• # A1.2 B1.2

(i) In Hamiltonian mechanics the action is written

$S=\int d t\left(p^{a} \dot{q}^{a}-H\left(q^{a}, p^{a}, t\right)\right)$

Starting from Maupertius' principle $\delta S=0$, derive Hamilton's equations

$\dot{q}^{a}=\frac{\partial H}{\partial p^{a}}, \quad \dot{p}^{a}=-\frac{\partial H}{\partial q^{a}} .$

Show that $H$ is a constant of the motion if $\partial H / \partial t=0$. When is $p^{a}$ a constant of the motion?

(ii) Consider the action $S$ given in Part (i), evaluated on a classical path, as a function of the final coordinates $q_{f}^{a}$ and final time $t_{f}$, with the initial coordinates and the initial time held fixed. Show that $S\left(q_{f}^{a}, t_{f}\right)$ obeys

$\frac{\partial S}{\partial q_{f}^{a}}=p_{f}^{a}, \quad \frac{\partial S}{\partial t_{f}}=-H\left(q_{f}^{a}, p_{f}^{a}, t_{f}\right)$

Now consider a simple harmonic oscillator with $H=\frac{1}{2}\left(p^{2}+q^{2}\right)$. Setting the initial time and the initial coordinate to zero, find the classical solution for $p$ and $q$ with final coordinate $q=q_{f}$ at time $t=t_{f}$. Hence calculate $S\left(t_{f}, q_{f}\right)$, and explicitly verify (2) in this case.

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• # A2.2 B2.1

(i) Consider a light rigid circular wire of radius $a$ and centre $O$. The wire lies in a vertical plane, which rotates about the vertical axis through $O$. At time $t$ the plane containing the wire makes an angle $\phi(t)$ with a fixed vertical plane. A bead of mass $m$ is threaded onto the wire. The bead slides without friction along the wire, and its location is denoted by $A$. The angle between the line $O A$ and the downward vertical is $\theta(t)$.

Show that the Lagrangian of the system is

$\frac{m a^{2}}{2} \dot{\theta}^{2}+\frac{m a^{2}}{2} \dot{\phi}^{2} \sin ^{2} \theta+m g a \cos \theta .$

Calculate two independent constants of the motion, and explain their physical significance.

(ii) A dynamical system has Hamiltonian $H(q, p, \lambda)$, where $\lambda$ is a parameter. Consider an ensemble of identical systems chosen so that the number density of systems, $f(q, p, t)$, in the phase space element $d q d p$ is either zero or one. Prove Liouville's Theorem, namely that the total area of phase space occupied by the ensemble is time-independent.

Now consider a single system undergoing periodic motion $q(t), p(t)$. Give a heuristic argument based on Liouville's Theorem to show that the area enclosed by the orbit,

$I=\oint p d q$

is approximately conserved as the parameter $\lambda$ is slowly varied (i.e. that $I$ is an adiabatic invariant).

Consider $H(q, p, \lambda)=\frac{1}{2} p^{2}+\lambda q^{2 n}$, with $n$ a positive integer. Show that as $\lambda$ is slowly varied the energy of the system, $E$, varies as

$E \propto \lambda^{1 /(n+1)} .$

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• # A3.2

(i) Explain the concept of a canonical transformation from coordinates $\left(q^{a}, p^{a}\right)$ to $\left(Q^{a}, P^{a}\right)$. Derive the transformations corresponding to generating functions $F_{1}\left(t, q^{a}, Q^{a}\right)$ and $F_{2}\left(t, q^{a}, P^{a}\right)$.

(ii) A particle moving in an electromagnetic field is described by the Lagrangian

$L=\frac{1}{2} m \dot{\mathbf{x}}^{2}-e\left(\phi-\frac{\dot{\mathbf{x}} \cdot \mathbf{A}}{c}\right)$

where $c$ is constant

(a) Derive the equations of motion in terms of the electric and magnetic fields $\mathbf{E}$ and $\mathbf{B}$.

