Classical Dynamics

• 1.I.9C

  Classical Dynamics

  Part II, 2005

  A particle of mass $m_{1}$ is constrained to move on a circle of radius $r_{1}$, centre $x=y=0$ in a horizontal plane $z=0$. A second particle of mass $m_{2}$ moves on a circle of radius $r_{2}$, centre $x=y=0$ in a horizontal plane $z=c$. The two particles are connected by a spring whose potential energy is

  $$V=\frac{1}{2} \omega^{2} d^{2}$$

  where $d$ is the distance between the particles. How many degrees of freedom are there? Identify suitable generalized coordinates and write down the Lagrangian of the system in terms of them.

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• 1.II.15C

  Classical Dynamics

  Part II, 2005

  (i) The action for a system with generalized coordinates $\left(q_{a}\right)$ is given by

  $$S=\int_{t_{1}}^{t_{2}} L\left(q_{a}, \dot{q}_{b}\right) d t$$

  Derive Lagrange's equations from the principle of least action by considering all paths with fixed endpoints, $\delta q_{a}\left(t_{1}\right)=\delta q_{a}\left(t_{2}\right)=0$.

  (ii) A pendulum consists of a point mass $m$ at the end of a light rod of length $l$. The pivot of the pendulum is attached to a mass $M$ which is free to slide without friction along a horizontal rail. Choose as generalized coordinates the position $x$ of the pivot and the angle $\theta$ that the pendulum makes with the vertical.

  Write down the Lagrangian and derive the equations of motion.

  Find the frequency of small oscillations around the stable equilibrium.

  Now suppose that a force acts on the pivot causing it to travel with constant acceleration in the $x$-direction. Find the equilibrium angle $\theta$ of the pendulum.

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• 2.I.9C

  Classical Dynamics

  Part II, 2005

  A rigid body has principal moments of inertia $I_{1}, I_{2}$ and $I_{3}$ and is moving under the action of no forces with angular velocity components $\left(\omega_{1}, \omega_{2}, \omega_{3}\right)$. Its motion is described by Euler's equations

  $$\begin{aligned} &I_{1} \dot{\omega}_{1}-\left(I_{2}-I_{3}\right) \omega_{2} \omega_{3}=0 \\ &I_{2} \dot{\omega}_{2}-\left(I_{3}-I_{1}\right) \omega_{3} \omega_{1}=0 \\ &I_{3} \dot{\omega}_{3}-\left(I_{1}-I_{2}\right) \omega_{1} \omega_{2}=0 \end{aligned}$$

  Are the components of the angular momentum to be evaluated in the body frame or the space frame?

  Now suppose that an asymmetric body is moving with constant angular velocity $(\Omega, 0,0)$. Show that this motion is stable if and only if $I_{1}$ is the largest or smallest principal moment.

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• 3.I.9C

  Classical Dynamics

  Part II, 2005

  Define the Poisson bracket $\{f, g\}$ between two functions $f\left(q_{a}, p_{a}\right)$ and $g\left(q_{a}, p_{a}\right)$ on phase space. If $f\left(q_{a}, p_{a}\right)$ has no explicit time dependence, and there is a Hamiltonian $H$, show that Hamilton's equations imply

  $$\frac{d f}{d t}=\{f, H\} .$$

  A particle with position vector $\mathbf{x}$ and momentum $\mathbf{p}$ has angular momentum $\mathbf{L}=\mathbf{x} \times \mathbf{p}$. Compute $\left\{p_{a}, L_{b}\right\}$ and $\left\{L_{a}, L_{b}\right\}$.

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• 3.II.15C

  Classical Dynamics

  Part II, 2005

  (i) A point mass $m$ with position $q$ and momentum $p$ undergoes one-dimensional periodic motion. Define the action variable $I$ in terms of $q$ and $p$. Prove that an orbit of energy $E$ has period

  $$T=2 \pi \frac{d I}{d E} .$$

  (ii) Such a system has Hamiltonian

  $$H(q, p)=\frac{p^{2}+q^{2}}{\mu^{2}-q^{2}}$$

  where $\mu$ is a positive constant and $|q|<\mu$ during the motion. Sketch the orbits in phase space both for energies $E \gg 1$ and $E \ll 1$. Show that the action variable $I$ is given in terms of the energy $E$ by

  $$I=\frac{\mu^{2}}{2} \frac{E}{\sqrt{E+1}} .$$

  Hence show that for $E \gg 1$ the period of the orbit is $T \approx \frac{1}{2} \pi \mu^{3} / p_{0}$, where $p_{0}$ is the greatest value of the momentum during the orbit.

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• 4.I.9C

  Classical Dynamics

  Part II, 2005

  Define a canonical transformation for a one-dimensional system with coordinates $(q, p) \rightarrow(Q, P)$. Show that if the transformation is canonical then $\{Q, P\}=1$.

  Find the values of constants $\alpha$ and $\beta$ such that the following transformations are canonical: (i) $Q=p q^{\beta}, P=\alpha q^{-1}$. (ii) $Q=q^{\alpha} \cos (\beta p), P=q^{\alpha} \sin (\beta p)$.

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• 1.I.9C

  Classical Dynamics

  Part II, 2006

  Hamilton's equations for a system with $n$ degrees of freedom can be written in vector form as

  $$\dot{\mathbf{x}}=J \frac{\partial H}{\partial \mathbf{x}}$$

  where $\mathbf{x}=\left(q_{1}, \ldots, q_{n}, p_{1}, \ldots, p_{n}\right)^{T}$ is a $2 n$-vector and the $2 n \times 2 n$ matrix $J$ takes the form

  $$J=\left(\begin{array}{cc} 0 & 1 \\ -1 & 0 \end{array}\right)$$

  where 1 is the $n \times n$ identity matrix. Derive the condition for a transformation of the form $x_{i} \rightarrow y_{i}(\mathbf{x})$ to be canonical. For a system with a single degree of freedom, show that the following transformation is canonical for all nonzero values of $\alpha$ :

  $$Q=\tan ^{-1}\left(\frac{\alpha q}{p}\right), \quad P=\frac{1}{2}\left(\alpha q^{2}+\frac{p^{2}}{\alpha}\right)$$

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• 1.II.15C

  Classical Dynamics

  Part II, 2006

  (a) In the Hamiltonian framework, the action is defined as

  $$S=\int\left(p_{a} \dot{q}_{a}-H\left(q_{a}, p_{a}, t\right)\right) d t$$

  Derive Hamilton's equations from the principle of least action. Briefly explain how the functional variations in this derivation differ from those in the derivation of Lagrange's equations from the principle of least action. Show that $H$ is a constant of the motion whenever $\partial H / \partial t=0$.

  (b) What is the invariant quantity arising in Liouville's theorem? Does the theorem depend on assuming $\partial H / \partial t=0$ ? State and prove Liouville's theorem for a system with a single degree of freedom.

  (c) A particle of mass $m$ bounces elastically along a perpendicular between two parallel walls a distance $b$ apart. Sketch the path of a single cycle in phase space, assuming that the velocity changes discontinuously at the wall. Compute the action $I=\oint p d q$ as a function of the energy $E$ and the constants $m, b$. Verify that the period of oscillation $T$ is given by $T=d I / d E$. Suppose now that the distance $b$ changes slowly. What is the relevant adiabatic invariant? How does $E$ change as a function of $b$ ?

