Mechanical similarity

Remarks

The key property: the image of a motion is again a motion. Indeed $\nabla V$ is homogeneous of degree $k - 1$, so $M \ddot{\tilde q}(t) = \frac{\alpha}{\beta^2} M \ddot q(t/\beta) = -\frac{\alpha}{\beta^2} \nabla V(q(t/\beta))$, while $-\nabla V(\tilde q(t)) = -\alpha^{k-1} \nabla V(q(t/\beta))$, and $\alpha/\beta^2 = \alpha^{k-1}$. Times scale by $\beta$ and the energy scales by $\alpha^k$. Example: for the Kepler potential $V = -c/\abs{q}$, $k = -1$ and $\beta = \alpha^{3/2}$, so periods $P$ and sizes $a$ of similar orbits satisfy $P^2 \propto a^3$, as in Kepler's third law. Non-example: for $V(q) = \abs{q}^2 + \abs{q}^4$, which is not homogeneous, no single exponent works, and the scaled curve of a motion is in general not a motion.