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Compute the Poisson brackets of \(x\)- component of angular momentum  \(L_x\) with the following quantities.

A body of mass $\mu$ is projected  from a  hypothetical planet, mass $16M$ and radius $2R$, towards another planet of mass $M$ and radius $R$. Plot the gravitational potential seen by the body  as function of distance from the center of the first planet. The distance between the  centers of the planet is given to be $10R.$

For a particle  moving in a spherically symmetric potential the three  components of angular momentum are conserved. For a potential depending on $r$ as well as $\vec{n}\cdot\vec{r}$ ( $\vec{n} = $  a fixed vector) prove that only the component of angular momentum along $\vec{n}$ is conserved.

Find the time dependence of coordinates $(x,y)$ of a particle with energy  $E=0$ moving in a parabola in a potential $$V(r)= - {k\over r}.$$

A particle moves in a central potential $$ V(r) = -{\alpha \over r^2} $$       and has energy $E$, angular momentum $L$. Show that the orbit of the particle is given by $$ {1\over r} = \sqrt{2mE\over L^2 -2m\alpha } \cos ( \lambda (\phi + \delta))$$  where $\lambda ^2 =1- {2m\alpha \over L^2}$

Consider the motion in a spherically symmetric potential $$ V(r) = - V_0  \left( {3 R\over r } + {R^3\over r^3} \right)$$ If orbital angular momentum of the particle is given by $l^2 = 10 m V_0 R^2$, plot the potential as a function of $r$ and answer the following questions.

1. What is the minimum energy so that particle coming from infinity may fall to the center?
2. What should be the energy of the particle so that it may move in a circular orbit? Find the radius of the orbit.

Sketch the equivalent one dimensional potential for the centralforce corresponding to \(F=-\frac{3a}{r^4}\). A particle  with energy \(E\) and angular momentum \(\ell\)  approaches the centre of force. Find condition(s) on energy and the angular momentum such that the values of \(E^\prime\) such that the particle may fall to the centre.