Liouville dynamical system

In classical mechanics, a Liouville dynamical system (named after Joseph Liouville) 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: T = 1 2 { u 1 ( q 1 ) + u 2 ( q 2 ) + ⋯ + u s ( q s ) } { v 1 ( q 1 ) q ˙ 1 2 + v 2 ( q 2 ) q ˙ 2 2 + ⋯ + v s ( q s ) q ˙ s 2 } {\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\}} V = w 1 ( q 1 ) + w 2 ( q 2 ) + ⋯ + w s ( q s ) u 1 ( q 1 ) + u 2 ( q 2 ) + ⋯ + u s ( q s ) {\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 2 Y d t = d φ 1 E χ 1 − ω 1 + γ 1 = d φ 2 E χ 2 − ω 2 + γ 2 = ⋯ = d φ s E χ s − ω s + γ s {\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 γ s {\displaystyle \gamma _{s}} are constants. As described below, the variables have been changed from qs to φs, and the functions us and ws substituted by their counterparts χs and ωs.

Source: Wikipedia — Liouville dynamical system (CC BY-SA 4.0)

Liouville dynamical system

In classical mechanics, a Liouville dynamical system (named after Joseph Liouville) 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: T = 1 2 { u 1 ( q 1 ) + u 2 ( q 2 ) + ⋯ + u s ( q s ) } { v 1 ( q 1 ) q ˙ 1 2 + v 2 ( q 2 ) q ˙ 2 2 + ⋯ + v s ( q s ) q ˙ s 2 } {\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\}} V = w 1 ( q 1 ) + w 2 ( q 2 ) + ⋯ + w s ( q s ) u 1 ( q 1 ) + u 2 ( q 2 ) + ⋯ + u s ( q s ) {\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 2 Y d t = d φ 1 E χ 1 − ω 1 + γ 1 = d φ 2 E χ 2 − ω 2 + γ 2 = ⋯ = d φ s E χ s − ω s + γ s {\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 γ s {\displaystyle \gamma _{s}} are constants. As described below, the variables have been changed from qs to φs, and the functions us and ws substituted by their counterparts χs and ωs.

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Source: Wikipedia "Liouville dynamical system" · CC BY-SA 4.0

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