In classical mechanics, a Liouville dynamical system is an exactly solvable dynamical system in which the kinetic energy T and potential energy V can be expressed in terms of the s generalized coordinates q as follows:^[1]

${\displaystyle T={\frac {1}{2}}\left\{u_{1}(q_{1})+u_{2}(q_{2})+\cdots +u_{s}(q_{s})\right\}\left\{v_{1}(q_{1}){\dot {q}}_{1}^{2}+v_{2}(q_{2}){\dot {q}}_{2}^{2}+\cdots +v_{s}(q_{s}){\dot {q}}_{s}^{2}\right\}}$

${\displaystyle V={\frac {w_{1}(q_{1})+w_{2}(q_{2})+\cdots +w_{s}(q_{s})}{u_{1}(q_{1})+u_{2}(q_{2})+\cdots +u_{s}(q_{s})}}}$

The solution of this system consists of a set of separably integrable equations

${\displaystyle {\frac {\sqrt {2}}{Y}}\,dt={\frac {d\varphi _{1}}{\sqrt {E\chi _{1}-\omega _{1}+\gamma _{1}}}}={\frac {d\varphi _{2}}{\sqrt {E\chi _{2}-\omega _{2}+\gamma _{2}}}}=\cdots ={\frac {d\varphi _{s}}{\sqrt {E\chi _{s}-\omega _{s}+\gamma _{s}}}}}$

where E = T + V is the conserved energy and the ${\displaystyle \gamma _{s}}$ are constants. As described below, the variables have been changed from q_s to φ_s, and the functions u_s and w_s substituted by their counterparts χ_s and ω_s. This solution has numerous applications, such as the orbit of a small planet about two fixed stars under the influence of Newtonian gravity. The Liouville dynamical system is one of several things named after Joseph Liouville, an eminent French mathematician.

Example of bicentric orbits

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 force such as Newtonian gravity or Coulomb's law. Examples of the bicenter problem include a planet moving around two slowly moving stars, or an electron moving in the electric field of two positively charged nuclei, such as the first ion of the hydrogen molecule H₂, namely the hydrogen molecular ion or H₂⁺. The strength of the two attractions need not be equal; thus, the two stars may have different masses or the nuclei two different charges.

Solution

Let the fixed centers of attraction be located along the x-axis at ±a. The potential energy of the moving particle is given by

${\displaystyle V(x,y)={\frac {-\mu _{1}}{\sqrt {\left(x-a\right)^{2}+y^{2}}}}-{\frac {\mu _{2}}{\sqrt {\left(x+a\right)^{2}+y^{2}}}}.}$

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.

Introducing elliptic coordinates,

${\displaystyle x=a\cosh \xi \cos \eta ,}$

${\displaystyle y=a\sinh \xi \sin \eta ,}$

the potential energy can be written as

${\displaystyle V(\xi ,\eta )={\frac {-\mu _{1}}{a\left(\cosh \xi -\cos \eta \right)}}-{\frac {\mu _{2}}{a\left(\cosh \xi +\cos \eta \right)}}={\frac {-\mu _{1}\left(\cosh \xi +\cos \eta \right)-\mu _{2}\left(\cosh \xi -\cos \eta \right)}{a\left(\cosh ^{2}\xi -\cos ^{2}\eta \right)}},}$

and the kinetic energy as

${\displaystyle T=\frac{m{a}^2}{2}\left({\cosh }^2<!--\; \; -->\mathrm{\xi <!--\; \xi \; -->}-<!--\; -\; -->{\cos }^2<!--\; \; -->\mathrm{\eta <!--\; \eta \; -->}\right)({\stackrel{\dot{}<!--\; \dot{}\; -->}{\mathrm{\xi <!--\; \xi \; -->}}}^{}}$Zdroj:https://en.wikipedia.org?pojem=Liouville_dynamical_system
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