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Lagrangian Dynamics and Orbital Mechanics

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1. Two blocks each of mass M are connected by an extension less, uniform string of length l. One block is placed on a smooth horizontal surface and the other block hangs over the side of the table with the string passing over a frictionless pulley. Describe the motion of the system:

a) when the mass of the string is negligible
b) when the string has mass m

Use lagrangian/Hamiltonian Dynamics to solve this problem.

2. The motion of two bodies interacting only with each other by central forces can be reduced to an equivalent one body problem. Show by explicit calculation that such a reduction is also possible for bodies moving in an external uniform gravitational field

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Solution Preview

Problem 1 is in the attachment.

Problem 2 is trivial. You have to switch to new variables. You go from the positions of the two particles r_{1} and r_{2} to the center of mass

R = [m_{1}r_{1} + m_{2}r_{2}]/[m_{1} + m_{2}]

and the relative position:

r_{1,2} = r_{1} - r_{2}

If you write down the Lagrangian, then the only new term is the potential energy in the external gravitational field, which can be written as:

V = [m_{1}z_{1} + m_{2}z_{2}]g = [m_{1}+m_{2}] Z g

where Z is the z-component of the center of mass.

So, the homogeneous gravitational potential is just that of a point mass with mass equal to the total mass located at the center of mass position. Therefore, if you consider only the relative motion of the particles, this term drops out.

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