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In classical mechanics, a central force on an object is a force that is directed towards or away from a point called center of force.[a][1]: 93
Not all central force fields are conservative or spherically symmetric. However, a central force is conservative if and only if it is spherically symmetric or rotationally invariant.[1]: 133–38
Properties
Central forces that are conservative can always be expressed as the negative gradient of a potential energy:
In a conservative field, the total mechanical energy (kinetic and potential) is conserved:
It can also be shown that an object that moves under the influence of any central force obeys Kepler's second law. However, the first and third laws depend on the inverse-square nature of Newton's law of universal gravitation and do not hold in general for other central forces.
As a consequence of being conservative, these specific central force fields are irrotational, that is, its curl is zero, except at the origin:
Examples
Gravitational force and Coulomb force are two familiar examples with being proportional to 1/r2 only. An object in such a force field with negative (corresponding to an attractive force) obeys Kepler's laws of planetary motion.
The force field of a spatial harmonic oscillator is central with proportional to r only and negative.
By Bertrand's theorem, these two, and , are the only possible central force fields where all bounded orbits are stable closed orbits. However, there exist other force fields, which have some closed orbits.
See also
Notes
References
- ^ a b c Taylor, John R. (2005). Classical Mechanics. Sausalito, CA.: Univ. Science Books. ISBN 1-891389-22-X.
- ^ Eric W. Weisstein (1996–2007). "Central Force". ScienceWorld. Wolfram Research. Retrieved 2008-08-18.
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