Effective potential

Important properties

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

To find the radius of a circular orbit, simply minimize the effective potential with respect to ${\displaystyle r}$, or equivalently set the net force to zero and then solve for ${\displaystyle r_0}$: $${\displaystyle \frac{d{U}_{\text{eff}}}{dr}=0}$$ After solving for ${\displaystyle r_0}$, plug this back into ${\displaystyle U_{\text{eff}}}$ to find the maximum value of the effective potential ${\displaystyle U_{\text{eff}}^{\text{max}}}$.

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: $${\displaystyle \frac{d^2{U}_{\text{eff}}}{d{r}^2}>0}$$

The frequency of small oscillations, using basic Hamiltonian analysis, is $${\displaystyle \omega =\sqrt{\frac{U_{\text{eff}}^{″}}{m}},}$$ where the double prime indicates the second derivative of the effective potential with respect to ${\displaystyle r}$ 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 $${\displaystyle E=\frac{1}{2}m\left({\dot{r}}^2+r^2{\dot{\varphi }}^2\right)-\frac{GmM}{r},}$$ $${\displaystyle L=m{r}^2\dot{\varphi }}$$ when the motion of the larger mass is negligible. In these expressions,

• ${\displaystyle \dot{r}}$ is the derivative of r with respect to time,
• ${\displaystyle \dot{\varphi }}$ 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 $${\displaystyle m{\dot{r}}^2=2E-\frac{L^2}{m{r}^2}+\frac{2GmM}{r}=2E-\frac{1}{r^2}\left(\frac{L^2}{m}-2GmMr\right),}$$ $${\displaystyle \frac{1}{2}m{\dot{r}}^2=E-U_{\text{eff}}(r),}$$ where $${\displaystyle U_{\text{eff}}(r)=\frac{L^2}{2m{r}^2}-\frac{GmM}{r}}$$ 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).