(b) Show that $\mathbf{E}$ and $\mathbf{B}$ are invariant under the gauge transformation

$\mathbf{A} \rightarrow \mathbf{A}+\nabla \Lambda, \quad \phi \rightarrow \phi-\frac{1}{c} \frac{\partial \Lambda}{\partial t}$

for $\operatorname{arbitrary} \Lambda(t, \mathbf{x})$.

(c) Construct the Hamiltonian. Find the generating function $F_{2}$ for the canonical transformation which implements the gauge transformation (1).

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• # A4.2

Consider a system of coordinates rotating with angular velocity $\boldsymbol{\omega}$ relative to an inertial coordinate system.

Show that if a vector $\mathbf{v}$ is changing at a rate $d \mathbf{v} / d t$ in the inertial system, then it is changing at a rate

$\left.\frac{d \mathbf{v}}{d t}\right|_{\text {rot }}=\frac{d \mathbf{v}}{d t}-\boldsymbol{\omega} \wedge \mathbf{v}$

with respect to the rotating system.

A solid body rotates with angular velocity $\omega$ in the absence of external torque. Consider the rotating coordinate system aligned with the principal axes of the body.

(a) Show that in this system the motion is described by the Euler equations

$\left.I_{1} \frac{d \omega_{1}}{d t}\right|_{\text {rot }}=\omega_{2} \omega_{3}\left(I_{2}-I_{3}\right) \quad,\left.\quad I_{2} \frac{d \omega_{2}}{d t}\right|_{\text {rot }}=\omega_{3} \omega_{1}\left(I_{3}-I_{1}\right) \quad,\left.\quad I_{3} \frac{d \omega_{3}}{d t}\right|_{\text {rot }}=\omega_{1} \omega_{2}\left(I_{1}-I_{2}\right)$, where $\left(\omega_{1}, \omega_{2}, \omega_{3}\right)$ are the components of the angular velocity in the rotating system and $I_{1,2,3}$ are the principal moments of inertia.

(b) Consider a body with three unequal moments of inertia, $I_{3}. Show that rotation about the 1 and 3 axes is stable to small perturbations, but rotation about the 2 axis is unstable.

(c) Use the Euler equations to show that the kinetic energy, $T$, and the magnitude of the angular momentum, $L$, are constants of the motion. Show further that

$2 T I_{3} \leq L^{2} \leq 2 T I_{1} \text {. }$

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• # A1.2 B1.2

(i) Consider $N$ particles moving in 3 dimensions. The Cartesian coordinates of these particles are $x^{A}(t), A=1, \ldots, 3 N$. Now consider an invertible change of coordinates to coordinates $q^{a}\left(x^{A}, t\right), \quad a=1, \ldots, 3 N$, so that one may express $x^{A}$ as $x^{A}\left(q^{a}, t\right)$. Show that the velocity of the system in Cartesian coordinates $\dot{x}^{A}(t)$ is given by the following expression:

$\dot{x}^{A}\left(\dot{q}^{a}, q^{a}, t\right)=\sum_{b=1}^{3 N} \dot{q}^{b} \frac{\partial x^{A}}{\partial q^{b}}\left(q^{a}, t\right)+\frac{\partial x^{A}}{\partial t}\left(q^{a}, t\right)$

Furthermore, show that Lagrange's equations in the two coordinate systems are related via

$\frac{\partial L}{\partial q^{a}}-\frac{d}{d t}\left(\frac{\partial L}{\partial \dot{q}^{a}}\right)=\sum_{A=1}^{3 N} \frac{\partial x^{A}}{\partial q^{a}}\left(\frac{\partial L}{\partial x^{A}}-\frac{d}{d t} \frac{\partial L}{\partial \dot{x}^{A}}\right)$

(ii) Now consider the case where there are $p<3 N$ constraints applied, $f^{\ell}\left(x^{A}, t\right)=$ $0, \ell=1, \ldots, p$. By considering the $f^{\ell}, \ell=1, \ldots, p$, and a set of independent coordinates $q^{a}, a=1, \ldots, 3 N-p$, as a set of $3 N$ new coordinates, show that the Lagrange equations of the constrained system, i.e.

