{{short description|Net potential energy encountered in orbital mechanics}} 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== thumb|Effective potential. ''E'' > 0: hyperbolic orbit (A<sub>1</sub> as pericenter), ''E'' = 0: parabolic orbit (A<sub>2</sub> as pericenter), ''E'' < 0: elliptic orbit (''A''<sub>3</sub> as pericenter, ''A''<sub>3</sub>' as apocenter), ''E'' = ''E''<sub>min</sub>: circular orbit (''A''<sub>4</sub> as radius). Points ''A''<sub>1</sub>, ..., ''A''<sub>4</sub> are called turning points. 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), : ''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, <math display="block"> \frac{d^2 U_\text{eff}}{dr^2} > 0, </math> the orbit is stable.
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 is evaluated at a minimum.
==Gravitational potential== {{Main|Gravitational potential}} thumb|Components of the effective potential of two rotating bodies: (top) the combined gravitational potentials; (btm) the combined gravitational and rotational potentials [[File:Lagrangian points equipotential.png|thumb|link={{filepath:Lagrangian_points_equipotential.gif}}|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)<ref>{{cite journal |title=The Roche Problem: Some Analytics |journal=The Astrophysical Journal |volume=603 |pages=283–284 |doi=10.1086/381315 |year=2004 |last1=Seidov |first1=Zakir F. |arxiv=astro-ph/0311272 |bibcode=2004ApJ...603..283S}}</ref>]]
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, : ''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.<ref group="Note">A similar derivation may be found in José & Saletan, ''Classical Dynamics: A Contemporary Approach'', pp. 31–33.</ref> 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== * Geopotential
==Notes== {{reflist|group=Note}}
==References== {{Reflist}}
==Further reading== * {{Cite book |last1=José |first1=J. V. |last2=Saletan |first2=E. J. |year=1998 |title=Classical Dynamics: A Contemporary Approach |edition=1st |publisher=Cambridge University Press |isbn=978-0-521-63636-0}}. * {{cite journal |url=http://jcp.aip.org/jcpsa6/v117/i4/p1869_s1?isAuthorized=no |last=Likos |first=C. N. |last2=Rosenfeldt |first2=S. |last3=Dingenouts |first3=N. |last4=Ballauff |first4=M. |authorlink4= Matthias Ballauff |last5=Lindner |first5=P. |last6=Werner |first6=N. |last7=Vögtle |first7=F. |title=Gaussian effective interaction between flexible dendrimers of fourth generation: a theoretical and experimental study |journal=J. Chem. Phys. |volume=117 |pages=1869–1877 |year=2002 |doi=10.1063/1.1486209 |bibcode=2002JChPh.117.1869L |issue=4 |display-authors=etal |url-status=dead |archiveurl=https://web.archive.org/web/20110719010918/http://jcp.aip.org/jcpsa6/v117/i4/p1869_s1?isAuthorized=no |archivedate=2011-07-19 |df= |url-access=subscription }} * {{cite journal |last = Baeurle |first = S. A. |author2=Kroener J. |title = Modeling Effective Interactions of Micellar Aggregates of Ionic Surfactants with the Gauss-Core Potential |journal = J. Math. Chem. |volume = 36 |pages = 409–421 |year = 2004 |doi = 10.1023/B:JOMC.0000044526.22457.bb |issue = 4}} * {{cite journal |last = Likos |first = C. N. |title = Effective interactions in soft condensed matter physics |journal = Physics Reports |volume = 348 |issue = 4–5 |pages = 267–439 |year = 2001 |doi = 10.1016/S0370-1573(00)00141-1 |bibcode = 2001PhR...348..267L |citeseerx = 10.1.1.473.7668 }}
{{DEFAULTSORT:Effective Potential}} Category:Mechanics Category:Potentials