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• 2.I.9C

  Classical Dynamics

  Part II, 2006

  Two point masses, each of mass $m$, are constrained to lie on a straight line and are connected to each other by a spring of force constant $k$. The left-hand mass is also connected to a wall on the left by a spring of force constant $j$. The right-hand mass is similarly connected to a wall on the right, by a spring of force constant $\ell$, so that the potential energy is

  $$V=\frac{1}{2} k\left(\eta_{1}-\eta_{2}\right)^{2}+\frac{1}{2} j \eta_{1}^{2}+\frac{1}{2} \ell \eta_{2}^{2}$$

  where $\eta_{i}$ is the distance from equilibrium of the $i^{\text {th }}$mass. Derive the equations of motion. Find the frequencies of the normal modes.

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• 3.I.9C

  Classical Dynamics

  Part II, 2006

  A pendulum of length $\ell$ oscillates in the $x y$ plane, making an angle $\theta(t)$ with the vertical $y$ axis. The pivot is attached to a moving lift that descends with constant acceleration $a$, so that the position of the bob is

  $$x=\ell \sin \theta, \quad y=\frac{1}{2} a t^{2}+\ell \cos \theta .$$

  Given that the Lagrangian for an unconstrained particle is

  $$L=\frac{1}{2} m\left(\dot{x}^{2}+\dot{y}^{2}\right)+m g y,$$

  determine the Lagrangian for the pendulum in terms of the generalized coordinate $\theta$. Derive the equation of motion in terms of $\theta$. What is the motion when $a=g$ ?

  Find the equilibrium configurations for arbitrary $a$. Determine which configuration is stable when

  $$\text { (i) } a<g$$

  and when

  $$\text { (ii) } a>g \text {. }$$

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• 3.II.15C

  Classical Dynamics

  Part II, 2006

  A particle of mass $m$ is constrained to move on the surface of a sphere of radius $\ell$.

  The Lagrangian is given in spherical polar coordinates by

  $$L=\frac{1}{2} m \ell^{2}\left(\dot{\theta}^{2}+\dot{\phi}^{2} \sin ^{2} \theta\right)+m g \ell \cos \theta,$$

  where gravity $g$ is constant. Find the two constants of the motion.

  The particle is projected horizontally with velocity $v$ from a point whose depth below the centre is $\ell \cos \theta=D$. Find $v$ such that the particle trajectory

  (i) just grazes the horizontal equatorial plane $\theta=\pi / 2$;

  (ii) remains at depth $D$ for all time $t$.

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• 4.I.9C

  Classical Dynamics

  Part II, 2006

  Calculate the principal moments of inertia for a uniform cylinder, of mass $M$, radius $R$ and height $2 h$, about its centre of mass. For what height-to-radius ratio does the cylinder spin like a sphere?

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• 1.I.9C

  Classical Dynamics

  Part II, 2007

  The action for a system with generalized coordinates, $q_{i}(t)$, for a time interval $\left[t_{1}, t_{2}\right]$ is given by

  $$S=\int_{t_{1}}^{t_{2}} L\left(q_{i}, \dot{q}_{i}\right) d t$$

  where $L$ is the Lagrangian, and where the end point values $q_{i}\left(t_{1}\right)$ and $q_{i}\left(t_{2}\right)$ are fixed at specified values. Derive Lagrange's equations from the principle of least action by considering the variation of $S$ for all possible paths.

  What is meant by the statement that a particular coordinate $q_{j}$ is ignorable? Show that there is an associated constant of the motion, to be specified in terms of $L$.

  A particle of mass $m$ is constrained to move on the surface of a sphere of radius $a$ under a potential, $V(\theta)$, for which the Lagrangian is given by

  $$L=\frac{m}{2} a^{2}\left(\dot{\theta}^{2}+\dot{\phi}^{2} \sin ^{2} \theta\right)-V(\theta)$$

  Identify an ignorable coordinate and find the associated constant of the motion, expressing it as a function of the generalized coordinates. Evaluate the quantity

  $$H=\dot{q}_{i} \frac{\partial L}{\partial \dot{q}_{i}}-L$$

  in terms of the same generalized coordinates, for this case. Is $H$ also a constant of the motion? If so, why?

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• 2.II.15C

  Classical Dynamics

  Part II, 2007

  (a) A Hamiltonian system with $n$ degrees of freedom is described by the phase space coordinates $\left(q_{1}, q_{2}, \ldots, q_{n}\right)$ and momenta $\left(p_{1}, p_{2}, \ldots, p_{n}\right)$. Show that the phase-space volume element

  $$d \tau=d q_{1} d q_{2} \ldots . d q_{n} d p_{1} d p_{2} \ldots . d p_{n}$$

  is conserved under time evolution.

  (b) The Hamiltonian, $H$, for the system in part (a) is independent of time. Show that if $F\left(q_{1}, \ldots, q_{n}, p_{1}, \ldots, p_{n}\right)$ is a constant of the motion, then the Poisson bracket $[F, H]$ vanishes. Evaluate $[F, H]$ when

  $$F=\sum_{k=1}^{n} p_{k}$$

  and

  $$H=\sum_{k=1}^{n} p_{k}^{2}+V\left(q_{1}, q_{2}, \ldots, q_{n}\right)$$

  where the potential $V$ depends on the $q_{k}(k=1,2, \ldots, n)$ only through quantities of the form $q_{i}-q_{j}$ for $i \neq j$.

  (c) For a system with one degree of freedom, state what is meant by the transformation

  $$(q, p) \rightarrow(Q(q, p), P(q, p))$$

  being canonical. Show that the transformation is canonical if and only if the Poisson bracket $[Q, P]=1$.

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• 2.I.9C

  Classical Dynamics

  Part II, 2007

  The Lagrangian for a particle of mass $m$ and charge $e$ moving in a magnetic field with position vector $\mathbf{r}=(x, y, z)$ is given by

  $$L=\frac{1}{2} m \dot{\mathbf{r}}^{2}+e \frac{\dot{\mathbf{r}} \cdot \mathbf{A}}{c}$$

  where the vector potential $\mathbf{A}(\mathbf{r})$, which does not depend on time explicitly, is related to the magnetic field $\mathbf{B}$ through

  $$\mathbf{B}=\nabla \times \mathbf{A}$$

  Write down Lagrange's equations and use them to show that the equation of motion of the particle can be written in the form

  $$m \ddot{\mathbf{r}}=e \frac{\dot{\mathbf{r}} \times \mathbf{B}}{c}$$

  Deduce that the kinetic energy, $T$, is constant.

  When the magnetic field is of the form $\mathbf{B}=(0,0, d F / d x)$ for some specified function $F(x)$, show further that

  $$\dot{x}^{2}=\frac{2 T}{m}-\frac{(e F(x)+C)^{2}}{m^{2} c^{2}}+D$$

  where $C$ and $D$ are constants.

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• 3.I.9C

  Classical Dynamics

  Part II, 2007

  A particle of mass $m_{1}$ is constrained to move in the horizontal $(x, y)$ plane, around a circle of fixed radius $r_{1}$ whose centre is at the origin of a Cartesian coordinate system $(x, y, z)$. A second particle of mass $m_{2}$ is constrained to move around a circle of fixed radius $r_{2}$ that also lies in a horizontal plane, but whose centre is at $(0,0, a)$. It is given that the Lagrangian $L$ of the system can be written as

  $$L=\frac{m_{1}}{2} r_{1}^{2} \dot{\phi}_{1}^{2}+\frac{m_{2}}{2} r_{2}^{2} \dot{\phi}_{2}^{2}+\omega^{2} r_{1} r_{2} \cos \left(\phi_{2}-\phi_{1}\right)$$

  using the particles' cylindrical polar angles $\phi_{1}$ and $\phi_{2}$ as generalized coordinates. Deduce the equations of motion and use them to show that $m_{1} r_{1}^{2} \dot{\phi}_{1}+m_{2} r_{2}^{2} \dot{\phi}_{2}$ is constant, and that $\psi=\phi_{2}-\phi_{1}$ obeys an equation of the form

  $$\ddot{\psi}=-k^{2} \sin \psi$$

  where $k$ is a constant to be determined.

