# Mechanical similarity

Let us consider a system of particles and assume that the interaction energy between any couple of particles has the form $U(r)\propto r^k$, where $r$ is the distance between the two particles. In such a case the equations of motion permit a series of geometrically similar paths, and the times of motion $t$ at corresponding points on the paths are related to the linear size $l$ of the path by

$t \propto l^{1-k/2}.$

## Examples

• The period of small oscillations ($k=2$) is independent of their amplitude.
• The time of free fall under gravity ($k=1$) is proportional to the square root of the initial altitude.
• The square of the time of revolution of the planets ($k=-1$) is proportional to the cube of the orbital size.