Effective potential

The effective potential, or effective potential energy is a mathematical expression combining the centrifugal potential energy with 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. The units of the effective potential are kg m^2/s^2, the same as energy.

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

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

$\frac{1}{2}m\dot{r}^2 = E - U_\text{eff}(r),$

where

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

is the effective potential. 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).

References

• José, JV; Saletan, EJ (1998). Classical Dynamics: A Contemporary Approach (1st ed.). Cambridge University Press. ISBN 0-521-63636-1 .

• Likos et al, C.N.; Rosenfeldt, S.; Dingenouts, N.; Ballauff, M.; Lindner, P.; Werner, N.; Vögtle, F. (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. http://jcp.aip.org/jcpsa6/v117/i4/p1869_s1?isAuthorized=no. 

• 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. http://www.springerlink.com/content/t238g8pk30606027/. 

• 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. http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6TVP-439VDH7-1&_user=616165&_coverDate=07%2F31%2F2001&_rdoc=1&_fmt=high&_orig=search&_sort=d&_docanchor=&view=c&_searchStrId=1343199058&_rerunOrigin=scholar.google&_acct=C000032338&_version=1&_urlVersion=0&_userid=616165&md5=09b16d8bfcfeecad13e87c345623bd4e.