  Find two values of $\psi$ corresponding to equilibria, and show that one of the two equilibria is stable. Find the period of small oscillations about the stable equilibrium.

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• 4.II.15C

  Classical Dynamics

  Part II, 2007

  The Hamiltonian for an oscillating particle with one degree of freedom is

  $$H=\frac{p^{2}}{2 m}+V(q, \lambda)$$

  The mass $m$ is a constant, and $\lambda$ is a function of time $t$ alone. Write down Hamilton's equations and use them to show that

  $$\frac{d H}{d t}=\frac{\partial H}{\partial \lambda} \frac{d \lambda}{d t}$$

  Now consider a case in which $\lambda$ is constant and the oscillation is exactly periodic. Denote the constant value of $H$ in that case by $E$. Consider the quantity $I=$ $(2 \pi)^{-1} \oint p d q$, where the integral is taken over a single oscillation cycle. For any given function $V(q, \lambda)$ show that $I$ can be expressed as a function of $E$ and $\lambda$ alone, namely

  $$I=I(E, \lambda)=\frac{(2 m)^{1 / 2}}{2 \pi} \oint(E-V(q, \lambda))^{1 / 2} d q$$

  where the sign of the integrand alternates between the two halves of the oscillation cycle. Let $\tau$ be the period of oscillation. Show that the function $I(E, \lambda)$ has partial derivatives

  $$\frac{\partial I}{\partial E}=\frac{\tau}{2 \pi} \quad \text { and } \quad \frac{\partial I}{\partial \lambda}=-\frac{1}{2 \pi} \oint \frac{\partial V}{\partial \lambda} d t$$

  You may assume without proof that $\partial / \partial E$ and $\partial / \partial \lambda$ may be taken inside the integral.

  Now let $\lambda$ change very slowly with time $t$, by a negligible amount during an oscillation cycle. Assuming that, to sufficient approximation,

  $$\frac{d\langle H\rangle}{d t}=\frac{\partial\langle H\rangle}{\partial \lambda} \frac{d \lambda}{d t}$$

  where $\langle H\rangle$ is the average value of $H$ over an oscillation cycle, and that

  $$\frac{d I}{d t}=\frac{\partial I}{\partial E} \frac{d\langle H\rangle}{d t}+\frac{\partial I}{\partial \lambda} \frac{d \lambda}{d t}$$

  deduce that $d I / d t=0$, carefully explaining your reasoning.

  When

  $$V(q, \lambda)=\lambda q^{2 n}$$

  with $n$ a positive integer and $\lambda$ positive, deduce that

  $$\langle H\rangle=C \lambda^{1 /(n+1)}$$

  for slowly-varying $\lambda$, where $C$ is a constant.

  [Do not try to solve Hamilton's equations. Rather, consider the form taken by $I$. ]

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• 4.I $9 \mathrm{C} \quad$

  Classical Dynamics

  Part II, 2007

  (a) Show that the principal moments of inertia for the oblate spheroid of mass $M$ defined by

  $$\frac{\left(x_{1}^{2}+x_{2}^{2}\right)}{a^{2}}+\frac{x_{3}^{2}}{a^{2}\left(1-e^{2}\right)} \leqslant 1$$

  are given by $\left(I_{1}, I_{2}, I_{3}\right)=\frac{2}{5} M a^{2}\left(1-\frac{1}{2} e^{2}, 1-\frac{1}{2} e^{2}, 1\right)$. Here $a$ is the semi-major axis and $e$ is the eccentricity.

  [You may assume that a sphere of radius a has principal moments of inertia $\frac{2}{5} M a^{2}$.]

  (b) The spheroid in part (a) rotates about an axis that is not a principal axis. Euler's equations governing the angular velocity $\left(\omega_{1}, \omega_{2}, \omega_{3}\right)$ as viewed in the body frame are

  $$\begin{aligned} &I_{1} \frac{d \omega_{1}}{d t}=\left(I_{2}-I_{3}\right) \omega_{2} \omega_{3} \\ &I_{2} \frac{d \omega_{2}}{d t}=\left(I_{3}-I_{1}\right) \omega_{3} \omega_{1} \end{aligned}$$

  and

  $$I_{3} \frac{d \omega_{3}}{d t}=\left(I_{1}-I_{2}\right) \omega_{1} \omega_{2}$$

  Show that $\omega_{3}$ is constant. Show further that the angular momentum vector precesses around the $x_{3}$ axis with period

  $$P=\frac{2 \pi\left(2-e^{2}\right)}{e^{2} \omega_{3}}$$

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• 1.I.9A

  Classical Dynamics

  Part II, 2008

  The action for a system with generalized coordinates $q_{i}(t)$ for a time interval $\left[t_{1}, t_{2}\right]$ is given by

  $$S=\int_{t_{1}}^{t_{2}} L\left(q_{i}, \dot{q}_{i}, t\right) d t$$

  where $L$ is the Lagrangian. The end point values $q_{i}\left(t_{1}\right)$ and $q_{i}\left(t_{2}\right)$ are fixed.

  Derive Lagrange's equations from the principle of least action by considering the variation of $S$ for all possible paths.

  Define the momentum $p_{i}$ conjugate to $q_{i}$. Derive a condition for $p_{i}$ to be a constant of the motion.

  A symmetric top moves under the action of a potential $V(\theta)$. The Lagrangian is given by

  $$L=\frac{1}{2} I_{1}\left(\dot{\theta}^{2}+\dot{\phi}^{2} \sin ^{2} \theta\right)+\frac{1}{2} I_{3}(\dot{\psi}+\dot{\phi} \cos \theta)^{2}-V$$

  where the generalized coordinates are the Euler angles $(\theta, \phi, \psi)$ and the principal moments of inertia are $I_{1}$ and $I_{3}$.

  Show that $\omega_{3}=\dot{\psi}+\dot{\phi} \cos \theta$ is a constant of the motion and give expressions for two others. Show further that it is possible for the top to move with both $\theta$ and $\dot{\phi}$ constant provided these satisfy the condition

  $$I_{1} \dot{\phi}^{2} \sin \theta \cos \theta-I_{3} \omega_{3} \dot{\phi} \sin \theta=\frac{d V}{d \theta}$$

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• 2.II.15B

  Classical Dynamics

  Part II, 2008

  A particle of mass $m$, charge $e$ and position vector $\mathbf{r}=\left(x_{1}, x_{2}, x_{3}\right) \equiv \mathbf{q}$ moves in a magnetic field whose vector potential is A. Its Hamiltonian is given by

  $$H(\mathbf{p}, \mathbf{q})=\frac{1}{2 m}\left(\mathbf{p}-e \frac{\mathbf{A}}{c}\right)^{2}$$

  Write down Hamilton's equations and use them to derive the equations of motion for the charged particle.