$\begin{gathered} \frac{\partial L}{\partial x^{A}}-\frac{d}{d t}\left(\frac{\partial L}{\partial \dot{x}^{A}}\right)+\sum_{\ell=1}^{p} \lambda^{\ell} \frac{\partial f^{\ell}}{\partial x^{A}}=0, \quad A=1, \ldots, 3 N \\ f^{\ell}=0, \quad \ell=1, \ldots, p \end{gathered}$

(where the $\lambda^{\ell}$ are Lagrange multipliers) imply Lagrange's equations for the unconstrained coordinates, i.e.

$\frac{\partial L}{\partial q^{a}}-\frac{d}{d t}\left(\frac{\partial L}{\partial \dot{q}^{a}}\right)=0, \quad a=1, \ldots, 3 N-p .$

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• # A2.2 B2.1

(i) The trajectory $\mathbf{x}(t)$ of a non-relativistic particle of mass $m$ and charge $q$ moving in an electromagnetic field obeys the Lorentz equation

$m \ddot{\mathbf{x}}=q\left(\mathbf{E}+\frac{\dot{\mathbf{x}}}{c} \wedge \mathbf{B}\right) .$

Show that this equation follows from the Lagrangian

$L=\frac{1}{2} m \dot{\mathbf{x}}^{2}-q\left(\phi-\frac{\dot{\mathbf{x}} \cdot \mathbf{A}}{c}\right)$

where $\phi(\mathbf{x}, t)$ is the electromagnetic scalar potential and $\mathbf{A}(\mathbf{x}, t)$ the vector potential, so that

$\mathbf{E}=-\frac{1}{c} \dot{\mathbf{A}}-\nabla \phi \text { and } \mathbf{B}=\nabla \wedge \mathbf{A}$

(ii) Let $\mathbf{E}=0$. Consider a particle moving in a constant magnetic field which points in the $z$ direction. Show that the particle moves in a helix about an axis pointing in the $z$ direction. Evaluate the radius of the helix.

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• # A3.2

(i) An axisymmetric bowling ball of mass $M$ has the shape of a sphere of radius $a$. However, it is biased so that the centre of mass is located a distance $a / 2$ away from the centre, along the symmetry axis.

The three principal moments of inertia about the centre of mass are $(A, A, C)$. The ball starts out in a stable equilibrium at rest on a perfectly frictionless flat surface with the symmetry axis vertical. The symmetry axis is then tilted through $\theta_{0}$, the ball is spun about this axis with an angular velocity $n$, and the ball is released.

Explain why the centre of mass of the ball moves only in the vertical direction during the subsequent motion. Write down the Lagrangian for the ball in terms of the usual Euler angles $\theta, \phi$ and $\psi$.

(ii) Show that there are three independent constants of the motion. Eliminate two of the angles from the Lagrangian and find the effective Lagrangian for the coordinate $\theta$.

Find the maximum and minimum values of $\theta$ in the motion of the ball when the quantity $\frac{C^{2} n^{2}}{A M g a}$ is (a) very small and (b) very large.

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• # A4.2

The action $S$ of a Hamiltonian system may be regarded as a function of the final coordinates $q^{a}, a=1, \ldots, N$, and the final time $t$ by setting

$S\left(q^{a}, t\right)=\int_{\left(q_{i}^{a}, t_{i}\right)}^{\left(q^{a}, t\right)} d t^{\prime}\left[p^{a}\left(t^{\prime}\right) \dot{q}^{a}\left(t^{\prime}\right)-H\left(p^{a}\left(t^{\prime}\right), q^{a}\left(t^{\prime}\right), t^{\prime}\right)\right]$

where the initial coordinates $q_{i}^{a}$ and time $t_{i}$ are held fixed, and $p^{a}\left(t^{\prime}\right), q^{a}\left(t^{\prime}\right)$ are the solutions to Hamilton's equations with Hamiltonian $H$, satisfying $q^{a}(t)=q^{a}, q^{a}\left(t_{i}\right)=q_{i}^{a}$.

