The Hamiltonian for a particle of mass \(m\), with arbitrary initial position and velocity, and subject to an inverse-square attractive force, can be written

\[H=\frac{p_{r}^{2}}{2 m}+\frac{p_{\theta}^{2}}{2 m r^{2}}-\frac{k}{r}\]

where \(k\) is a positive constant.

(a) Write the Hamilton-Jacobi equation for the particle.

(b) By separating variables, show that Hamilton's principal function can be written in the form

\[S=S_{r}+S_{\theta}+S_{t}=S_{r}+C_{1} t+C_{2} \theta\]

where \(C_{1}\) and \(C_{2}\) are constants, and \(S_{r}\) depends only upon \(r\).

(c) Write an expression for \(S_{r}\) in the form of an integral over \(r\).

(d) The new coordinate \(Q_{r}=\partial S / \partial C_{2}=\) \(\partial\left(S_{r}+S_{\theta}\right) / \partial C_{2}=\alpha\), a constant, since the new coordinates in Hamilton-Jacobi theory are necessarily constants. Take this partial derivative (right through the integral sign!), to show that (with an appropriate choice of signs)

\[\theta-\alpha=\int \frac{C_{2} d r}{r^{2} \sqrt{-2 m C_{1}+2 m k / r-C_{2}^{2} / r^{2}}}\]

(e) Evaluate the integral with the help of the substitution \(u=1 / r\); then find an expression for \(r(\theta)\). This gives the possible orbital shapes: circles, ellipses, parabolas, and hyperbolas.