  Define the Poisson bracket $[F, G]$ for general $F(\mathbf{p}, \mathbf{q})$ and $G(\mathbf{p}, \mathbf{q})$. Show that for motion governed by the above Hamiltonian

  $$\left[m \dot{x}_{i}, x_{j}\right]=-\delta_{i j}, \quad \text { and } \quad\left[m \dot{x}_{i}, m \dot{x}_{j}\right]=\frac{e}{c}\left(\frac{\partial A_{j}}{\partial x_{i}}-\frac{\partial A_{i}}{\partial x_{j}}\right)$$

  Consider the vector potential to be given by $\mathbf{A}=(0,0, F(r))$, where $r=\sqrt{x_{1}^{2}+x_{2}^{2}}$. Use Hamilton's equations to show that $p_{3}$ is constant and that circular motion at radius $r$ with angular frequency $\Omega$ is possible provided that

  $$\Omega^{2}=-\left(p_{3}-\frac{e F}{c}\right) \frac{e}{m^{2} c r} \frac{d F}{d r}$$

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• 2.I.9A

  Classical Dynamics

  Part II, 2008

  A system of $N$ particles $i=1,2,3, \ldots, N$, with mass $m_{i}$, moves around a circle of radius $a$. The angle between the radius to particle $i$ and a fixed reference radius is $\theta_{i}$. The interaction potential for the system is

  $$V=\frac{1}{2} k \sum_{j=1}^{N}\left(\theta_{j+1}-\theta_{j}\right)^{2}$$

  where $k$ is a constant and $\theta_{N+1}=\theta_{1}+2 \pi$.

  The Lagrangian for the system is

  $$L=\frac{1}{2} a^{2} \sum_{j=1}^{N} m_{j} \dot{\theta}_{j}^{2}-V$$

  Write down the equation of motion for particle $i$ and show that the system is in equilibrium when the particles are equally spaced around the circle.

  Show further that the system always has a normal mode of oscillation with zero frequency. What is the form of the motion associated with this?

  Find all the frequencies and modes of oscillation when $N=2, m_{1}=k m / a^{2}$ and $m_{2}=2 \mathrm{~km} / \mathrm{a}^{2}$, where $m$ is a constant.

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• 3.I.9E

  Classical Dynamics

  Part II, 2008

  Writing $\mathbf{x}=\left(p_{1}, p_{2}, p_{3}, \ldots, p_{n}, q_{1}, q_{2}, q_{3}, \ldots, q_{n}\right)$, Hamilton's equations may be written in the form

  $$\dot{\mathbf{x}}=\mathbf{J} \frac{\partial H}{\partial \mathbf{x}} \text {, }$$

  where the $2 n \times 2 n$ matrix

  $$\mathbf{J}=\left(\begin{array}{rr} 0 & -I \\ I & 0 \end{array}\right)$$

  and $I$ and 0 denote the $n \times n$ unit and zero matrices respectively.

  Explain what is meant by the statement that the transformation $\mathbf{x} \rightarrow \mathbf{y}$,

  $$\left(p_{1}, p_{2}, p_{3}, \ldots, p_{n}, q_{1}, q_{2}, q_{3}, \ldots, q_{n}\right) \rightarrow\left(P_{1}, P_{2}, P_{3}, \ldots, P_{n}, Q_{1}, Q_{2}, Q_{3}, \ldots, Q_{n}\right)$$

  is canonical, and show that the condition for this is that

  $$\mathbf{J}=\mathcal{J} \mathbf{J} \mathcal{J}^{T}$$

  where $\mathcal{J}$ is the Jacobian matrix with elements

  $$\mathcal{J}_{i j}=\frac{\partial y_{i}}{\partial x_{j}}$$

  Use this condition to show that for a system with $n=1$ the transformation given by

  $$P=p+2 q, \quad Q=\frac{1}{2} q-\frac{1}{4} p$$

  is canonical.

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• 4.II.15B

  Classical Dynamics

  Part II, 2008

  (a) A Hamiltonian system with $n$ degrees of freedom has Hamiltonian $H=H(\mathbf{p}, \mathbf{q})$, where the coordinates $\mathbf{q}=\left(q_{1}, q_{2}, q_{3}, \ldots, q_{n}\right)$ and the momenta $\mathbf{p}=\left(p_{1}, p_{2}, p_{3}, \ldots, p_{n}\right)$ respectively.

  Show from Hamilton's equations that when $H$ does not depend on time explicitly, for any function $F=F(\mathbf{p}, \mathbf{q})$,

  $$\frac{d F}{d t}=[F, H],$$

  where $[F, H]$ denotes the Poisson bracket.

  For a system of $N$ interacting vortices

  $$H(\mathbf{p}, \mathbf{q})=-\frac{\kappa}{4} \sum_{\substack{i=1 \\ N}}^{N} \sum_{\substack{j=1 \\ j \neq i}}^{N} \ln \left[\left(p_{i}-p_{j}\right)^{2}+\left(q_{i}-q_{j}\right)^{2}\right]$$

  where $\kappa$ is a constant. Show that the quantity defined by

  $$F=\sum_{j=1}^{N}\left(q_{j}^{2}+p_{j}^{2}\right)$$

  is a constant of the motion.

  (b) The action for a Hamiltonian system with one degree of freedom with $H=H(p, q)$ for which the motion is periodic is

  $$I=\frac{1}{2 \pi} \oint p(H, q) d q .$$

  Show without assuming any specific form for $H$ that the period of the motion $T$ is given by

  $$\frac{2 \pi}{T}=\frac{d H}{d I}$$

  Suppose now that the system has a parameter that is allowed to vary slowly with time. Explain briefly what is meant by the statement that the action is an adiabatic invariant. Suppose that when this parameter is fixed, $H=0$ when $I=0$. Deduce that, if $T$ decreases on an orbit with any $I$ when the parameter is slowly varied, then $H$ increases.

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• 4.I.9B

  Classical Dynamics

  Part II, 2008

  (a) Show that the principal moments of inertia for an infinitesimally thin uniform rectangular sheet of mass $M$ with sides of length $a$ and $b$ (with $b<a$ ) about its centre of mass are $I_{1}=M b^{2} / 12, I_{2}=M a^{2} / 12$ and $I_{3}=M\left(a^{2}+b^{2}\right) / 12$.

  (b) Euler's equations governing the angular velocity $\left(\omega_{1}, \omega_{2}, \omega_{3}\right)$ of the sheet as viewed in the body frame are

  $$\begin{aligned} &I_{1} \frac{d \omega_{1}}{d t}=\left(I_{2}-I_{3}\right) \omega_{2} \omega_{3} \\ &I_{2} \frac{d \omega_{2}}{d t}=\left(I_{3}-I_{1}\right) \omega_{3} \omega_{1} \end{aligned}$$

  and

  $$I_{3} \frac{d \omega_{3}}{d t}=\left(I_{1}-I_{2}\right) \omega_{1} \omega_{2} .$$

  A possible solution of these equations is such that the sheet rotates with $\omega_{1}=\omega_{3}=0$, and $\omega_{2}=\Omega=$ constant.

  By linearizing, find the equations governing small motions in the neighbourhood of this solution that have $\left(\omega_{1}, \omega_{3}\right) \neq 0$. Use these to show that there are solutions corresponding to instability such that $\omega_{1}$ and $\omega_{3}$ are both proportional to exp $(\beta \Omega t)$, with $\beta=\sqrt{\left(a^{2}-b^{2}\right) /\left(a^{2}+b^{2}\right)} .$

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• Paper 3, Section I, E

  Classical Dynamics

  Part II, 2009

  (a) Show that the principal moments of inertia of a uniform circular cylinder of radius $a$, length $h$ and mass $M$ about its centre of mass are $I_{1}=I_{2}=M\left(a^{2} / 4+h^{2} / 12\right)$ and $I_{3}=M a^{2} / 2$, with the $x_{3}$ axis being directed along the length of the cylinder.

