Chapter 12

A particle of mass \(m\) slides on the inside of a smooth cone of semivertical angle \(\alpha\), whose axis points vertically upwards. Obtain the Hamiltonian function, using the distance \(r\) from the vertex, and the azimuth angle \(\varphi\) as generalized co-ordinates. Show that stable circular motion is possible for any value of \(r\), and determine the corresponding angular velocity, \(\omega\). Find the angle \(\alpha\) if the frequency of small oscillations about this circular motion is also \(\omega\).

Find the Hamiltonian function for the forced pendulum considered in \(\S 10.4\), and verify that it is equal to \(T^{\prime}+V^{\prime} .\) Determine the frequency of small oscillations about the stable 'equilibrium' position when \(\omega^{2}>g / l\).

A light, inextensible string passes over a small pulley and carries a mass \(2 \mathrm{~m}\) on one end. On the other end is a mass \(m\), and beneath it, supported by a spring with spring constant \(k\), a second mass \(m\). Find the Hamiltonian function, using the distance \(x\) of the first mass beneath the pulley, and the extension \(y\) in the spring, as generalized coordinates. Show that \(x\) is ignorable. To what symmetry property does this correspond? (In other words, what operation can be performed on the system without changing its energy?) If the system is released from rest with the spring unextended, find the positions of the particles at any later time.

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.

To prove that the effective potential energy function \(U(\theta)\) of the symmetric top (see \(\S\) 12.4) has only a single minimum, show that the equation \(U(\theta)=E\) can be written as a cubic equation in the variable \(z=\cos \theta\), with three roots in general. Show, however, that \(f(z)\) has the same sign at both \(z=\pm 1\), and hence that there are either two roots or none between these points: for every \(E\) there are at most two values of \(\theta\) for which \(U(\theta)=E\)

Find the Hamiltonian for a charged particle in electric and magnetic fields in cylindrical polars, starting from the Lagrangian function (10.29). Show that in the case of an axially symmetric, static magnetic field, described by the single component \(A_{\varphi}(\rho, z)\) of the vector potential, it takes the form \(H=\frac{1}{2 m}\left(p_{z}^{2}+p_{\rho}^{2}+\frac{\left(p_{\varphi}-q \rho A_{\varphi}\right)^{2}}{\rho^{2}}\right)\) (Note: Remember that the subscripts \(\varphi\) on the generalized momentum \(p_{\varphi}\) and on the component \(A_{\varphi}\) mean different things.)

A particle of mass \(m\) and charge \(q\) is moving around a fixed point charge \(-q^{\prime}\left(q q^{\prime}>0\right)\), and in a uniform magnetic field \(\boldsymbol{B}\). The motion is confined to the plane perpendicular to \(\boldsymbol{B}\). Write down the Lagrangian function in polar co-ordinates rotating with the Larmor angular velocity \(\omega_{\mathrm{L}}=-q B / 2 m\) (see \(\left.\S 5.5\right) .\) Hence find the Hamiltonian function. Show that \(\varphi\) is ignorable, and interpret the conservation law. (Note that \(J_{z}\) is not a constant of the motion.)

A particle of mass \(m\) and charge \(q\) is moving in the equatorial plane \(z=0\) of a magnetic dipole of moment \(\mu\), described (see Appendix A, Problem 12) by a vector potential with the single non-zero component \(A_{\varphi}=\mu_{0} \mu \sin \theta / 4 \pi r^{2}\). Show that it will continue to move in this plane. Initially, it is approaching from a great distance with velocity \(v\) and impact parameter \(b\), whose sign is defined to be that of \(p_{\varphi}\). Show that \(v\) and \(p_{\varphi}\) are constants of the motion, and that the distance of closest approach to the dipole is \(\frac{1}{2}\left(\sqrt{b^{2} \mp a^{2}} \pm b\right)\), according as \(b>a\) or \(b

To investigate the stability of the motion described in the preceding question, evaluate the second derivatives of \(U\) at \(\rho=a, z=0\), and show that they may be written \(\begin{gathered} \frac{\partial^{2} U}{\partial \rho^{2}}=\frac{q^{2}}{m}\left[B_{z}\left(B_{z}+\rho \frac{\partial B_{z}}{\partial \rho}\right)\right]_{\rho=a, z=0} \\ \frac{\partial^{2} U}{\partial \rho \partial z}=0, \quad \frac{\partial^{2} U}{\partial z^{2}}=-\frac{q^{2}}{m}\left[B_{z} \rho \frac{\partial B_{z}}{\partial \rho}\right]_{\rho=a, z=0} \end{gathered}\) (Hint: You will need to use the \(\varphi\) component of the equation \(\boldsymbol{\nabla} \wedge \boldsymbol{B}=\mathbf{0}\), and the fact that, since \(B_{\rho}=0\) for all \(\rho, \partial B_{\rho} / \partial \rho=0\) also.) Given that the dependence of \(B_{z}\) on \(\rho\) near the equilibrium orbit is described by \(B_{z} \propto(a / \rho)^{n}\), show that the orbit is stable if \(0

Show that the Poisson brackets of the components of angular momentum are $$\left[J_{x}, J_{y}\right]=J_{z}$$ (together with two other relations obtained by cyclic permutation of \(x, y, z)\). Interpret this result in terms of the transformation of one component generated by another.

Show that the condition that Hamilton's equations remain unchanged under the transformation generated by \(G\) is \(\mathrm{d} G / \mathrm{d} t=0\) even in the case when \(G\) has an explicit time-dependence, in addition to its dependence via \(q(t)\) and \(p(t)\). Proceed as follows. The first set of Hamilton's equations, \((12.6)\), will be unchanged provided that $$\frac{\mathrm{d}}{\mathrm{d} t}\left(\delta q_{\alpha}\right)=\delta\left(\frac{\partial H}{\partial p_{\alpha}}\right)$$ Write both sides of this equation in terms of \(G\) and use \((12.33)\) applied both to \(\partial G / \partial p_{\alpha}\) and to \(G\) itself to show that it is equivalent to the condition $$\frac{\partial}{\partial p_{\alpha}}\left(\frac{\mathrm{d} G}{\mathrm{~d} t}\right)=0$$ Thus \(\mathrm{d} G / \mathrm{d} t\) is independent of each \(p_{\alpha} .\) Similarly, by using the other set of Hamilton's equations, \((12.7)\), show that it is independent of each \(q_{\alpha}\). Thus \(\mathrm{d} G / \mathrm{d} t\) must be a function of \(t\) alone. But since we can always add to \(G\) any function of \(t\) alone without affecting the transformation it generates, this means we can choose it so that \(\mathrm{d} G / \mathrm{d} t=0\).