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cahill8
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Homework Statement
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
Homework Equations
L = T-V
Euler-Lagrange equations
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
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
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
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
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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