  (b) Euler's equations governing the angular velocity $\left(\omega_{1}, \omega_{2}, \omega_{3}\right)$ of an arbitrary rigid body as viewed in the body frame are

  $$\begin{aligned} &I_{1} \frac{d \omega_{1}}{d t}=\left(I_{2}-I_{3}\right) \omega_{2} \omega_{3} \\ &I_{2} \frac{d \omega_{2}}{d t}=\left(I_{3}-I_{1}\right) \omega_{3} \omega_{1} \end{aligned}$$

  and

  $$I_{3} \frac{d \omega_{3}}{d t}=\left(I_{1}-I_{2}\right) \omega_{1} \omega_{2}$$

  Show that, for the cylinder of part $(\mathrm{a}), \omega_{3}$ is constant. Show further that, when $\omega_{3} \neq 0$, the angular momentum vector precesses about the $x_{3}$ axis with angular velocity $\Omega$ given by

  $$\Omega=\left(\frac{3 a^{2}-h^{2}}{3 a^{2}+h^{2}}\right) \omega_{3}$$

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• Paper 1, Section I, E

  Classical Dynamics

  Part II, 2009

  Lagrange's equations for a system with generalized coordinates $q_{i}(t)$ are given by

  $$\frac{d}{d t}\left(\frac{\partial L}{\partial \dot{q}_{i}}\right)-\frac{\partial L}{\partial q_{i}}=0$$

  where $L$ is the Lagrangian. The Hamiltonian is given by

  $$H=\sum_{j} p_{j} \dot{q}_{j}-L,$$

  where the momentum conjugate to $q_{j}$ is

  $$p_{j}=\frac{\partial L}{\partial \dot{q}_{j}}$$

  Derive Hamilton's equations in the form

  $$\dot{q}_{i}=\frac{\partial H}{\partial p_{i}}, \quad \dot{p}_{i}=-\frac{\partial H}{\partial q_{i}}$$

  Explain what is meant by the statement that $q_{k}$ is an ignorable coordinate and give an associated constant of the motion in this case.

  The Hamiltonian for a particle of mass $m$ moving on the surface of a sphere of radius $a$ under a potential $V(\theta)$ is given by

  $$H=\frac{1}{2 m a^{2}}\left(p_{\theta}^{2}+\frac{p_{\phi}^{2}}{\sin ^{2} \theta}\right)+V(\theta)$$

  where the generalized coordinates are the spherical polar angles $(\theta, \phi)$. Write down two constants of the motion and show that it is possible for the particle to move with constant $\theta$ provided that

  $$p_{\phi}^{2}=\left(\frac{m a^{2} \sin ^{3} \theta}{\cos \theta}\right) \frac{d V}{d \theta} .$$

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• Paper 4, Section I, E

  Classical Dynamics

  Part II, 2009

  (a) A Hamiltonian system with $n$ degrees of freedom has the Hamiltonian $H(\mathbf{p}, \mathbf{q})$, where $\mathbf{q}=\left(q_{1}, q_{2}, q_{3}, \ldots, q_{n}\right)$ are the coordinates and $\mathbf{p}=\left(p_{1}, p_{2}, p_{3}, \ldots, p_{n}\right)$ are the momenta.

  A second Hamiltonian system has the Hamiltonian $G=G(\mathbf{p}, \mathbf{q})$. Neither $H$ nor $G$ contains the time explicitly. Show that the condition for $H(\mathbf{p}, \mathbf{q})$ to be invariant under the evolution of the coordinates and momenta generated by the Hamiltonian $G(\mathbf{p}, \mathbf{q})$ is that the Poisson bracket $[H, G]$ vanishes. Deduce that $G$ is a constant of the motion for evolution under $H$.

  Show that, when $G=\alpha \sum_{k=1}^{n} p_{k}$, where $\alpha$ is constant, the motion it generates is a translation of each $q_{k}$ by an amount $\alpha t$, while the corresponding $p_{k}$ remains fixed. What do you infer is conserved when $H$ is invariant under this transformation?

  (b) When $n=3$ and $H$ is a function of $p_{1}^{2}+p_{2}^{2}+p_{3}^{2}$ and $q_{1}^{2}+q_{2}^{2}+q_{3}^{2}$ only, find $\left[H, L_{i}\right]$ when

  $$L_{i}=\epsilon_{i j k} q_{j} p_{k}$$

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• Paper 2, Section $\mathbf{I}$, E

  Classical Dynamics

  Part II, 2009

  A system of three particles of equal mass $m$ moves along the $x$ axis with $x_{i}$ denoting the $x$ coordinate of particle $i$. There is an equilibrium configuration for which $x_{1}=0$, $x_{2}=a$ and $x_{3}=2 a$.

  Particles 1 and 2, and particles 2 and 3, are connected by springs with spring constant $\mu$ that provide restoring forces when the respective particle separations deviate from their equilibrium values. In addition, particle 1 is connected to the origin by a spring with spring constant $16 \mu / 3$. The Lagrangian for the system is

  $$L=\frac{m}{2}\left(\dot{x}_{1}^{2}+\dot{\eta}_{1}^{2}+\dot{\eta}_{2}^{2}\right)-\frac{\mu}{2}\left(\frac{16}{3} x_{1}^{2}+\left(\eta_{1}-x_{1}\right)^{2}+\left(\eta_{2}-\eta_{1}\right)^{2}\right)$$

  where the generalized coordinates are $x_{1}, \eta_{1}=x_{2}-a$ and $\eta_{2}=x_{3}-2 a$.

  Write down the equations of motion. Show that the generalized coordinates can oscillate with a period $P=2 \pi / \omega$, where

  $$\omega^{2}=\frac{\mu}{3 m}$$

  and find the form of the corresponding normal mode in this case.

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• Paper 2, Section II, E

  Classical Dynamics

  Part II, 2009

  A symmetric top of unit mass moves under the action of gravity. The Lagrangian is given by

  $$L=\frac{1}{2} I_{1}\left(\dot{\theta}^{2}+\dot{\phi}^{2} \sin ^{2} \theta\right)+\frac{1}{2} I_{3}(\dot{\psi}+\dot{\phi} \cos \theta)^{2}-g l \cos \theta$$

  where the generalized coordinates are the Euler angles $(\theta, \phi, \psi)$, the principal moments of inertia are $I_{1}$ and $I_{3}$ and the distance from the centre of gravity of the top to the origin is $l$.

  Show that $\omega_{3}=\dot{\psi}+\dot{\phi} \cos \theta$ and $p_{\phi}=I_{1} \dot{\phi} \sin ^{2} \theta+I_{3} \omega_{3} \cos \theta$ are constants of the motion. Show further that, when $p_{\phi}=I_{3} \omega_{3}$, with $\omega_{3}>0$, the equation of motion for $\theta$ is

  $$\frac{d^{2} \theta}{d t^{2}}=\frac{g l \sin \theta}{I_{1}}\left(1-\frac{I_{3}^{2} \omega_{3}^{2}}{4 I_{1} g l \cos ^{4}(\theta / 2)}\right)$$

  Find the possible equilibrium values of $\theta$ in the two cases:

  (i) $I_{3}^{2} \omega_{3}^{2}>4 I_{1} g l$,

  (ii) $I_{3}^{2} \omega_{3}^{2}<4 I_{1} g l$.

  By considering linear perturbations in the neighbourhoods of the equilibria in each case, find which are unstable and give expressions for the periods of small oscillations about the stable equilibria.

