= Nondimensionalization
{wiki}
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 $t_* =\sqrt{\ell/g}$ and put $\tau=t/t_*$. 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.
Back to article page
