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.

See also

• Virial theorem

References

• Landau LD and Lifshitz EM (1976) Mechanics, 3rd. ed., Pergamon Press. ISBN 0-08-021022-8 (hardcover) and ISBN 0-08-029141-4 (softcover).