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• Paper 4, Section II, E

  Classical Dynamics

  Part II, 2009

  The Hamiltonian for a particle of mass $m$, charge $e$ and position vector $\mathbf{q}=(x, y, z)$, moving in an electromagnetic field, is given by

  $$H(\mathbf{p}, \mathbf{q}, t)=\frac{1}{2 m}\left(\mathbf{p}-\frac{e \mathbf{A}}{c}\right)^{2}$$

  where $\mathbf{A}(\mathbf{q}, t)$ is the vector potential. Write down Hamilton's equations and use them to derive the equations of motion for the charged particle.

  Show that, when $\mathbf{A}=\left(-y B_{0}(z, t), 0,0\right)$, there are solutions for which $p_{x}=0$ and for which the particle motion is such that

  $$\frac{d^{2} y}{d t^{2}}=-\Omega^{2} y$$

  where $\Omega=e B_{0} /(m c)$. Show in addition that the Hamiltonian may be written as

  $$H=\frac{m}{2}\left(\frac{d z}{d t}\right)^{2}+E^{\prime}$$

  where

  $$E^{\prime}=\frac{m}{2}\left(\left(\frac{d y}{d t}\right)^{2}+\Omega^{2} y^{2}\right)$$

  Assuming that $B_{0}$ is constant, find the action

  $$I\left(E^{\prime}, B_{0}\right)=\frac{1}{2 \pi} \oint m\left(\frac{d y}{d t}\right) d y$$

  associated with the $y$ motion.

  It is now supposed that $B_{0}$ varies on a time-scale much longer than $\Omega^{-1}$ and thus is slowly varying. Show by applying the theory of adiabatic invariance that the motion in the $z$ direction takes place under an effective potential and give an expression for it.

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• Paper 1, Section I, D

  Classical Dynamics

  Part II, 2010

  A system with coordinates $q_{i}, i=1, \ldots, n$, has the Lagrangian $L\left(q_{i}, \dot{q}_{i}\right)$. Define the energy $E$.

  Consider a charged particle, of mass $m$ and charge $e$, moving with velocity $\mathbf{v}$ in the presence of a magnetic field $\mathbf{B}=\boldsymbol{\nabla} \times \mathbf{A}$. The usual vector equation of motion can be derived from the Lagrangian

  $$L=\frac{1}{2} m \mathbf{v}^{2}+e \mathbf{v} \cdot \mathbf{A}$$

  where $\mathbf{A}$ is the vector potential.

  The particle moves in the presence of a field such that

  $$\mathbf{A}=(0, r g(z), 0), \quad g(z)>0$$

  referred to cylindrical polar coordinates $(r, \phi, z)$. Obtain two constants of the motion, and write down the Lagrangian equations of motion obtained by variation of $r, \phi$ and $z$.

  Show that, if the particle is projected from the point $\left(r_{0}, \phi_{0}, z_{0}\right)$ with velocity $\left(0,-2(e / m) r_{0} g\left(z_{0}\right), 0\right)$, it will describe a circular orbit provided that $g^{\prime}\left(z_{0}\right)=0$.

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• Paper 2, Section I, D

  Classical Dynamics

  Part II, 2010

  Given the form

  $$T=\frac{1}{2} T_{i j} \dot{q}_{i} \dot{q}_{j}, \quad V=\frac{1}{2} V_{i j} q_{i} q_{j}$$

  for the kinetic energy $T$ and potential energy $V$ of a mechanical system, deduce Lagrange's equations of motion.

  A light elastic string of length $4 b$, fixed at both ends, has three particles, each of mass $m$, attached at distances $b, 2 b, 3 b$ from one end. Gravity can be neglected. The particles vibrate with small oscillations transversely to the string, the tension $S$ in the string providing the restoring force. Take the displacements of the particles, $q_{i}, i=1,2,3$, to be the generalized coordinates. Take units such that $m=1, S / b=1$ and show that

  $$V=\frac{1}{2}\left[q_{1}^{2}+\left(q_{1}-q_{2}\right)^{2}+\left(q_{2}-q_{3}\right)^{2}+{q_{3}}^{2}\right]$$

  Find the normal-mode frequencies for this system.

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• Paper 3, Section I, D

  Classical Dynamics

  Part II, 2010

  Euler's equations for the angular velocity $\boldsymbol{\omega}=\left(\omega_{1}, \omega_{2}, \omega_{3}\right)$ of a rigid body, viewed in the body frame, are

  $$I_{1} \frac{d \omega_{1}}{d t}=\left(I_{2}-I_{3}\right) \omega_{2} \omega_{3}$$

  and cyclic permutations, where the principal moments of inertia are assumed to obey $I_{1}<I_{2}<I_{3}$.

  Write down two quadratic first integrals of the motion.

  There is a family of solutions $\boldsymbol{\omega}(t)$, unique up to time-translations $t \rightarrow\left(t-t_{0}\right)$, which obey the boundary conditions $\boldsymbol{\omega} \rightarrow(0, \Omega, 0)$ as $t \rightarrow-\infty$ and $\boldsymbol{\omega} \rightarrow(0,-\Omega, 0)$ as $t \rightarrow \infty$, for a given positive constant $\Omega$. Show that, for such a solution, one has

  $$\mathbf{L}^{2}=2 E I_{2},$$

  where $\mathbf{L}$ is the angular momentum and $E$ is the kinetic energy.

  By eliminating $\omega_{1}$ and $\omega_{3}$ in favour of $\omega_{2}$, or otherwise, show that, in this case, the second Euler equation reduces to

  $$\frac{d s}{d \tau}=1-s^{2}$$

  where $s=\omega_{2} / \Omega$ and $\tau=\Omega t\left[\left(I_{1}-I_{2}\right)\left(I_{2}-I_{3}\right) / I_{1} I_{3}\right]^{1 / 2}$. Find the general solution $s(\tau)$.

  [You are not expected to calculate $\omega_{1}(t)$ or $\left.\omega_{3}(t) .\right]$

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• Paper 4, Section I, D

  Classical Dynamics

  Part II, 2010

  A system with one degree of freedom has Lagrangian $L(q, \dot{q})$. Define the canonical momentum $p$ and the energy $E$. Show that $E$ is constant along any classical path.

  Consider a classical path $q_{c}(t)$ with the boundary-value data

  $$q_{c}(0)=q_{I}, \quad q_{c}(T)=q_{F}, \quad T>0$$

  Define the action $S_{c}\left(q_{I}, q_{F}, T\right)$ of the path. Show that the total derivative $d S_{c} / d T$ along the classical path obeys

  $$\frac{d S_{c}}{d T}=L$$

  Using Lagrange's equations, or otherwise, deduce that

  $$\frac{\partial S_{c}}{\partial q_{F}}=p_{F}, \quad \frac{\partial S_{c}}{\partial T}=-E,$$

  where $p_{F}$ is the final momentum.

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• Paper 2, Section II, D

  Classical Dynamics

  Part II, 2010

  An axially-symmetric top of mass $m$ is free to rotate about a fixed point $O$ on its axis. The principal moments of inertia about $O$ are $A, A, C$, and the centre of gravity $G$ is at a distance $\ell$ from $O$. Define Euler angles $\theta, \phi$ and $\psi$ which specify the orientation of the top, where $\theta$ is the inclination of $O G$ to the upward vertical. Show that there are three conserved quantities for the motion, and give their physical meaning.

