Wikipedia / Effective potential
Wikipedia / Effective potential
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
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),
U(r) is the general form of the potential.
The effective force, then, is the negative gradient denotes a unit vector in the radial direction.
Important properties
To find the radius of a circular orbit.
the orbit is stable.
The frequency of small oscillations, using basic Hamiltonian and is evaluated at a minimum.
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 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 massm,
G is the gravitational constant,
E is the total energy,
L is the angular momentum.
is the effective potential. 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
Notes
References
Further reading
.