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Lagrangian points in circular restricted threebody problem 
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#1
Feb2712, 12:32 AM

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QUESTION:
In the circular restricted 3body problem, if we consider motion confined to the xy plane and adopt units such that G(m_{1} + m_{2}) = 1 [m_{1} and m_{2} are the masses of the two heavy bodies], the semimajor axis of the relative orbit of the massive bodies = 1, and n = 1 (n is mean motion, 2∏/period), the effective potential associated with Jacobi's constant is U = 1/2(x^{2} + y^{2})  μ^{1}/r_{1}  μ_{2}/r_{2}, where μ[SUP]1 = m_{1}/(m_{1}+m_{2}), μ[SUP]2 = m_{2}/(m_{1}+m_{2}), r_{1}^{2} = (x + μ_{2})^{2} + y^{2} and r_{2}^{2} = (x  μ_{1})^{2} + y^{2}. [Note  it doesn't say so in the question, but I assume x and y are in the rotating frame of the two massive bodies, with both bodies along the x axis, and the origin at their centre of mass.] (a) The Lagrangian points are locations where a test particle could remain stationary in the rotating frame. Show that these are stationary points of U. (b) Find the positions of the triangular Lagrangian points L_{4} and L_{5} by showing that dU/dr_{1} = 0 and dU/dr_{2} = 0 [partial derivatives] would give stationary points. MY PROGRESS: (a) I am unsure of the overall approach to take. I calculated the partial derivatives of U with respect to x and y. I know that to find stationary points, I set both of these to zero, then solve the two equations simultaneously. But this will be very messy and it doesn't seem like the right approach here. I also know [itex]\nabla[/itex]U should equal the force per unit mass on the test particle, which must be zero if the particle is stationary, as is the case at Lagrangian points. So we get 0 = dU/dx dU/dy dU/dz [partial derivatives]. dU/dz = 0, so we get dU/dx = dU/dy. As above, I know what dU/dx and dU/dy are. But subbing them in gives an extremely messy equation. And anyway, where would I go next?? Would solving the equation for x and y give me sets of coordinates that I could sub into dU/dx and dU/dy to show they would be zero? (b) I have written x and y as functions of r_{1} and r_{2}. Then I could sub these into the expression for U and take dU/dr_{1} and dU/dr_{2}. But these are also very messy... 


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