  Initially, the top is spinning with angular velocity $n$ about $O G$, with $G$ vertically above $O$, before being disturbed slightly. Show that, in the subsequent motion, $\theta$ will remain close to zero provided $C^{2} n^{2}>4 m g \ell A$, but that if $C^{2} n^{2}<4 m g \ell A$, then $\theta$ will attain a maximum value given by

  $$\cos \theta \simeq\left(C^{2} n^{2} / 2 m g \ell A\right)-1$$

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• Paper 4, Section II, D

  Classical Dynamics

  Part II, 2010

  A system is described by the Hamiltonian $H(q, p)$. Define the Poisson bracket $\{f, g\}$ of two functions $f(q, p, t), g(q, p, t)$, and show from Hamilton's equations that

  $$\frac{d f}{d t}=\{f, H\}+\frac{\partial f}{\partial t}$$

  Consider the Hamiltonian

  $$H=\frac{1}{2}\left(p^{2}+\omega^{2} q^{2}\right)$$

  and define

  $$a=(p-i \omega q) /(2 \omega)^{1 / 2}, \quad a^{*}=(p+i \omega q) /(2 \omega)^{1 / 2},$$

  where $i=\sqrt{-1}$. Evaluate $\{a, a\}$ and $\left\{a, a^{*}\right\}$, and show that $\{a, H\}=-i \omega a$ and $\left\{a^{*}, H\right\}=i \omega a^{*}$. Show further that, when $f(q, p, t)$ is regarded as a function of the independent complex variables $a, a^{*}$ and of $t$, one has

  $$\frac{d f}{d t}=i \omega\left(a^{*} \frac{\partial f}{\partial a^{*}}-a \frac{\partial f}{\partial a}\right)+\frac{\partial f}{\partial t}$$

  Deduce that both $\log a^{*}-i \omega t$ and $\log a+i \omega t$ are constant during the motion.

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• Paper 1, Section I, C

  Classical Dynamics

  Part II, 2011

  (i) A particle of mass $m$ and charge $q$, at position $\mathbf{x}$, moves in an electromagnetic field with scalar potential $\phi(\mathbf{x}, t)$ and vector potential $\mathbf{A}(\mathbf{x}, t)$. Verify that the Lagrangian

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

  gives the correct equations of motion.

  [Note that $\mathbf{E}=-\nabla \phi-\dot{\mathbf{A}}$ and $\mathbf{B}=\boldsymbol{\nabla} \times \mathbf{A}$.]

  (ii) Consider the case of a constant uniform magnetic field, with $\mathbf{E}=\mathbf{0}$, given by $\phi=0$, $\mathbf{A}=(0, x B, 0)$, where $(x, y, z)$ are Cartesian coordinates and $B$ is a constant. Find the motion of the particle, and describe it carefully.

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• Paper 2, Section I, C

  Classical Dynamics

  Part II, 2011

  Three particles, each of mass $m$, move along a straight line. Their positions on the line containing the origin, $O$, are $x_{1}, x_{2}$ and $x_{3}$. They are subject to forces derived from the potential energy function

  $$V=\frac{1}{2} m \Omega^{2}\left[\left(x_{1}-x_{2}\right)^{2}+\left(x_{2}-x_{3}\right)^{2}+\left(x_{3}-x_{1}\right)^{2}+x_{1}^{2}+x_{2}^{2}+x_{3}^{2}\right]$$

  Obtain Lagrange's equations for the system, and show that the frequency, $\omega$, of a normal mode satisfies

  $$f^{3}-9 f^{2}+24 f-16=0$$

  where $f=\left(\omega^{2} / \Omega^{2}\right)$. Find a complete set of normal modes for the system, and draw a diagram indicating the nature of the corresponding motions.

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• Paper 3, Section I, $\mathbf{9 C}$

  Classical Dynamics

  Part II, 2011

  The Lagrangian for a heavy symmetric top is

  $$L=\frac{1}{2} I_{1}\left(\dot{\theta}^{2}+\dot{\phi}^{2} \sin ^{2} \theta\right)+\frac{1}{2} I_{3}(\dot{\psi}+\dot{\phi} \cos \theta)^{2}-M g l \cos \theta$$

  State Noether's Theorem. Hence, or otherwise, find two conserved quantities linear in momenta, and a third conserved quantity quadratic in momenta.

  Writing $\mu=\cos \theta$, deduce that $\mu$ obeys an equation of the form

  $$\dot{\mu}^{2}=F(\mu)$$

  where $F(\mu)$ is cubic in $\mu$. [You need not determine the explicit form of $F(\mu) .$ ]

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• Paper 4, Section I, C

  Classical Dynamics

  Part II, 2011

  (i) A dynamical system is described by the Hamiltonian $H\left(q_{i}, p_{i}\right)$. Define the Poisson bracket $\{f, g\}$ of two functions $f\left(q_{i}, p_{i}, t\right), g\left(q_{i}, p_{i}, t\right)$. Assuming the Hamiltonian equations of motion, find an expression for $d f / d t$ in terms of the Poisson bracket.

  (ii) A one-dimensional system has the Hamiltonian

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

  Show that $u=p q-2 H t$ is a constant of the motion. Deduce the form of $(q(t), p(t))$ along a classical path, in terms of the constants $u$ and $H$.

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• Paper 2, Section II, C

  Classical Dynamics

  Part II, 2011

  Derive Euler's equations governing the torque-free and force-free motion of a rigid body with principal moments of inertia $I_{1}, I_{2}$ and $I_{3}$, where $I_{1}<I_{2}<I_{3}$. Identify two constants of the motion. Hence, or otherwise, find the equilibrium configurations such that the angular-momentum vector, as measured with respect to axes fixed in the body, remains constant. Discuss the stability of these configurations.

  A spacecraft may be regarded as moving in a torque-free and force-free environment. Nevertheless, flexing of various parts of the frame can cause significant dissipation of energy. How does the angular-momentum vector ultimately align itself within the body?

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• Paper 4, Section II, C

  Classical Dynamics

  Part II, 2011

  Given a Hamiltonian system with variables $\left(q_{i}, p_{i}\right), i=1, \ldots, n$, state the definition of a canonical transformation

  $$\left(q_{i}, p_{i}\right) \rightarrow\left(Q_{i}, P_{i}\right),$$

  where $\mathbf{Q}=\mathbf{Q}(\mathbf{q}, \mathbf{p}, t)$ and $\mathbf{P}=\mathbf{P}(\mathbf{q}, \mathbf{p}, t)$. Write down a matrix equation that is equivalent to the condition that the transformation is canonical.

  Consider a harmonic oscillator of unit mass, with Hamiltonian

  $$H=\frac{1}{2}\left(p^{2}+\omega^{2} q^{2}\right) .$$

  Write down the Hamilton-Jacobi equation for Hamilton's principal function $S(q, E, t)$, and deduce the Hamilton-Jacobi equation

  $$\frac{1}{2}\left[\left(\frac{\partial W}{\partial q}\right)^{2}+\omega^{2} q^{2}\right]=E$$

  for Hamilton's characteristic function $W(q, E)$.

  Solve (1) to obtain an integral expression for $W$, and deduce that, at energy $E$,

  $$S=\sqrt{2 E} \int d q \sqrt{\left(1-\frac{\omega^{2} q^{2}}{2 E}\right)}-E t$$

  Let $\alpha=E$, and define the angular coordinate

  $$\beta=\left(\frac{\partial S}{\partial E}\right)_{q, t}$$

  You may assume that (2) implies

  $$t+\beta=\left(\frac{1}{\omega}\right) \arcsin \left(\frac{\omega q}{\sqrt{2 E}}\right)$$

  Deduce that

  $$p=\frac{\partial S}{\partial q}=\frac{\partial W}{\partial q}=\sqrt{\left(2 E-\omega^{2} q^{2}\right)}$$

  from which

  $$p=\sqrt{2 E} \cos [\omega(t+\beta)] .$$

  Hence, or otherwise, show that the transformation from variables $(q, p)$ to $(\alpha, \beta)$ is canonical.

