en>M-le-mot-dit: /* Technical details */ disambiguation error

2010-02-21T15:52:25Z

Technical details: disambiguation error

New page

The '''effective potential''' (also known as '''effective potential energy''') is a mathematical expression combining multiple (perhaps opposing) effects into a single potential. In [[classical mechanics]] it is defined as the sum of the 'opposing' [[centrifugal potential energy]] with the [[potential energy]] of a [[dynamical system]]. It is commonly used in calculating the [[orbit]]s of planets (both [[Newtonian mechanics|Newtonian]] and [[general relativity|relativistic]]) and in semi-classical atomic calculations, and often allows problems to be reduced to fewer [[dimension]]s.

==Definition==
The effective potential <math>U_\text{eff}</math> is defined in the following way:

<math> U_\text{eff}(\mathbf{r}) = \frac{L^2}{2mr^2} + U(\mathbf{r}) </math>
:''L'' is the [[angular momentum]]
:''r'' is the distance between the two masses
:''m'' is the [[mass]] of the orbiting body
:''U(r)'' is the general form of the [[potential]]

The effective force, then, is the negative [[gradient]] of the effective potential:

<math>\begin{align}
\mathbf{F}_\text{eff} &= -\nabla U_\text{eff}(\mathbf{r}) \\
&= \frac{L^2}{mr^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.
The condition for a particle of energy ''E'' flying by to be `trapped' and go into an orbit:
:<math> U_\text{eff} < E </math>

To find the radius of a circular orbit, we 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> \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>

To find the frequency of small oscillations:

:<math> \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>.

==Example: gravitational potential==

For example, consider a particle of mass ''m'' orbiting a much heavier object of mass ''M''. Assuming Newtonian mechanics can be used, and the motion of the larger mass is negligible, then the conservation of [[energy]] and [[angular momentum]] give two constants ''E'' and ''L'', with values

:<math>E = \frac{1}{2}m\left(\dot{r}^2 + r^2\dot{\phi}^2\right) - \frac{GmM}{r},</math>
:<math>L = mr^2\dot{\phi} \, </math>

where

:<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

Only two variables are needed, since the motion occurs in a plane. Substituting the second expression into the first and rearranging gives

:<math>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>\frac{1}{2}m\dot{r}^2 = E - U_\text{eff}(r),</math>

where

:<math>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'', pgs. 31–33</ref> As is evident from the above equation, 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 [[Mechanical equilibrium|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 fields of condensed matter, like e.g. the Gauss-core potential (Likos 2002, Baeurle 2004) and the screened [[Coulomb potential]] (Likos 2001).

==Notes==
<references group="Note"/>

==References==
{{Reflist}}

* {{Cite book|last1=José|first1=JV|last2=Saletan|first2=EJ|year=1998|title=Classical Dynamics: A Contemporary Approach|edition=1st|publisher=Cambridge University Press|isbn=0-521-63636-1|postscript={{inconsistent citations}}}}.

* {{cite journal
|url = http://jcp.aip.org/jcpsa6/v117/i4/p1869_s1?isAuthorized=no
|last = Likos et al
|first = C.N.
|last2 = Rosenfeldt
|first2 = S.
|last3 = Dingenouts
|first3 = N.
|last4 = Ballauff
|first4 = M.
|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 }}

* {{cite journal
|url = http://www.springerlink.com/content/t238g8pk30606027/
|last = Baeurle
|first = S.A.
|coauthors = 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
|url = http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6TVP-439VDH7-1&_user=616165&_coverDate=07%2F31%2F2001&_rdoc=1&_fmt=high&_orig=search&_sort=d&_docanchor=&view=c&_searchStrId=1343199058&_rerunOrigin=scholar.google&_acct=C000032338&_version=1&_urlVersion=0&_userid=616165&md5=09b16d8bfcfeecad13e87c345623bd4e
|last = Likos
|first = C.N.
|title = Effective interactions in soft condensed matter physics
|journal = Physics Reports
|volume = 348
|pages = 267–439
|year = 2001
|doi = 10.1016/S0370-1573(00)00141-1|bibcode = 2001PhR...348..267L
|issue = 4–5 }}

{{DEFAULTSORT:Effective Potential}}
[[Category:Mechanics]]

PDE surface - Revision history

en>M-le-mot-dit: /* Technical details */ disambiguation error