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Effective potential

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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.


Effective potential. E > 0: hyperbolic orbit (A1 as pericenter), E = 0: parabolic orbit (A2 as pericenter), E < 0: elliptic orbit (A3 as pericenter, A3' as apocenter), E = Emin: circular orbit (A4 as radius). Points A1, ..., A4 are called turning points.

The basic form of potential is defined as:


  • 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 denotes a unit vector in the radial direction.

Important properties[edit]

There are many useful features of the effective potential, such as

To find the radius of a circular orbit, 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 .

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:

The frequency of small oscillations, using basic Hamiltonian analysis, is

where the double prime indicates the second derivative of the effective potential with respect to and it is evaluated at a minimum.

Gravitational potential[edit]

Components of the effective potential of two rotating bodies: (top) the combined gravitational potentials; (btm) the combined gravitational and rotational potentials
Visualisation of the effective potential in a plane containing the orbit (grey rubber-sheet model with purple contours of equal potential), the Lagrangian points (red) and a planet (blue) orbiting a star (yellow)[1]

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

when the motion of the larger mass is negligible. In these expressions,

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

is the effective potential.[Note 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[edit]


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


  1. ^ Seidov, Zakir F. (2004). "Seidov, Roche Problem". The Astrophysical Journal. 603: 283–284. arXiv:astro-ph/0311272. Bibcode:2004ApJ...603..283S. doi:10.1086/381315.

Further reading[edit]

  • 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.