Non-Dimensionalization

Differential equations may well have many constants in them. Often we can reduce the number of constants in an equation by choosing the right units for the various quantities in the equation. Suppose we have the ODE (Ordinary Differential Equation)

$$\frac{dx}{dt} = ax^2 + bx + c$$

The coefficients $a$, $b$ and $c$ must all have different units, as the terms $ax^2$, $bx$ and $c$ must all have dimensions of length/time. To simplify the equation we introduce a reference scale $x_0$ for $x$, and define the dimensionless variable

$$\phi = \frac{x}{x_0}$$

The variable $ \phi $ measures $ x $ relative to the characteristic scale $ x_0 $.

Similarly, we introduce a reference scale $t_0$ for $t$, and define the dimensionless variable

$$\tau = \frac{t}{t_0}$$

$\phi$ and $\tau$ are non-dimensionalized versions of $x$ and $t$, we can now derive a differential equation for $\phi$ and $\tau$, and then afterwards figure out which choice of $x_0$ and $t_0$ simplifies things most. We have

$$ x(t) = x_0 \, \phi(\tau) \quad t = t_0 \, \tau $$

By the chain rule, we have

$$\frac{dx}{dt} = \frac{dx}{d\tau} \frac{d\tau}{dt}$$ $$\Rightarrow \frac{dx}{dt} = \frac{x_0}{t_0} \frac{d\phi}{d\tau}$$

and we also have

$$ax^2 + bx + c = (a{x_0}^2) \phi^2 + (bx_0)\phi + c$$ $$\Rightarrow \frac{x_0}{t_0} \frac{d\phi}{d\tau} = a{x_0}^2 \phi^2 + bx_0\phi + c$$ $$\frac{d\phi}{d\tau} = ax_0t_0 \phi^2 + bt_0\phi + \frac{ct_0}{x_0}$$

We now choose $x_0$ and $t_0$ so that as many coefficients as possible become 1. If we let

$$t_0 = \frac{1}{b}, \quad x_0 = ct_0 = \frac{c}{b}$$

(assuming $b$ and $c$ are not 0), the coefficient of $\phi^2$ then becomes

$$\alpha = ax_0t_0 = \frac{ac}{b^2}$$ $$\Rightarrow \frac{d\phi}{d\tau} = \alpha \phi^2 + \phi + 1$$

This is a dimensionless version of our original equation. Non-dimensionalization reduces dependence on arbitrary unit choices, reveals the relative importance of different terms, and often combines several physical parameters into a smaller number of dimensionless quantities. In our example, the three parameters $a$, $b$, and $c$ combine into the single dimensionless parameter $ \displaystyle \alpha = \frac{ac}{b^2}. $ Different physical systems with the same value of $ \alpha $ therefore exhibit the same dimensionless behavior, even if their original scales are very different.

This is especially useful in computational physics, where rescaling variables so that important quantities are dimensionless and typically of order one can improve numerical stability. It also makes it easier to identify dominant terms in an equation: if $ \alpha \gg 1 $, the quadratic term dominates, whereas if $ \alpha \ll 1 $, it may be neglected.