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
The basic form of potential $$U_\text{eff}$$ is defined as: $$ U_\text{eff}(\mathbf{r}) = \frac{L^2}{2 \mu r^2} + U(\mathbf{r}), $$ 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: $$ \begin{align} \mathbf{F}_\text{eff} &= -\nabla U_\text{eff}(\mathbf{r}) \\ &= \frac{L^2}{ \mu r^3} \hat{\mathbf{r}} - \nabla U(\mathbf{r}) \end{align}$$ where $$\hat{\mathbf{r}}$$ denotes a unit vector in the radial direction.

Important properties
There are many useful features of the effective potential, such as $$ U_\text{eff} \leq E .$$

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 $$r_0$$: $$ \frac{d U_\text{eff}}{dr} = 0 $$ After solving for $$r_0$$, plug this back into $$U_\text{eff}$$ to find the maximum value of the effective potential $$U_\text{eff}^\text{max}$$.

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: $$ \frac{d^2 U_\text{eff}}{dr^2} > 0 $$

The frequency of small oscillations, using basic Hamiltonian analysis, is $$ \omega = \sqrt{\frac{U_\text{eff}''}{m}} ,$$ 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


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 $$E = \frac{1}{2}m \left(\dot{r}^2 + r^2\dot{\phi}^2\right) - \frac{GmM}{r},$$ $$L = mr^2\dot{\phi} $$ when the motion of the larger mass is negligible. In these expressions,
 * $$\dot{r}$$ 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. 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).