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| In [[classical mechanics]], a '''Liouville dynamical system''' is an exactly soluble [[dynamical system]] in which the [[kinetic energy]] ''T'' and [[potential energy]] ''V'' can be expressed in terms of the ''s'' [[generalized coordinate]]s ''q'' as follows:<ref name="liouville_1849">{{cite journal | last = Liouville | year = 1849 | title = Mémoire sur l'intégration des équations différentielles du mouvement d'un nombre quelconque de points matériels | journal = Journal de Mathématiques Pures et Appliquées | volume = 14 | pages = 257–299 | url = http://visualiseur.bnf.fr/ConsulterElementNum?O=NUMM-16393&Deb=263&Fin=305&E=PDF}}</ref>
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| :<math>
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| T = \frac{1}{2} \left\{ u_{1}(q_{1}) + u_{2}(q_{2}) + \cdots + u_{s}(q_{s}) \right\}
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| \left\{ v_{1}(q_{1}) \dot{q}_{1}^{2} + v_{2}(q_{2}) \dot{q}_{2}^{2} + \cdots + v_{s}(q_{s}) \dot{q}_{s}^{2} \right\}
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| </math>
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| :<math>
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| V = \frac{w_{1}(q_{1}) + w_{2}(q_{2}) + \cdots + w_{s}(q_{s}) }{u_{1}(q_{1}) + u_{2}(q_{2}) + \cdots + u_{s}(q_{s}) }
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| </math>
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| The solution of this system consists of a set of separably integrable equations
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| :<math>
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| \frac{\sqrt{2}}{Y}\, dt = \frac{d\varphi_{1}}{\sqrt{E \chi_{1} - \omega_{1} + \gamma_{1}}} =
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| \frac{d\varphi_{2}}{\sqrt{E \chi_{2} - \omega_{2} + \gamma_{2}}} = \cdots =
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| \frac{d\varphi_{s}}{\sqrt{E \chi_{s} - \omega_{s} + \gamma_{s}}}
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| </math>
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| where ''E = T + V'' is the conserved energy and the <math>\gamma_{s}</math> are constants. As described below, the variables have been changed from ''q<sub>s</sub>'' to φ<sub>s</sub>, and the functions ''u<sub>s</sub>'' and ''w<sub>s</sub>'' substituted by their counterparts ''χ<sub>s</sub>'' and ''ω<sub>s</sub>''. This solution has numerous applications, such as the orbit of a small planet about two fixed stars under the influence of [[gravitation|Newtonian gravity]]. The Liouville dynamical system is one of several things named after [[Joseph Liouville]], an eminent French mathematician. | |
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| ==Example of bicentric orbits==
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| In [[classical mechanics]], [[Euler's three-body problem]] describes the motion of a particle in a plane under the influence of two fixed centers, each of which attract the particle with an [[inverse-square law|inverse-square force]] such as [[gravitation|Newtonian gravity]] or [[Coulomb's law]]. Examples of the bicenter problem include a [[planet]] moving around two slowly moving [[star]]s, or an [[electron]] moving in the [[electric field]] of two positively charged [[atomic nucleus|nuclei]], such as the first [[ion]] of the hydrogen molecule H<sub>2</sub>, namely the [[Dihydrogen cation|hydrogen molecular ion]] or H<sub>2</sub><sup>+</sup>. The strength of the two attractions need not be equal; thus, the two stars may have different masses or the nuclei two different charges.
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| ===Solution===
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| Let the fixed centers of attraction be located along the ''x''-axis at ±''a''. The potential energy of the moving particle is given by
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| :<math> | |
| V(x, y) = \frac{-\mu_{1}}{\sqrt{\left( x - a \right)^{2} + y^{2}}} - \frac{\mu_{2}}{\sqrt{\left( x + a \right)^{2} + y^{2}}} .
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| </math>
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| The two centers of attraction can be considered as the foci of a set of ellipses. If either center were absent, the particle would move on one of these ellipses, as a solution of the [[Kepler problem]]. Therefore, according to [[Bonnet's theorem]], the same ellipses are the solutions for the bicenter problem.
