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:
The solution of this system consists of a set of separably integrable equations
where E = T + V is the conserved energy and the 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. 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.
Let the fixed centers of attraction be located along the x-axis at ±a. The potential energy of the moving particle is given by
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,
the potential energy can be written as
and the kinetic energy as
This is a Liouville dynamical system if ξ and η are taken as φ1 and φ2, respectively; thus, the function Y equals
and the function W equals
Using the general solution for a Liouville dynamical system below, one obtains
Introducing a parameter u by the formula
gives the parametric solution
Since these are elliptic integrals, the coordinates ξ and η can be expressed as elliptic functions of u.
Constant of motion
The bicentric problem has a constant of motion, namely,
from which the problem can be solved using the method of the last multiplier.
To eliminate the v functions, the variables are changed to an equivalent set
giving the relation
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,
the kinetic energy can be written as
Similarly, denoting the sum of the ω functions by W
the potential energy V can be written as
The Lagrange equation for the rth variable is
Multiplying both sides by , re-arranging, and exploiting the relation 2T = YF yields the equation
which may be written as
where E = T + V is the (conserved) total energy. It follows that
which may be integrated once to yield
where the are constants of integration subject to the energy conservation
Inverting, taking the square root and separating the variables yields a set of separably integrable equations:
- Liouville (1849). "Mémoire sur l'intégration des équations différentielles du mouvement d'un nombre quelconque de points matériels". Journal de Mathématiques Pures et Appliquées 14: 257–299.
- Whittaker, ET (1937). A Treatise on the Analytical Dynamics of Particles and Rigid Bodies, with an Introduction to the Problem of Three Bodies (4th ed.). New York: Dover Publications. ISBN.