(a) Show that under an infinitesimal change of the final coordinates $\delta q^{a}$ and time $\delta t$, the change in $S$ is

$\delta S=p_{a}(t) \delta q_{a}-H\left(p^{a}(t), q^{a}(t), t\right) \delta t$

(b) Hence derive the Hamilton-Jacobi equation

$\frac{\partial S}{\partial t}\left(q^{a}, t\right)+H\left(\frac{\partial S}{\partial q^{a}}\left(q^{a}, t\right), q^{a}, t\right)=0$

(c) If we can find a solution to $(*)$,

$S=S\left(q^{a}, t ; P^{a}\right),$

where $P^{a}$ are $N$ integration constants, then we can use $S$ as a generating function of type $I I$, where

$p^{a}=\frac{\partial S}{\partial q^{a}} \quad, \quad Q^{a}=-\frac{\partial S}{\partial P^{a}}$

Show that the Hamiltonian $K$ in the new coordinates $Q^{a}, P^{a}$ vanishes.

(d) Write down and solve the Hamilton-Jacobi equation for the one-dimensional simple harmonic oscillator, where $H=\frac{1}{2}\left(p^{2}+q^{2}\right)$. Show the solution takes the form $S(q, t ; E)=W(q, E)-E t$. Using this as a generating function $F_{I I}(q, t, P)$ show that the new coordinates $Q, P$ are constants of the motion and give their physical interpretation.

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• # A1.2 B1.2

(i) Derive Hamilton's equations from Lagrange's equations. Show that the Hamiltonian $H$ is constant if the Lagrangian $L$ does not depend explicitly on time.

(ii) A particle of mass $m$ is constrained to move under gravity, which acts in the negative $z$-direction, on the spheroidal surface $\epsilon^{-2}\left(x^{2}+y^{2}\right)+z^{2}=l^{2}$, with $0<\epsilon \leqslant 1$. If $\theta, \phi$ parametrize the surface so that

$x=\epsilon l \sin \theta \cos \phi, y=\epsilon l \sin \theta \sin \phi, z=l \cos \theta,$

find the Hamiltonian $H\left(\theta, \phi, p_{\theta}, p_{\phi}\right)$.

Show that the energy

$E=\frac{p_{\theta}^{2}}{2 m l^{2}\left(\epsilon^{2} \cos ^{2} \theta+\sin ^{2} \theta\right)}+\frac{\alpha}{\sin ^{2} \theta}+m g l \cos \theta$

is a constant of the motion, where $\alpha$ is a non-negative constant.

Rewrite this equation as

$\frac{1}{2} \dot{\theta}^{2}+V_{\mathrm{eff}}(\theta)=0$

and sketch $V_{\mathrm{eff}}(\theta)$ for $\epsilon=1$ and $\alpha>0$, identifying the maximal and minimal values of $\theta(t)$ for fixed $\alpha$ and $E$. If $\epsilon$ is now taken not to be unity, how do these values depend on $\epsilon$ ?

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• # A2.2 B2.1

(i) A number $N$ of non-interacting particles move in one dimension in a potential $V(x, t)$. Write down the Hamiltonian and Hamilton's equations for one particle.

At time $t$, the number density of particles in phase space is $f(x, p, t)$. Write down the time derivative of $f$ along a particle's trajectory. By equating the rate of change of the number of particles in a fixed domain $V$ in phase space to the flux into $V$ across its boundary, deduce that $f$ is a constant along any particle's trajectory.

(ii) Suppose that $V(x)=\frac{1}{2} m \omega^{2} x^{2}$, and particles are injected in such a manner that the phase space density is a constant $f_{1}$ at any point of phase space corresponding to a particle energy being smaller than $E_{1}$ and zero elsewhere. How many particles are present?

Suppose now that the potential is very slowly altered to the square well form

$V(x)=\left\{\begin{array}{cc} 0, & -L

Show that the greatest particle energy is now

$E_{2}=\frac{\pi^{2}}{8} \frac{E_{1}^{2}}{m L^{2} \omega^{2}} .$

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• # A3.2

(i) Show that Hamilton's equations follow from the variational principle

$\delta \int_{t_{1}}^{t_{2}}[p \dot{q}-H(q, p, t)] d t=0$

under the restrictions $\delta q\left(t_{1}\right)=\delta q\left(t_{2}\right)=\delta p\left(t_{1}\right)=\delta p\left(t_{2}\right)=0$. Comment on the difference from the variational principle for Lagrange's equations.

