# 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 is defined in the following way:

*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:

Where denotes a unit vector in the radial direction.

## Important properties

There are many useful features of the effective potential:

To find the radius of a circular orbit, we 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

To find the frequency of small oscillations:

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

## 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

where

- is the derivative of r with respect to time,
- 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

where

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

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

## References

- José, JV; Saletan, EJ (1998).
*Classical Dynamics: A Contemporary Approach*(1st ed.). Cambridge University Press. ISBN 0-521-63636-1. - Likos, C.N.; Rosenfeldt, S.; Dingenouts, N.; Ballauff, M.; Lindner, P.; Werner, N.; Vögtle, F.; et al. (2002). "Gaussian effective interaction between flexible dendrimers of fourth generation: a theoretical and experimental study".
*J. Chem. Phys*.**117**(4): 1869–1877. Bibcode:2002JChPh.117.1869L. doi:10.1063/1.1486209.

- Baeurle, S.A.; Kroener J. (2004). "Modeling Effective Interactions of Micellar Aggregates of Ionic Surfactants with the Gauss-Core Potential".
*J. Math. Chem*.**36**(4): 409–421. doi:10.1023/B:JOMC.0000044526.22457.bb.

- Likos, C.N. (2001). "Effective interactions in soft condensed matter physics".
*Physics Reports*.**348**(4–5): 267–439. Bibcode:2001PhR...348..267L. doi:10.1016/S0370-1573(00)00141-1.