Nondimensionalization

Rescaling dimensional quantities by chosen characteristic scales produces dimensionless variables. The remaining dimensionless ratios identify independent parameters and possible asymptotic regimes.

In the simple pendulum equation, choose the time scale $t_{\ast }=\sqrt{\ell /g}$ and put $\tau =t/t_{\ast }$. The chain rule gives $\ddot{\theta }=(g/\ell )d^2\theta /d{\tau }^2$, so the equation becomes $d^2\theta /d{\tau }^2+\sin \theta =0$. This removes $\ell$ and $g$ from the equation, while the initial conditions still include angular amplitude and dimensionless initial angular speed. A solution can therefore be calculated once and rescaled to many physical lengths and gravity values with the same dimensionless initial data.