In
classical mechanics
Classical mechanics is a physical theory describing the motion of macroscopic objects, from projectiles to parts of machinery, and astronomical objects, such as spacecraft, planets, stars, and galaxies. For objects governed by classical ...
, a branch overlapping in
physics
Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which r ...
and
applied mathematics
Applied mathematics is the application of mathematical methods by different fields such as physics, engineering, medicine, biology, finance, business, computer science, and industry. Thus, applied mathematics is a combination of mathematical s ...
, mechanical similarity occurs when the
potential energy
In physics, potential energy is the energy held by an object because of its position relative to other objects, stresses within itself, its electric charge, or other factors.
Common types of potential energy include the gravitational potentia ...
is a
homogeneous function
In mathematics, a homogeneous function is a function of several variables such that, if all its arguments are multiplied by a scalar, then its value is multiplied by some power of this scalar, called the degree of homogeneity, or simply the ''deg ...
of the positions of the particles, with the result that the trajectories of the particles in the system are
geometrically similar
In Euclidean geometry, two objects are similar if they have the same shape, or one has the same shape as the mirror image of the other. More precisely, one can be obtained from the other by uniformly scaling (enlarging or reducing), possibly wit ...
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
In physics, equations of motion are equations that describe the behavior of a physical system in terms of its motion as a function of time.''Encyclopaedia of Physics'' (second Edition), R.G. Lerner, G.L. Trigg, VHC Publishers, 1991, ISBN (Ver ...
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
In mechanics, the virial theorem provides a general equation that relates the average over time of the total kinetic energy of a stable system of discrete particles, bound by potential forces, with that of the total potential energy of the system. ...
References
* Landau LD and Lifshitz EM (1976) ''Mechanics'' §10, 3rd. ed., Pergamon Press. (hardcover) and (softcover).
Classical mechanics
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