(ii) Suppose we transform from $p$ and $q$ to $p^{\prime}=p^{\prime}(q, p, t)$ and $q^{\prime}=q^{\prime}(q, p, t)$, with

$p^{\prime} \dot{q}^{\prime}-H^{\prime}=p \dot{q}-H+\frac{\mathrm{d}}{\mathrm{d} t} F\left(q, p, q^{\prime}, p^{\prime}, t\right)$

where $H^{\prime}$ is the new Hamiltonian. Show that $p^{\prime}$ and $q^{\prime}$ obey Hamilton's equations with Hamiltonian $H^{\prime}$.

Show that the time independent generating function $F=F_{1}\left(q, q^{\prime}\right)=q^{\prime} / q$ takes the Hamiltonian

$H=\frac{1}{2 q^{2}}+\frac{1}{2} p^{2} q^{4}$

to harmonic oscillator form. Show that $q^{\prime}$ and $p^{\prime}$ obey the Poisson bracket relation

$\left\{q^{\prime}, p^{\prime}\right\}=1$

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• # A4.2

Explain how the orientation of a rigid body can be specified by means of the three Eulerian angles, $\theta, \phi$ and $\psi$.

An axisymmetric top of mass $M$ has principal moments of inertia $A, A$ and $C$, and is spinning with angular speed $n$ about its axis of symmetry. Its centre of mass lies a distance $h$ from the fixed point of support. Initially the axis of symmetry points vertically upwards. It then suffers a small disturbance. For what values of the spin is the initial configuration stable?

If the spin is such that the initial configuration is unstable, what is the lowest angle reached by the symmetry axis in the nutation of the top? Find the maximum and minimum values of the precessional angular velocity $\dot{\phi}$.

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• # A1.2 B1.2

(i) Show that Newton's equations in Cartesian coordinates, for a system of $N$ particles at positions $\mathbf{x}_{i}(t), i=1,2 \ldots N$, in a potential $V(\mathbf{x}, t)$, imply Lagrange's equations in a generalised coordinate system

$q_{j}=q_{j}\left(\mathbf{x}_{i}, t\right) \quad, \quad j=1,2 \ldots 3 N$

that is,

$\frac{d}{d t}\left(\frac{\partial L}{\partial \dot{q}_{j}}\right)=\frac{\partial L}{\partial q_{j}} \quad, \quad j=1,2 \ldots 3 N$

where $L=T-V, T(q, \dot{q}, t)$ being the total kinetic energy and $V(q, t)$ the total potential energy.

(ii) Consider a light rod of length $L$, free to rotate in a vertical plane (the $x z$ plane), but with one end $P$ forced to move in the $x$-direction. The other end of the rod is attached to a heavy mass $M$ upon which gravity acts in the negative $z$ direction.

(a) Write down the Lagrangian for the system.

(b) Show that, if $P$ is stationary, the rod has two equilibrium positions, one stable and the other unstable.

(c) The end at $P$ is now forced to move with constant acceleration, $\ddot{x}=A$. Show that, once more, there is one stable equilibrium value of the angle the rod makes with the vertical, and find it.

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• # A2.2 B2.1

(i) An axially symmetric top rotates freely about a fixed point $O$ on its axis. The principal moments of inertia are $A, A, C$ and the centre of gravity $G$ is a distance $h$ from $O .$

Define the three Euler angles $\theta, \phi$ and $\psi$, specifying the orientation of the top. Use Lagrange's equations to show that there are three conserved quantities in the motion. Interpret them physically.

(ii) Initially the top is spinning with angular speed $n$ about $O G$, with $O G$ vertical, before it is slightly disturbed.

Show that, in the subsequent motion, $\theta$ stays close to zero if $C^{2} n^{2}>4 m g h A$, but if this condition fails then $\theta$ attains a maximum value given approximately by

$\cos \theta \approx \frac{C^{2} n^{2}}{2 m g h A}-1$

Why is this only an approximation?

