Английская Википедия:Effective potential
Шаблон:Short description 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 <math>U_\text{eff}</math> is defined as: <math display="block"> U_\text{eff}(\mathbf{r}) = \frac{L^2}{2 \mu r^2} + U(\mathbf{r}), </math> 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: <math display="block"> \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}</math> where <math>\hat{\mathbf{r}}</math> denotes a unit vector in the radial direction.
Important properties
There are many useful features of the effective potential, such as <math display="block"> U_\text{eff} \leq E .</math>
To find the radius of a circular orbit, simply minimize the effective potential with respect to <math>r</math>, or equivalently set the net force to zero and then solve for <math>r_0</math>: <math display="block"> \frac{d U_\text{eff}}{dr} = 0 </math> After solving for <math>r_0</math>, plug this back into <math>U_\text{eff}</math> to find the maximum value of the effective potential <math>U_\text{eff}^\text{max}</math>.
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: <math display="block"> \frac{d^2 U_\text{eff}}{dr^2} > 0 </math>
The frequency of small oscillations, using basic Hamiltonian analysis, is <math display="block"> \omega = \sqrt{\frac{U_\text{eff}}{m}} ,</math> where the double prime indicates the second derivative of the effective potential with respect to <math>r</math> 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 <math display="block">E = \frac{1}{2}m \left(\dot{r}^2 + r^2\dot{\phi}^2\right) - \frac{GmM}{r},</math> <math display="block">L = mr^2\dot{\phi} </math> when the motion of the larger mass is negligible. In these expressions,
- <math>\dot{r}</math> is the derivative of r with respect to time,
- <math>\dot{\phi}</math> 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 <math display="block">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),</math> <math display="block">\frac{1}{2} m \dot{r}^2 = E - U_\text{eff}(r),</math> where <math display="block">U_\text{eff}(r) = \frac{L^2}{2mr^2} - \frac{GmM}{r} </math> 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
Notes
References
Further reading
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