Write down the Lagrangian of the system using the polar coordinates (with O as origin) of the two planets, (r1, θ1) and (r2, θ2), as the generalized coordinates [Note that the potential energy includes the gravitational potential between the planets as well as the gravitational potential between the sun and each planet]. Hence, Using Lagrange’s equations, write down the equations of motion of the planets in terms of the generalized coordinates r1, θ1, r2 or θ2
1.Write down the Lagrangian of the system using the polar coordinates (with O as origin) of the two planets, (r1, θ1) and (r2, θ2), as the generalized coordinates [Note that the potential energy includes the gravitational potential between the planets as well as the gravitational potential between the sun and each planet]. Hence, Using Lagrange’s equations, write down the equations of motion of the planets in terms of the generalized coordinates r1, θ1, r2 or θ2.
2.Is any of the generalized coordinates, r1, θ1, r2 or θ2, ignorable (i.e., the Lagrangian does not explicitly
depend on the coordinate itself)? If yes, write down the corresponding conserved generalized momentum and describe what it is. If not, choose an appropriate change /transformation of the generalized coordinates to obtain a new set of four generalized coordinates such that at least one of the new generalized coordinates is ignorable, and hence write down the corresponding conserved generalized momentum and describe what it is.
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