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 followscite journal | last = Liouville | year = 1849 | title = Mémoire sur l'intégration des équations différentielles du mouvement d'un nombre quelconque de points matériels | journal = Journal de Mathématiques Pures et Appliquées | volume = 14 | pages = 257–299 | url = http://visualiseur.bnf.fr/ConsulterElementNum?O=NUMM-16393&Deb=263&Fin=305&E=PDF] :

:$$T = frac{1}{2} left{ u_{1}(q_{1}) + u_{2}(q_{2}) + cdots + u_{s}(q_{s}) ight}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} ight}

:$$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{dvarphi_{1{sqrt{E chi_{1} - omega_{1} + gamma_{1} = frac{dvarphi_{2{sqrt{E chi_{2} - omega_{2} + gamma_{2} = cdots =frac{dvarphi_{s{sqrt{E chi_{s} - omega_{s} + gamma_{s}

where "E = T + V" is the conserved energy and the $$gamma_{r} are constants. As described below, the variables have been changed from "q_r" to φ_r, and the functions "u_r" and "w_r" substituted by their counterparts "χ_r" and "ω_r". 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. The strength of the two attractions need not be equal; thus, the two stars may have different masses or the nuclei two different charges.

olution

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) = frac{-mu_{1{sqrt{left( x - a ight)^{2} + y^{2} - frac{mu_{2{sqrt{left( x + a ight)^{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,

:$$x = a cosh xi cos eta,

:$$y = a sinh xi sin eta,

the potential energy can be written as

:$$V(xi, eta) = frac{-mu_{1{aleft( cosh xi - cos eta ight)} - frac{mu_{2{aleft( cosh xi + cos eta ight)}= frac{-mu_{1} left( cosh xi + cos eta ight) - mu_{2} left( cosh xi - cos eta ight)}{aleft( cosh^{2} xi - cos^{2} eta ight)},

and the kinetic energy as

:$$T = frac{ma^{2{2} left( cosh^{2} xi - cos^{2} eta ight) left( dot{xi}^{2} + dot{eta}^{2} ight).

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

:$$Y = cosh^{2} xi - cos^{2} eta

and the function "W" equals

:$$W = -mu_{1} left( cosh xi + cos eta ight) - mu_{2} left( cosh xi - cos eta ight)

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

:$$frac{ma^{2{2} left( cosh^{2} xi - cos^{2} eta ight) dot{xi}^{2} = E cosh^{2} xi + left( frac{mu_{1} + mu_{2{a} ight) cosh xi - gamma

:$$frac{ma^{2{2} left( cosh^{2} xi - cos^{2} eta ight)^{2} dot{eta}^{2} = -E cos^{2} eta + left( frac{mu_{1} - mu_{2{a} ight) cos eta + gamma

Introducing a parameter "u" by the formula

:$$du = frac{dxi}{sqrt{E cosh^{2} xi + left( frac{mu_{1} + mu_{2{a} ight) cosh xi - gamma = frac{deta}{sqrt{-E cos^{2} eta + left( frac{mu_{1} - mu_{2{a} ight) cos eta + gamma,

gives the parametric solution

:$$u = int frac{dxi}{sqrt{E cosh^{2} xi + left( frac{mu_{1} + mu_{2{a} ight) cosh xi - gamma = int frac{deta}{sqrt{-E cos^{2} eta + left( frac{mu_{1} - mu_{2{a} ight) cos eta + gamma.

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,

:$$r_{1}^{2} r_{2}^{2} left( frac{d heta_{1{dt} ight) left( frac{d heta_{2{dt} ight) - 2c left [ mu_{1} cos heta_{1} + mu_{2} cos heta_{2} ight] ,

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

:$$varphi_{r} = int dq_{r} sqrt{v_{r}(q_{r})},

giving the relation

:$$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} =dot{varphi}_{1}^{2} + dot{varphi}_{2}^{2} + cdots + dot{varphi}_{s}^{2} = 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 = chi_{1}(varphi_{1}) + chi_{2}(varphi_{2}) + cdots + chi_{s}(varphi_{s}),

the kinetic energy can be written as

:$$T = frac{1}{2} Y F.

Similarly, denoting the sum of the ω functions by "W"

:$$W = omega_{1}(varphi_{1}) + omega_{2}(varphi_{2}) + cdots + omega_{s}(varphi_{s}),

the potential energy "V" can be written as

:$$V = frac{W}{Y}.

Laplace equation

The Laplace equation for the "r"^th variable $$varphi_{r} is

:$$frac{d}{dt} left( frac{partial T}{partial dot{varphi}_{r ight) - frac{partial T}{partial varphi_{r = frac{d}{dt} left( Y dot{varphi}_{r} ight) - frac{1}{2} F frac{partial Y}{partial varphi_{r =-frac{partial V}{partial varphi_{r.

Multiplying both sides by $$2 Y dot{varphi}_{r}, re-arranging, and exploiting the relation 2"T = YF" yields the equation

:$$2 Y dot{varphi}_{r} frac{d}{dt} left(Y dot{varphi}_{r} ight) = 2Tdot{varphi}_{r} frac{partial Y}{partial varphi_{r - 2 Y dot{varphi}_{r} frac{partial V}{partial varphi_{r = 2 dot{varphi}_{r} frac{partial}{partial varphi_{r left [ (E-V) Y ight] ,

which may be written as

:$$frac{d}{dt} left(Y^{2} dot{varphi}_{r}^{2} ight) = 2 E dot{varphi}_{r} frac{partial Y}{partial varphi_{r - 2 dot{varphi}_{r} frac{partial W}{partial varphi_{r = 2E dot{varphi}_{r} frac{dchi_{r} }{dvarphi_{r - 2 dot{varphi}_{r} frac{domega_{r{dvarphi_{r,

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

:$$frac{d}{dt} left(Y^{2} dot{varphi}_{r}^{2} ight) = 2frac{d}{dt} left( E chi_{r} - omega_{r} ight),

which may be integrated once to yield

:$$frac{1}{2} Y^{2} dot{varphi}_{r}^{2} = E chi_{r} - omega_{r} + gamma_{r},

where the $$gamma_{r} are constants of integration subject to the energy conservation

:$$sum_{r=1}^{s} gamma_{r} = 0.

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

:$$frac{sqrt{2{Y} dt = frac{dvarphi_{1{sqrt{E chi_{1} - omega_{1} + gamma_{1} = frac{dvarphi_{2{sqrt{E chi_{2} - omega_{2} + gamma_{2} = cdots =frac{dvarphi_{s{sqrt{E chi_{s} - omega_{s} + gamma_{s}.

References

Further reading

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