Effective potential explained

The effective potential (also known as effective potential energy) combines multiple, perhaps opposing, effects into a single potential. In its basic form, it is the sum of the 'opposing' centrifugal potential energy with the potential energy of a dynamical system. It may be used to determine the orbits of planets (both Newtonian and relativistic) and to perform semi-classical atomic calculations, and often allows problems to be reduced to fewer dimensions.

Definition

The basic form of potential

U _{eff} is defined as:$$U\_\backslash text(\backslash mathbf)\; =\; \backslash frac\; +\; U(\backslash mathbf),$$where

• 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); and
• U(r) is the general form of the potential.

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

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

Important properties

There are many useful features of the effective potential, such as$$U\_\backslash text\; \backslash leq\; E\; .$$

To find the radius of a circular orbit, simply minimize the effective potential with respect to

r , or equivalently set the net force to zero and then solve for r _{0} :$$\backslash frac\; =\; 0$$After solving for r _{0} , plug this back into U _{eff} to find the maximum value of the effective potential U ^{max} _{eff} .

A circular orbit may be either stable or unstable. If it is unstable, a small perturbation could destabilize the orbit, but a stable orbit would return to equilibrium. To determine the stability of a circular orbit, determine the concavity of the effective potential. If the concavity is positive, the orbit is stable:$$\backslash frac\; >\; 0$$

The frequency of small oscillations, using basic Hamiltonian analysis, is$$\backslash omega\; =\; \backslash sqrt,$$where the double prime indicates the second derivative of the effective potential with respect to

r and it is evaluated at a minimum.

Gravitational potential

See main article: Gravitational potential.

Consider a particle of mass m orbiting a much heavier object of mass M. Assume Newtonian mechanics, which is both classical and non-relativistic. The conservation of energy and angular momentum give two constants E and L, which have values$$E\; =\; \backslash fracm\; \backslash left(\backslash dot^2\; +\; r^2\backslash dot^2\backslash right)\; -\; \backslash frac,$$$$L\; =\; mr^2\backslash dot$$when the motion of the larger mass is negligible. In these expressions,

• r is the derivative of r with respect to time,• \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$$m\backslash dot^2\; =\; 2E\; -\; \backslash frac\; +\; \backslash frac\; =\; 2E\; -\; \backslash frac\; \backslash left(\backslash frac\; -\; 2GmMr\backslash right),$$$$\backslash frac\; m\; \backslash dot^2\; =\; E\; -\; U\_\backslash text(r),$$where$$U\_\backslash text(r)\; =\; \backslash frac\; -\; \backslash frac$$is the effective potential.^[1] 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 condensed matter subfields, e.g. the Gauss-core potential (Likos 2002, Baeurle 2004) and the screened Coulomb potential (Likos 2001).

See also

• Geopotential

Further reading

• Book: José. JV. Saletan. EJ. 1998. Classical Dynamics: A Contemporary Approach. 1st. Cambridge University Press. 978-0-521-63636-0. .
• Gaussian effective interaction between flexible dendrimers of fourth generation: a theoretical and experimental study . Likos . C.N. . Rosenfeldt . S. . Dingenouts . N. . Ballauff . M. . Matthias Ballauff . Lindner . P. . Werner . N. . Vögtle . F. . J. Chem. Phys. . 117 . 1869 - 1877 . 2002 . 10.1063/1.1486209 . 2002JChPh.117.1869L . 4 . etal . dead . https://web.archive.org/web/20110719010918/http://jcp.aip.org/jcpsa6/v117/i4/p1869_s1?isAuthorized=no . 2011-07-19 .
• Baeurle . S.A. . Kroener J. . Modeling Effective Interactions of Micellar Aggregates of Ionic Surfactants with the Gauss-Core Potential . J. Math. Chem. . 36 . 409 - 421 . 2004 . 10.1023/B:JOMC.0000044526.22457.bb . 4.
• Likos . C.N. . Effective interactions in soft condensed matter physics . Physics Reports . 348 . 267 - 439 . 2001 . 10.1016/S0370-1573(00)00141-1. 2001PhR...348..267L . 4–5 . 10.1.1.473.7668 .

Notes and References

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