Scaling law

There are similar physical laws among different scales.

Notes

1. Mechanical Similarity For a Lagrange function:

$$L(q,\dot{q},t)$$

we change the $q$ to $\alpha q$, change $t$ to $\beta t$ if it obey the same mechanical restriction, it is the mechanical similarity.

$$\begin{array}{r}T\to \frac{{\alpha }^2}{{\beta }^2}T\\ U\to {\alpha }^{k}U\\ {\alpha }^{1-\frac{k}{2}}=\beta \end{array}\text{\hspace{1em}(1)}$$

some example here:

• oscillation, k=2, so $\beta =1$
• gravity, k=-1, so $\beta ={\alpha }^{3/2}$
• uniform force, k=1, so $\beta ={\alpha }^{1/2}$

Related

• Renormalization group