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.

Definition[edit]

The basic form of effective potential is defined as:

,

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 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 .

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 .

Gravitational potential[edit]

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,

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
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

where

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).

Notes[edit]

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

References[edit]

  1. ^ "Seidov, Roche Problem".