André–Quillen cohomology: Difference between revisions
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Let us consider a system of particles and assume that the interaction energy between any couple of particles has the form <math>U(r)\propto r^k</math>, where <math>r</math> 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 <math>t</math> at corresponding points on the paths are related to the linear size <math>l</math> of the path by | |||
:<math>t \propto l^{1-k/2}.</math> | |||
==Examples== | |||
* The period of [[harmonic oscillator|small oscillations]] (<math>k=2</math>) is independent of their amplitude. | |||
* The time of [[free fall]] under gravity (<math>k=1</math>) is proportional to the square root of the initial altitude. | |||
* The square of the time of [[Kepler's laws of planetary motion|revolution of the planets]] (<math>k=-1</math>) 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). | |||
{{classicalmechanics-stub}} | |||
[[Category:Classical mechanics]] | |||
Revision as of 08:24, 20 August 2013
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
Examples
- 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.
See also
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).