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| Introducing [[elliptic coordinates]],
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| :<math>
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| x = a \cosh \xi \cos \eta,
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| </math>
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| :<math>
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| y = a \sinh \xi \sin \eta,
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| </math>
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| the potential energy can be written as
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| :<math>
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| V(\xi, \eta) = \frac{-\mu_{1}}{a\left( \cosh \xi - \cos \eta \right)} - \frac{\mu_{2}}{a\left( \cosh \xi + \cos \eta \right)}
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| = \frac{-\mu_{1} \left( \cosh \xi + \cos \eta \right) - \mu_{2} \left( \cosh \xi - \cos \eta \right)}{a\left( \cosh^{2} \xi - \cos^{2} \eta \right)},</math>
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| and the kinetic energy as
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| :<math>
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| T = \frac{ma^{2}}{2} \left( \cosh^{2} \xi - \cos^{2} \eta \right) \left( \dot{\xi}^{2} + \dot{\eta}^{2} \right).
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| </math>
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| This is a Liouville dynamical system if ξ and η are taken as φ<sub>1</sub> and φ<sub>2</sub>, respectively; thus, the function ''Y'' equals
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| :<math>
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| Y = \cosh^{2} \xi - \cos^{2} \eta
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| </math>
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| and the function ''W'' equals
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| :<math>
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| W = -\mu_{1} \left( \cosh \xi + \cos \eta \right) - \mu_{2} \left( \cosh \xi - \cos \eta \right)
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| </math>
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| Using the general solution for a Liouville dynamical system below, one obtains
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| :<math>
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| \frac{ma^{2}}{2} \left( \cosh^{2} \xi - \cos^{2} \eta \right)^{2} \dot{\xi}^{2} = E \cosh^{2} \xi + \left( \frac{\mu_{1} + \mu_{2}}{a} \right) \cosh \xi - \gamma
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| </math>
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| :<math>
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| \frac{ma^{2}}{2} \left( \cosh^{2} \xi - \cos^{2} \eta \right)^{2} \dot{\eta}^{2} = -E \cos^{2} \eta + \left( \frac{\mu_{1} - \mu_{2}}{a} \right) \cos \eta + \gamma
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| </math>
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| Introducing a parameter ''u'' by the formula
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| :<math> | |
| du = \frac{d\xi}{\sqrt{E \cosh^{2} \xi + \left( \frac{\mu_{1} + \mu_{2}}{a} \right) \cosh \xi - \gamma}} =
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| \frac{d\eta}{\sqrt{-E \cos^{2} \eta + \left( \frac{\mu_{1} - \mu_{2}}{a} \right) \cos \eta + \gamma}},
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| </math>
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| gives the [[parametric solution]]
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| :<math>
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| u = \int \frac{d\xi}{\sqrt{E \cosh^{2} \xi + \left( \frac{\mu_{1} + \mu_{2}}{a} \right) \cosh \xi - \gamma}} =
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| \int \frac{d\eta}{\sqrt{-E \cos^{2} \eta + \left( \frac{\mu_{1} - \mu_{2}}{a} \right) \cos \eta + \gamma}}.
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| </math>
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| Since these are [[elliptic integral]]s, the coordinates ξ and η can be expressed as elliptic functions of ''u''.
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| ===Constant of motion===
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| The bicentric problem has a constant of motion, namely,
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| :<math>
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| r_{1}^{2} r_{2}^{2} \left( \frac{d\theta_{1}}{dt} \right) \left( \frac{d\theta_{2}}{dt} \right) -
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| 2c \left[ \mu_{1} \cos \theta_{1} + \mu_{2} \cos \theta_{2} \right],
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| </math>
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| from which the problem can be solved using the method of the last multiplier.
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| ==Derivation==
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| ===New variables===
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| To eliminate the ''v'' functions, the variables are changed to an equivalent set
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| :<math>
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| \varphi_{r} = \int dq_{r} \sqrt{v_{r}(q_{r})},
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| </math>
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| giving the relation
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| :<math>
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| v_{1}(q_{1}) \dot{q}_{1}^{2} + v_{2}(q_{2}) \dot{q}_{2}^{2} + \cdots + v_{s}(q_{s}) \dot{q}_{s}^{2} =
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| \dot{\varphi}_{1}^{2} + \dot{\varphi}_{2}^{2} + \cdots + \dot{\varphi}_{s}^{2} = F,
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| </math>
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| which defines a new variable ''F''. Using the new variables, the u and w functions can be expressed by equivalent functions χ and ω. Denoting the sum of the χ functions by ''Y'',
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| :<math>
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| Y = \chi_{1}(\varphi_{1}) + \chi_{2}(\varphi_{2}) + \cdots + \chi_{s}(\varphi_{s}),
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| </math>
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| the kinetic energy can be written as
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| :<math>
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| T = \frac{1}{2} Y F.
