathbf A(\mathbf x_n).
\end{align}</math>

''Verlet integration'' in the form used as the ''Størmer method''<ref>[http://www.fisica.uniud.it/~ercolessi/md/md/node21.html webpage] {{Webarchive|url=https://web.archive.org/web/20040803212552/http://www.fisica.uniud.it/~ercolessi/md/md/node21.html |date=2004-08-03 }} with a description of the Størmer method.</ref> uses this equation to obtain the next position vector from the previous two without using the velocity as
:<math>\begin{align}
\mathbf x_{n+1} &= 2 \mathbf x_n - \mathbf x_{n-1} + \mathbf a_n\,\Delta t^2, \\[6pt]
\mathbf a_n &= \mathbf A(\mathbf x_n).
\end{align}</math>