Effective potential

The effective potential (also known as effective potential energy) is a mathematical expression combining multiple (perhaps opposing) effects into a single potential. In classical mechanics it is defined as the sum of the 'opposing' centrifugal potential energy with the potential energy of a dynamical system. It is commonly used in calculating the orbits of planets (both Newtonian and relativistic) and in semi-classical atomic calculations, and often allows problems to be reduced to fewer dimensions.

Definition

The effective potential U_\text{eff} is defined in the following way:

U_\text{eff}(\mathbf{r}) = \frac{L^2}{2mr^2} + U(\mathbf{r})

L is the angular momentum
r is the distance between the two masses
m is the mass of the orbiting body
U(r) is the general form of the potential

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

\begin{align} \mathbf{F}_\text{eff} &= -\nabla U_\text{eff}(\mathbf{r}) \\ &= \frac{L^2}{mr^3}\hat{\mathbf{r}} - \nabla U(\mathbf{r}) \end{align}

Where \hat{\mathbf{r}} denotes a unit vector in the radial direction.

Important properties

There are many useful features of the effective potential:

U_\text{eff} \leq 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:

\frac{d U_\text{eff}}{dr} = 0

After solving for r_0, plug this back into U_\text{eff} to find the maximum value of the effective potential U_\text{eff}^\text{max}

To find the frequency of small oscillations:

\omega = \sqrt{\frac{U_\text{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 = \frac{1}{2}m\left(\dot{r}^2 + r^2\dot{\phi}^2\right) - \frac{GmM}{r},
L = mr^2\dot{\phi} \,

where

\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

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

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

where

U_\text{eff}(r) = \frac{L^2}{2mr^2} - \frac{GmM}{r}

is the effective potential.[Note 1] 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).

Notes

  1. ↑ A similar derivation may be found in José & Saletan, Classical Dynamics: A Contemporary Approach, pgs. 31–33

References