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| </math>
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| Similarly, denoting the sum of the ω functions by ''W''
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| :<math>
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| W = \omega_{1}(\varphi_{1}) + \omega_{2}(\varphi_{2}) + \cdots + \omega_{s}(\varphi_{s}),
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| </math>
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| the potential energy ''V'' can be written as
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| :<math>
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| V = \frac{W}{Y}.
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| </math>
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| ===Lagrange equation===
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| The Lagrange equation for the ''r''<sup>th</sup> variable <math>\varphi_{r}</math> is
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| :<math>
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| \frac{d}{dt} \left( \frac{\partial T}{\partial \dot{\varphi}_{r}} \right) =
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| \frac{d}{dt} \left( Y \dot{\varphi}_{r} \right) = \frac{1}{2} F \frac{\partial Y}{\partial \varphi_{r}}
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| -\frac{\partial V}{\partial \varphi_{r}}. | |
| </math>
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| Multiplying both sides by <math>2 Y \dot{\varphi}_{r}</math>, re-arranging, and exploiting the relation 2''T = YF'' yields the equation
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| :<math>
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| 2 Y \dot{\varphi}_{r} \frac{d}{dt} \left(Y \dot{\varphi}_{r}\right) =
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| 2T\dot{\varphi}_{r} \frac{\partial Y}{\partial \varphi_{r}} - 2 Y \dot{\varphi}_{r} \frac{\partial V}{\partial \varphi_{r}} =
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| 2 \dot{\varphi}_{r} \frac{\partial}{\partial \varphi_{r}} \left[ (E-V) Y \right],
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| </math>
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| which may be written as
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| :<math>
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| \frac{d}{dt} \left(Y^{2} \dot{\varphi}_{r}^{2} \right) =
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| 2 E \dot{\varphi}_{r} \frac{\partial Y}{\partial \varphi_{r}} - 2 \dot{\varphi}_{r} \frac{\partial W}{\partial \varphi_{r}} =
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| 2E \dot{\varphi}_{r} \frac{d\chi_{r} }{d\varphi_{r}} - 2 \dot{\varphi}_{r} \frac{d\omega_{r}}{d\varphi_{r}},
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| </math>
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| where ''E = T + V'' is the (conserved) total energy. It follows that
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| :<math>
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| \frac{d}{dt} \left(Y^{2} \dot{\varphi}_{r}^{2} \right) =
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| 2\frac{d}{dt} \left( E \chi_{r} - \omega_{r} \right),
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| </math>
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| which may be integrated once to yield
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| :<math>
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| \frac{1}{2} Y^{2} \dot{\varphi}_{r}^{2} = E \chi_{r} - \omega_{r} + \gamma_{r},
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| </math>
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| where the <math>\gamma_{r}</math> are constants of integration subject to the energy conservation
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| :<math>
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| \sum_{r=1}^{s} \gamma_{r} = 0.
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| </math>
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| Inverting, taking the square root and separating the variables yields a set of separably integrable equations:
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| :<math>
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| \frac{\sqrt{2}}{Y} dt = \frac{d\varphi_{1}}{\sqrt{E \chi_{1} - \omega_{1} + \gamma_{1}}} =
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| \frac{d\varphi_{2}}{\sqrt{E \chi_{2} - \omega_{2} + \gamma_{2}}} = \cdots =
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| \frac{d\varphi_{s}}{\sqrt{E \chi_{s} - \omega_{s} + \gamma_{s}}}.
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| </math>
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| ==References==
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| {{reflist|1}}
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| ==Further reading==
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| * {{cite book | last = Whittaker | first = ET | year = 1937 | title = A Treatise on the Analytical Dynamics of Particles and Rigid Bodies, with an Introduction to the Problem of Three Bodies | edition = 4th | publisher = Dover Publications | location = New York | id = ISBN }}
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| [[Category:Classical mechanics]]
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