Liouville Dynamical System

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 coordinates q as follows:

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

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

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

$\frac{\sqrt{2}}{Y}\, dt = \frac{d\varphi_{1}}{\sqrt{E \chi_{1} - \omega_{1} + \gamma_{1}}} = \frac{d\varphi_{2}}{\sqrt{E \chi_{2} - \omega_{2} + \gamma_{2}}} = \cdots = \frac{d\varphi_{s}}{\sqrt{E \chi_{s} - \omega_{s} + \gamma_{s}}}$

where E = T + V is the conserved energy and the 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.

Read more about Liouville Dynamical System: Example of Bicentric Orbits

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