Let us consider a system of particles and assume that the interaction energy between any couple of particles has the form , where 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 at corresponding points on the paths are related to the linear size of the path by
- The period of small oscillations () is independent of their amplitude.
- The time of free fall under gravity () is proportional to the square root of the initial altitude.
- The square of the time of revolution of the planets () is proportional to the cube of the orbital size.
- Landau LD and Lifshitz EM (1976) Mechanics, 3rd. ed., Pergamon Press. ISBN 0-08-021022-8 (hardcover) and ISBN 0-08-029141-4 (softcover).
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