Effective potential

The effective potential is a mathematical expression integrating angular momentum into the potential energy of a dynamical system. Commonly used in calculating the orbits of planets (both Newtonian and relativistic), the effective potential allows one to reduce a problem to fewer dimensions.

One way of thinking of this concept is in terms of the minimum kinetic energy required for an object to escape a gravitational field. For a stationary object, the minimum energy equates to the potential of the object, but an orbiting body already has an amount of kinetic energy. Therefore the minimum amount of energy that would have to be supplied for the body to escape this field has a lower magnitude, by an amount equal to the kinetic energy that the body posseses.

For example, consider a particle of mass m orbiting a much heavier object of mass M. Assuming Newtonian mechanics can be used, and the motion of the larger mass is negligible, then the conservation of energy and angular momentum give two constants E and L, with values

$E={\frac {1}{2}}m\left({\dot {r}}^{2}+r^{2}{\dot {\phi }}^{2}\right)-{\frac {GmM}{r}},$

$L=mr^{2}{\dot {\phi }}$

where

r is the distance between the two masses,

${\dot {r}}$ is the derivative of r with respect to time,

${\dot {\phi }}$ is the angular velocity of mass m,

G is the gravitational constant,

E is the total energy, and

L is the angular momentum.

Only two variables are needed, since the motion occurs in a plane. Substituting the second expression into the first and rearranging gives

${\dot {r}}^{2}=2E-{\frac {L^{2}}{mr^{2}}}+{\frac {2GmM}{r}}=2E-{\frac {1}{r^{2}}}\left({\frac {L^{2}}{m}}-2GmMr\right),$

${\dot {r}}^{2}=2E-V_{E}(r),$

where

$V_{E}(r)={\frac {L^{2}}{2mr^{2}}}-{\frac {GmM}{r}}$

is the effective potential. As is evident from the above equation, 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 equilibrium.

A similar method may be used in other applications, for instance determining orbits in a general relativistic Schwarzschild metric.