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• # A3.2

(i) (a) Write down Hamilton's equations for a dynamical system. Under what condition is the Hamiltonian a constant of the motion? What is the condition for one of the momenta to be a constant of the motion?

(b) Explain the notion of an adiabatic invariant. Give an expression, in terms of Hamiltonian variables, for one such invariant.

(ii) A mass $m$ is attached to one end of a straight spring with potential energy $\frac{1}{2} k r^{2}$, where $k$ is a constant and $r$ is the length. The other end is fixed at a point $O$. Neglecting gravity, consider a general motion of the mass in a plane containing $O$. Show that the Hamiltonian is given by

$H=\frac{1}{2} \frac{p_{\theta}^{2}}{m r^{2}}+\frac{1}{2} \frac{p_{r}^{2}}{m}+\frac{1}{2} k r^{2},$

where $\theta$ is the angle made by the spring relative to a fixed direction, and $p_{\theta}, p_{r}$ are the generalised momenta. Show that $p_{\theta}$ and the energy $E$ are constants of the motion, using Hamilton's equations.

If the mass moves in a non-circular orbit, and the spring constant $k$ is slowly varied, the orbit gradually changes. Write down the appropriate adiabatic invariant $J\left(E, p_{\theta}, k, m\right)$. Show that $J$ is proportional to

$\sqrt{m k}\left(r_{+}-r_{-}\right)^{2},$

where

$r_{\pm}^{2}=\frac{E}{k} \pm \sqrt{\left(\frac{E}{k}\right)^{2}-\frac{p_{\theta}^{2}}{m k}}$

Consider an orbit for which $p_{\theta}$ is zero. Show that, as $k$ is slowly varied, the energy $E \propto k^{\alpha}$, for a constant $\alpha$ which should be found.

[You may assume without proof that

$\left.\int_{r_{-}}^{r_{+}} d r \sqrt{\left(1-\frac{r^{2}}{r_{+}^{2}}\right)\left(1-\frac{r_{-}^{2}}{r^{2}}\right)}=\frac{\pi}{4 r_{+}}\left(r_{+}-r_{-}\right)^{2} \cdot\right]$

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• # A4.2

(i) Consider a particle of charge $q$ and mass $m$, moving in a stationary magnetic field B. Show that Lagrange's equations applied to the Lagrangian

$L=\frac{1}{2} m \dot{\mathbf{r}}^{2}+q \dot{\mathbf{r}} \cdot \mathbf{A}(\mathbf{r})$

where $\mathbf{A}$ is the vector potential such that $\mathbf{B}=\operatorname{curl} \mathbf{A}$, lead to the correct Lorentz force law. Compute the canonical momentum $\mathbf{p}$, and show that the Hamiltonian is $H=\frac{1}{2} m \dot{\mathbf{r}}^{2}$.

(ii) Expressing the velocity components $\dot{r}_{i}$ in terms of the canonical momenta and co-ordinates for the above system, derive the following formulae for Poisson brackets: (b) $\left\{m \dot{r}_{i}, m \dot{r}_{j}\right\}=q \epsilon_{i j k} B_{k}$; (c) $\left\{m \dot{r}_{i}, r_{j}\right\}=-\delta_{i j}$; (d) $\left\{m \dot{r}_{i}, f\left(r_{j}\right)\right\}=-\frac{\partial}{\partial r_{i}} f\left(r_{j}\right)$.

(a) $\{F G, H\}=F\{G, H\}+\{F, H\} G$, for any functions $F, G, H$;

Now consider a particle moving in the field of a magnetic monopole,

$B_{i}=g \frac{r_{i}}{r^{3}} .$

Show that $\{H, \mathbf{J}\}=0$, where $\mathbf{J}=m \mathbf{r} \wedge \dot{\mathbf{r}}-g q \hat{\mathbf{r}}$. Explain why this means that $\mathbf{J}$ is conserved.

Show that, if $g=0$, conservation of $\mathbf{J}$ implies that the particle moves in a plane perpendicular to $\mathbf{J}$. What type of surface does the particle move on if $g \neq 0$ ?

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