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| In [[classical mechanics]], a '''central force''' on an object is a [[force (physics)|force]] whose magnitude only depends on the [[distance]] ''r'' of the object from the [[origin (mathematics)|origin]] and is directed along the line joining them: <ref name="wolfram">
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| {{cite web
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| |url= http://scienceworld.wolfram.com/physics/CentralForce.html
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| |title= Central Force
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| |accessdate= 2008-08-18
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| |author= Eric W. Weisstein
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| |authorlink= Eric W. Weisstein
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| |year= 1996–2007
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| |work= ScienceWorld
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| |publisher= Wolfram Research
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| }}
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| </ref> | |
| :<math> \vec{F} = \mathbf{F}(\mathbf{r}) = F( ||\mathbf{r}|| ) \hat{\mathbf{r}} </math>
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| where <math> \scriptstyle \vec{ \text{ F } } </math> is the force, '''F''' is a [[vector field|vector valued force function]], ''F'' is a scalar valued force function, '''r''' is the [[position vector]], ||'''r'''|| is its length, and <math> \scriptstyle \hat{\mathbf{r}}</math> = '''r'''/||'''r'''|| is the corresponding [[unit vector]].
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| Equivalently, a force field is central if and only if it is [[spherically symmetric]].
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| ==Properties==
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| A central force is a [[conservative field]], that is, it can always be expressed as the negative [[gradient]] of a [[potential]]:
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| :<math> \mathbf{F}(\mathbf{r}) = - \mathbf{\nabla} V(\mathbf{r})\text{, where }V(\mathbf{r}) = \int_{|\mathbf{r}|}^{+\infin} F(r)\,\mathrm{d}r</math> | |
| (the upper bound of integration is arbitrary, as the potential is defined [[up to]] an additive constant).
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| In a conservative field, the total [[mechanical energy]] ([[kinetic energy|kinetic]] and potential) is conserved:
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| :<math>E = \frac{1}{2} m |\mathbf{\dot{r}}|^2 + V(\mathbf{r}) = \text{constant}</math> | |
| (where '''ṙ''' denotes the [[derivative]] of '''r''' with respect to time, that is the [[velocity]]), and in a central force field, so is the [[angular momentum]]:
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| :<math>\mathbf{L} = \mathbf{r} \times m\mathbf{\dot{r}} = \text{constant}</math>
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| because the [[torque]] exerted by the force is zero. As a consequence, the body moves on the plane perpendicular to the angular momentum vector and containing the origin, and obeys [[Kepler's laws of planetary motion|Kepler's second law]]. (If the angular momentum is zero, the body moves along the line joining it with the origin.)
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| As a consequence of being conservative, a central force field is irrotational, that is, its [[curl (mathematics)|curl]] is zero, ''except at the origin'':
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| :<math> \nabla\times\mathbf{F} (\mathbf{r}) = \mathbf{0}\text{.}</math>
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| ==Examples==
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| [[Gravitational force]] and [[Coulomb force]] are two familiar examples with ''F''(''r'') being [[Inverse-square law|proportional to 1/''r''<sup>2</sup>]]. An object in such a force field with negative ''F'' (corresponding to an attractive force) obeys [[Kepler's laws of planetary motion]].
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| The force field of a spatial [[harmonic oscillator]] is central with ''F''(''r'') proportional to ''r'' and negative. | |
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| By [[Bertrand's theorem]], these two, ''F''(''r'') = −''k''/''r''<sup>2</sup> and ''F''(''r'') = −''kr'', are the only possible central force field with stable closed orbits.
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| ==See also==
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| * [[Classical central-force problem]]
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| ==References==
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| <references/>
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| [[Category:Force]]
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| [[Category:Classical mechanics]]
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