A particle of mass \(m\) moves in three dimensions under the action of a central, conservative force with potential energy \(V(r)\). Find the Hamiltonian function in terms of spherical polar co-ordinates, and show that \(\varphi\), but \(\operatorname{not} \theta\), is ignorable. Express the quantity \(J^{2}=\) \(m^{2} r^{4}\left(\dot{\theta}^{2}+\sin ^{2} \theta \dot{\varphi}^{2}\right)\) in terms of the generalized momenta, and show that it is a second constant of the motion.

First, we need to express the kinetic energy of the particle in spherical polar coordinates. To do this, we can recall the transformation between Cartesian and spherical polar coordinates: \(x = r\sin\theta\cos\varphi\) \(y = r\sin\theta\sin\varphi\) \(z = r\cos\theta\) Now, we differentiate these equations with respect to time to find the velocity components in Cartesian coordinates: \(\dot{x} = \dot{r}\sin\theta\cos\varphi + r\cos\theta\cos\varphi\dot{\theta} - r\sin\theta\sin\varphi\dot{\varphi}\) \(\dot{y} = \dot{r}\sin\theta\sin\varphi + r\cos\theta\sin\varphi\dot{\theta} + r\sin\theta\cos\varphi\dot{\varphi}\) \(\dot{z} = \dot{r}\cos\theta - r\sin\theta\dot{\theta}\) Now we can find the kinetic energy T in terms of spherical polar coordinates by using the velocity components: \(T = \dfrac{1}{2}m(\dot{x}^2 + \dot{y}^2 + \dot{z}^2)\) After some algebraic manipulations, we get: \(T=\dfrac{1}{2}m\left(\dot{r}^{2}+r^2\left(\dot{\theta}^{2}+\sin^2\theta\dot{\varphi}^{2}\right)\right)\)

Now that we have found the kinetic energy in spherical polar coordinates, let us write down the Hamiltonian function, which is the sum of the kinetic and potential energies: \(H = T + V = \dfrac{1}{2}m\left(\dot{r}^{2}+r^2\left(\dot{\theta}^{2}+\sin^2\theta\dot{\varphi}^{2}\right)\right) + V(r)\)

To determine whether \(\varphi\) and \(\theta\) are ignorable coordinates, we need to find the generalized momenta associated with these coordinates. The generalized momentum associated with a generalized coordinate \(q\) is defined as: \(p_q = \dfrac{\partial L}{\partial\dot{q}}\) where \(L\) is the Lagrangian, which can be written as \(L = T - V\). First, let us find the generalized momentum associated with \(\varphi\): \(p_{\varphi} = \dfrac{\partial}{\partial\dot{\varphi}}\left(\dfrac{1}{2}m\left(\dot{r}^{2}+r^2\left(\dot{\theta}^{2}+\sin^2\theta\dot{\varphi}^{2}\right)\right) - V(r)\right) = mr^2\sin^2\theta\dot{\varphi}\) Next, let's find the generalized momentum associated with \(\theta\): \(p_{\theta} = \dfrac{\partial}{\partial\dot{\theta}}\left(\dfrac{1}{2}m\left(\dot{r}^{2}+r^2\left(\dot{\theta}^{2}+\sin^2\theta\dot{\varphi}^{2}\right)\right) - V(r)\right) = mr^2\dot{\theta}\)

A coordinate is ignorable if its generalized momentum does not depend on the coordinate itself. From our results above, we see that: \(p_{\varphi} = mr^2\sin^2\theta\dot{\varphi}\) does not depend on \(\varphi\). Therefore, \(\varphi\) is an ignorable coordinate. On the other hand, \(p_{\theta} = mr^2\dot{\theta}\) does depend on \(\theta\) through the term \(\sin^2\theta\). Thus, \(\theta\) is not an ignorable coordinate.

We are asked to express the given quantity \(J^2\) in terms of generalized momenta: \(J^2 = m^2r^4\left(\dot{\theta}^2 + \sin^2\theta\dot{\varphi}^2\right)\) Using the expressions for \(p_{\varphi}\) and \(p_{\theta}\) we found earlier, we can write \(J^2\) as: \(J^2 = \dfrac{1}{m^2r^4}\left(p_{\theta}^2 + \dfrac{p_{\varphi}^2}{\sin^2\theta}\right)\)

A quantity is a constant of the motion if its time derivative is zero. To show that \(J^2\) is a constant of the motion, we need to differentiate it with respect to time and show that the derivative is zero. Differentiating \(J^2\) with respect to time gives: \(\dfrac{dJ^2}{dt} = \dfrac{d}{dt}\left(\dfrac{1}{m^2r^4}\left(p_{\theta}^2 + \dfrac{p_{\varphi}^2}{\sin^2\theta}\right)\right)\) Using the chain rule and noting that the time derivatives of \(p_{\theta}\) and \(p_{\varphi}\) are zero (since they are constants of motion), we find that the time derivative of \(J^2\) is indeed zero: \(\dfrac{dJ^2}{dt} = 0\) Therefore, \(J^2\) is a constant of the motion.