# Forest UCM Energy CentralForce

A central force is defined as a force depends only on separation distance

ie

Coulomb force and gravitation force.

# Spherical Coordinates

The differential change of in spherical coordinates occurs in three directions.

In the polar angle direction

In the azimuthal angle direction

The differential force of the displacement vector in spherical coordinates is

The derivative may be represented as

in three dimensional cartesian coordinates this may be written in terms of the gradient as

To determine the gradient in sperical coordinates on just compares the two equations

comparing terms of the above with

or

# The Central Force

## Spherical symmetry

A central force needs to be spherically symmetric in order to be conservative.

If the force is conservative then a potential exists such that

in order to be only a function of then

If is a conservative central force then and are spherically symmetric.

The converse is also true

if the central force is spherically symmetric then it is a conservative force.