Mechanical similarity
Introduction In classical mechanics, a branch overlapping in physics and applied mathematics, '''mechanical similarity''' occurs when the potential energy is a homogeneous function of the positions of the particles, with the result that the trajectories of the particles in the system are geometrically similar paths, differing in size but retaining shape.Consider a system of any number of particles and assume that the interaction energy between any pair of particles has the form :where ''r'' is the distance between the two particles. In such a case the solutions to the equations of motion are 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:
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'' §10, 3rd. ed., Pergamon Press. (hardcover) and  (softcover).