# Central force Information

*https://en.wikipedia.org/wiki/Central_force*

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]}

where is the force, **F** is a
vector valued force function, *F* is a scalar valued force function, **r** is the
position vector, ||**r**|| is its length, and is the corresponding
unit vector.

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.^{
[2]}

## Properties

Central forces that are conservative can always be expressed as the negative gradient of a potential energy:-

(the upper bound of integration is arbitrary, as the potential is defined up to an additive constant).

In a conservative field, the total mechanical energy ( kinetic and potential) is conserved:

(where '**ṙ'** denotes the
derivative of '**r'** with respect to time, that is the
velocity,'**I'** denotes
moment of inertia of that body and '**ω'** denotes
angular velocity), and in a central force field, so is the
angular momentum:

because the torque exerted by the force is zero. As a consequence, the body moves on the plane perpendicular to the angular momentum vector and containing the origin, and obeys Kepler's second law. (If the angular momentum is zero, the body moves along the line joining it with the origin.)

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/*r*^{2} 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.
*

## Notes

** ^{a}** This article uses the definition of central force given in Taylor.

^{ [1]}Another common definition (used in

*ScienceWorld*

^{ [3]}) adds the constraint that the force be spherically symmetric, i.e. .

## See also

## References

- ^
^{a}^{b}Taylor, John R. (2005).*Classical Mechanics*. Sausalito, Calif.: Univ. Science Books. p. 93. ISBN 1-891389-22-X. **^**Taylor, John R. (2005).*Classical Mechanics*. Sausalito, California: Univ. Science Books. pp. 133–38. ISBN 1-891389-22-X.**^**Eric W. Weisstein (1996–2007). "Central Force".*ScienceWorld*. Wolfram Research. Retrieved 2008-08-18.