Symmetric Central Force

A central force is spherically if the magnitude of the force does not depend on the direction (angles θ, or Ø) of the particle but only on distance r from the center of the force, that is F is spherically symmetric if,

F = F ( r ) e_r

A spherically symmetric force is conservative, conversely, if a central force F during a small displacement d l of the particle:

dW = F . d l = ( F ( r ) e_r ) . ( e_r dr + e_θ r dθ)

= F ( r ) dr

where we used dl in polar co-ordinates. The work done in moving the particle from r₁ to r₂, therefore, is

where, G ( r ) is the integral function of F ( r ); that is, F ( r ) = dG ( r )/dr

The work done thus depends only on the end co-ordinates and not on the path followed. Force F ( r ) e_r is therefore conservative.

G ( r ), in fact expresses the negative of potential energy function U ( r ):

W = - ( U ( r₂ ) - U ( r₁ ))

For example, the gravitational and electrostatic forces are spherically symmetric central forces. These are expressed as

 

We discussed these forces earlier; the potential energy of interaction for the two particles in case of above forces is:

 

The central forces (like gravitational or electrostatic) which vary as (1/r²) are called inverse square forces.

The conservation of energy principal for a particle moving in a spherically symmetric central force is expressed as,

½ mv2 + U ( r ) = E , constant

 

On the other hand, the angular momentum of the particle is also a constant of motion:

 

Hence we get

 

represents an effective potential energy. (Remember, L²/2 mr² is really a part of kinetic energy coming from transverse motion of the particle.)

If the particle motion is a one-dimensional motion along the radial direction under effective potential energy function U' ( r ). The entire effect of transverse motion of the particle is incorporated in the potential energy as additional L²/2 mr² term.

The term L²/2mr² is sometimes referred to as the 'centrifugal' potential energy. This is because the corresponding force is,

 

which is same as centrifugal force mr ω² in a co-ordinate frame rotating with instantaneous angular velocity ω = dθ/dt.