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:[1]

T=12{u1(q1)+u2(q2)+⋯+us(qs)}{v1(q1)q˙12+v2(q2)q˙22+⋯+vs(qs)q˙s2}

V=w1(q1)+w2(q2)+⋯+ws(qs)u1(q1)+u2(q2)+⋯+us(qs)

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

2Ydt=dφ1Eχ1−ω1+γ1=dφ2Eχ2−ω2+γ2=⋯=dφsEχs−ωs+γs

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

## Contents

* 1 Example of bicentric orbits
* 1.1 Solution
* 1.2 Constant of motion
* 2 Derivation
* 2.1 New variables
* 2.2 Lagrange equation
* 3 References
* 4 Further reading

## 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

V(x,y)=−μ1(x−a)2+y2−μ2(x+a)2+y2.

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,

x=acosh⁡ξcos⁡η,

y=asinh⁡ξsin⁡η,

the potential energy can be written as

V(ξ,η)=−μ1a(cosh⁡ξ−cos⁡η)−μ2a(cosh⁡ξ+cos⁡η)=−μ1(cosh⁡ξ+cos⁡η)−μ2(cosh⁡ξ−cos⁡η)a(cosh2ξ−cos2η),

and the kinetic energy as

T=ma22(cosh2ξ−cos2η)(ξ˙2+η˙2).

This is a Liouville dynamical system if ξ and η are taken as φ1 and φ2, respectively; thus, the function Y equals

Y=cosh2ξ−cos2η

and the function W equals

W=−μ1(cosh⁡ξ+cos⁡η)−μ2(cosh⁡ξ−cos⁡η)

Using the general solution for a Liouville dynamical system below, one obtains

ma22(cosh2ξ−cos2η)2ξ˙2=Ecosh2ξ+(μ1+μ2a)cosh⁡ξ−γ

ma22(cosh2ξ−cos2η)2η˙2=−Ecos2η+(μ1−μ2a)cos⁡η+γ

Introducing a parameter u by the formula

du=dξEcosh2ξ+(μ1+μ2a)cosh⁡ξ−γ=dη−Ecos2η+(μ1−μ2a)cos⁡η+γ,

gives the parametric solution

u=∫dξEcosh2ξ+(μ1+μ2a)cosh⁡ξ−γ=∫dη−Ecos2η+(μ1−μ2a)cos⁡η+γ.

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,

r12r22dθ1dtdθ2dt+2c(μ1cos⁡θ1−μ2cos⁡θ2),

from which the problem can be solved using the method of the last multiplier.

## Derivation

### New variables

To eliminate the v functions, the variables are changed to an equivalent set

φr=∫dqrvr(qr),

giving the relation

v1(q1)q˙12+v2(q2)q˙22+⋯+vs(qs)q˙s2=φ˙12+φ˙22+⋯+φ˙s2=F,

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,

Y=χ1(φ1)+χ2(φ2)+⋯+χs(φs),

the kinetic energy can be written as

T=12YF.

Similarly, denoting the sum of the ω functions by W

W=ω1(φ1)+ω2(φ2)+⋯+ωs(φs),

the potential energy V can be written as

V=WY.

### Lagrange equation

The Lagrange equation for the rth variable φr is

ddt(∂T∂φ˙r)=ddt(Yφ˙r)=12F∂Y∂φr−∂V∂φr.

Multiplying both sides by 2Yφ˙r, re-arranging, and exploiting the relation 2T = YF yields the equation

2Yφ˙rddt(Yφ˙r)=2Tφ˙r∂Y∂φr−2Yφ˙r∂V∂φr=2φ˙r∂∂φr[(E−V)Y],

which may be written as

ddt(Y2φ˙r2)=2Eφ˙r∂Y∂φr−2φ˙r∂W∂φr=2Eφ˙rdχrdφr−2φ˙rdωrdφr,

where E = T + V is the (conserved) total energy. It follows that

ddt(Y2φ˙r2)=2ddt(Eχr−ωr),

which may be integrated once to yield

12Y2φ˙r2=Eχr−ωr+γr,

where the γr are constants of integration subject to the energy conservation

∑r=1sγr=0.

Inverting, taking the square root and separating the variables yields a set of separably integrable equations:

2Ydt=dφ1Eχ1−ω1+γ1=dφ2Eχ2−ω2+γ2=⋯=dφsEχs−ωs+γs.

## References

1. ↑ 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. http://visualiseur.bnf.fr/ConsulterElementNum?O=NUMM-16393&Deb=263&Fin=305&E=PDF.

## Further reading

* 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.

0.00

(0 votes)

Original source: https://en.wikipedia.org/wiki/Liouville dynamical system. Read more

|

Retrieved from "https://handwiki.org/wiki/index.php?title=Physics:Liouville_dynamical_system&oldid=4369533"