Effective potential

Definition

The effective potential U eff is defined in the following way:

U eff ( r ) = L 2 2 μ r 2 + U ( r )

L is the angular momentum r is the distance between the two masses μ is the reduced mass of the two bodies (approximately equal to the mass of the orbiting body if one mass is much larger than the other) U(r) is the general form of the potential

The effective force, then, is the negative gradient of the effective potential:

F eff = − ∇ U eff ( r ) = L 2 μ r 3 r ^ − ∇ U ( r )

Where r ^ denotes a unit vector in the radial direction.

Important properties

There are many useful features of the effective potential:

U eff ≤ E

To find the radius of a circular orbit, we simply minimize the effective potential with respect to r , or equivalently set the net force to zero and then solve for r 0 :

d U eff d r = 0

After solving for r 0 , plug this back into U eff to find the maximum value of the effective potential U eff max

To find the frequency of small oscillations:

ω = U eff ″ m

where the double prime indicates the second derivative of the effective potential with respect to r .

Example: gravitational potential

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 = 1 2 m ( r ˙ 2 + r 2 ϕ ˙ 2 ) − G m M r , L = m r 2 ϕ ˙

where

r ˙ is the derivative of r with respect to time, ϕ ˙ 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

m r ˙ 2 = 2 E − L 2 m r 2 + 2 G m M r = 2 E − 1 r 2 ( L 2 m − 2 G m M r ) , 1 2 m r ˙ 2 = E − U eff ( r ) ,

where

U eff ( r ) = L 2 2 m r 2 − G m M 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 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 fields of condensed matter, like e.g. the Gauss-core potential (Likos 2002, Baeurle 2004) and the screened Coulomb potential (Likos 2001).