Effective potential

The effective potential combines multiple, perhaps opposing, effects into a single potential. In its basic form, it is the sum of the "opposing" centrifugal potential energy with the potential energy of a dynamical system. It may be used to determine the orbits of planets and to perform semi-classical atomic calculations, and often allows problems to be reduced to fewer dimensions.

Definition

The basic form of potential is defined as
where
The effective force, then, is the negative gradient of the effective potential:
where denotes a unit vector in the radial direction.

Important properties

There are many useful features of the effective potential, such as
To find the radius of a circular orbit, simply minimize the effective potential with respect to, or equivalently set the net force to zero and then solve for :
After solving for, plug this back into to find the maximum value of the effective potential.
A circular orbit may be either stable or unstable. If it is unstable, a small perturbation could destabilize the orbit, but a stable orbit would return to equilibrium. To determine the stability of a circular orbit, determine the concavity of the effective potential. If the concavity is positive,
the orbit is stable.
The frequency of small oscillations, using basic Hamiltonian analysis, is
where the double prime indicates the second derivative of the effective potential with respect to and is evaluated at a minimum.

Gravitational potential

Consider a particle of mass m orbiting a much heavier object of mass M. Assume Newtonian mechanics, which is both classical and non-relativistic. The conservation of energy and angular momentum give two constants E and L, which have values
when the motion of the larger mass is negligible. In these expressions,
Only two variables are needed, since the motion occurs in a plane. Substituting the second expression into the first and rearranging gives
where
is the effective potential. The original two-variable problem has been reduced to a one-variable problem. For many applications the effective potential can be treated exactly like the potential energy of a one-dimensional system: for instance, an energy diagram using the effective potential determines turning points and locations of stable and unstable equilibria. A similar method may be used in other applications, for instance, determining orbits in a general relativistic Schwarzschild metric.
Effective potentials are widely used in various condensed matter subfields, e.g. the Gauss-core potential and the screened Coulomb potential.