(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

From pages 124-125 in edition 3.

This is about the restricted three body problem (m3 << m1,m2)

[PLAIN]http://img718.imageshack.us/img718/7012/3bdy.jpg [Broken]

2. Relevant equations

L = T-V

Euler-Lagrange equations

3. The attempt at a solution

I'm interested in m3, the negligible mass object.

Latex isn't showing up in the preview so note that by xdot I mean d/dt(x), abs = absolute value, and r1,r2,r3 are vectors.

L = 1/2 m3 (xdot^2+ydot^2) - V

my first question is: is the potential simply this?

[PLAIN]http://img408.imageshack.us/img408/9834/potential.jpg [Broken]

and I'm calling this V(r,theta,t)

changing to polar coordinates (x=rcos(theta), y=rsin(theta)) and simplifying gives

[PLAIN]http://img11.imageshack.us/img11/6784/14361215.jpg [Broken]

This is where the example starts, except the potential is not stated, it is just given as V(r,theta,t)

the vectors in my potential depend on r, theta and t so I guess that is right?

Next in the example, the coordinate system is made to rotate at the same frequency as m3 so that m1 and m2 appear stationary.

theta' = theta + wt (I'm using w for omega)

theta = theta' - wt

thetadot = theta'dot - w

so I replace thetadot above in the equation for L and I eventually get

[PLAIN]http://img198.imageshack.us/img198/2047/lchanged1.jpg [Broken]

now my main confusion comes. In the example, as well as making the coordinate system rotate, they say switch to cylindrical coordinates and state that this becomes

[PLAIN]http://img685.imageshack.us/img685/421/lchanged2.jpg [Broken]

questions:

The t dependency in the potential changes into a z dependency?

z seems a strange choice as it is a two-dimensional problem?

The only reason I can see for this is to get the lagrangian in the form L = (rho,theta,z,rhodot,thetadot,zdot)

The book does not say what the transformations were except for theta. Just that it was a cylindrical transform using rho, theta and z

comparing the two equations (the one I derived and the one given) the transformation must have been

rho = r (since x and y are already in polar coordinates x=rcos(theta)=rhocos(theta))

theta = theta' - wt

z = t

and then they added the zdot^2 to the kinetic energy term which is fine since it is 0 anyway.

but can you simply make a change like this? z = t doesn't feel right.

So I'm stuck up to this point.

The book then says to find the euler-lagrange equations and look for solutions where rhodot=zdot=thetadot=0 and these will be the five lagrange points.

So the first thing I need is a complete lagrangian with a defined and transformed potential in order to get the derivatives for the euler-lagrange equations.

Hopefully someone can help me out, thanks

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# Homework Help: Help with example from goldstein (lagrangian)

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