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• Paper 4, Section I, A

  Classical Dynamics

  Part II, 2012

  Consider a one-dimensional dynamical system with generalized coordinate and momentum $(q, p)$.

  (a) Define the Poisson bracket $\{f, g\}$ of two functions $f(q, p, t)$ and $g(q, p, t)$.

  (b) Find the Poisson brackets $\{q, q\},\{p, p\}$ and $\{q, p\}$.

  (c) Assuming Hamilton's equations of motion prove that

  $$\frac{d f}{d t}=\{f, H\}+\frac{\partial f}{\partial t}$$

  (d) State the condition for a transformation $(q, p) \rightarrow(Q, P)$ to be canonical in terms of the Poisson brackets found in (b). Use this to determine whether or not the following transformations are canonical:

  (i) $Q=\sin q, P=\frac{p-a}{\cos q}$,

  (ii) $Q=\cos q, P=\frac{p-a}{\sin q}$,

  where $a$ is constant.

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• Paper 3, Section I, A

  Classical Dynamics

  Part II, 2012

  The motion of a particle of charge $q$ and mass $m$ in an electromagnetic field with scalar potential $\phi(\mathbf{r}, t)$ and vector potential $\mathbf{A}(\mathbf{r}, t)$ is characterized by the Lagrangian

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

  (a) Show that the Euler-Lagrange equation is invariant under the gauge transformation

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

  for an arbitrary function $\Lambda(\mathbf{r}, t)$.

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

  [Recall that $\mathbf{B}=\nabla \times \mathbf{A}$ and $\mathbf{E}=-\nabla \phi-\frac{\partial \mathbf{A}}{\partial t}$.]

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• Paper 2, Section I, A

  Classical Dynamics

  Part II, 2012

  (a) The action for a system with a generalized coordinate $q$ is given by

  $$S=\int_{t_{1}}^{t_{2}} L(q, \dot{q}, t) d t$$

  State the Principle of Least Action and state the Euler-Lagrange equation.

  (b) 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 this 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.

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• Paper 1, Section I, A

  Classical Dynamics

  Part II, 2012

  Consider a heavy symmetric top of mass $M$, pinned at point $P$, which is a distance $l$ from the centre of mass.

  (a) Working in the body frame $\left(\mathbf{e}_{1}, \mathbf{e}_{2}, \mathbf{e}_{3}\right)$ (where $\mathbf{e}_{3}$ is the symmetry axis of the top) define the Euler angles $(\psi, \theta, \phi)$ and show that the components of the angular velocity can be expressed in terms of the Euler angles as

  $$\boldsymbol{\omega}=(\dot{\phi} \sin \theta \sin \psi+\dot{\theta} \cos \psi, \dot{\phi} \sin \theta \cos \psi-\dot{\theta} \sin \psi, \dot{\psi}+\dot{\phi} \cos \theta)$$

  (b) Write down the Lagrangian of the top in terms of the Euler angles and the principal moments of inertia $I_{1}, I_{3}$.

  (c) Find the three constants of motion.

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• Paper 4, Section II, A

  Classical Dynamics

  Part II, 2012

  A homogenous thin rod of mass $M$ and length $l$ is constrained to rotate in a horizontal plane about its centre $O$. A bead of mass $m$ is set to slide along the rod without friction. The bead is attracted to $O$ by a force resulting in a potential $k x^{2} / 2$, where $x$ is the distance from $O$.

  (a) Identify suitable generalized coordinates and write down the Lagrangian of the system.

  (b) Identify all conserved quantities.

  (c) Derive the equations of motion and show that one of them can be written as

  $$m \ddot{x}=-\frac{\partial V_{\mathrm{eff}}(x)}{\partial x}$$

  where the form of the effective potential $V_{\text {eff }}(x)$ should be found explicitly.

  (d) Sketch the effective potential. Find and characterize all points of equilibrium.

  (e) Find the frequencies of small oscillations around the stable equilibria.

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• Paper 2, Section II, A

  Classical Dynamics

  Part II, 2012

  Consider a rigid body with principal moments of inertia $I_{1}, I_{2}, I_{3}$.

  (a) Derive Euler's equations of torque-free motion

  $$\begin{aligned} &I_{1} \dot{\omega}_{1}=\left(I_{2}-I_{3}\right) \omega_{2} \omega_{3}, \\ &I_{2} \dot{\omega}_{2}=\left(I_{3}-I_{1}\right) \omega_{3} \omega_{1} \\ &I_{3} \dot{\omega}_{3}=\left(I_{1}-I_{2}\right) \omega_{1} \omega_{2} \end{aligned}$$

  with components of the angular velocity $\boldsymbol{\omega}=\left(\omega_{1}, \omega_{2}, \omega_{3}\right)$ given in the body frame.

  (b) Show that rotation about the second principal axis is unstable if $\left(I_{2}-I_{3}\right)\left(I_{1}-I_{2}\right)>0$.

  (c) The principal moments of inertia of a uniform cylinder of radius $R$, height $h$ and mass $M$ about its centre of mass are

  $$I_{1}=I_{2}=\frac{M R^{2}}{4}+\frac{M h^{2}}{12} \quad ; \quad I_{3}=\frac{M R^{2}}{2} .$$

  The cylinder has two identical cylindrical holes of radius $r$ drilled along its length. The axes of symmetry of the holes are at a distance $a$ from the axis of symmetry of the cylinder such that $r<R / 2$ and $r<a<R-r$. All three axes lie in a single plane. Compute the principal moments of inertia of the body.

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• Paper 4, Section I, B

  Classical Dynamics

  Part II, 2013

  The Lagrangian for a heavy symmetric top of mass $M$, pinned at point $O$ which is a distance $l$ from the centre of mass, is

  $$L=\frac{1}{2} I_{1}\left(\dot{\theta}^{2}+\dot{\phi}^{2} \sin ^{2} \theta\right)+\frac{1}{2} I_{3}(\dot{\psi}+\dot{\phi} \cos \theta)^{2}-M g l \cos \theta$$

  (i) Starting with the fixed space frame $\left(\tilde{\mathbf{e}}_{\mathbf{1}}, \tilde{\mathbf{e}}_{2}, \tilde{\mathbf{e}}_{3}\right)$ and choosing $O$ at its origin, sketch the top with embedded body frame axis $\mathbf{e}_{3}$ being the symmetry axis. Clearly identify the Euler angles $(\theta, \phi, \psi)$.

  (ii) Obtain the momenta $p_{\theta}, p_{\phi}$ and $p_{\psi}$ and the Hamiltonian $H\left(\theta, \phi, \psi, p_{\theta}, p_{\phi}, p_{\psi}\right)$. Derive Hamilton's equations. Identify the three conserved quantities.

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• Paper 3, Section I, B

  Classical Dynamics

  Part II, 2013

  Two equal masses $m$ are connected to each other and to fixed points by three springs of force constant $5 k, k$ and $5 k$ as shown in the figure.

  

  (i) Write down the Lagrangian and derive the equations describing the motion of the system in the direction parallel to the springs.

  (ii) Find the normal modes and their frequencies. Comment on your results.

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• Paper 2, Section I, B

  Classical Dynamics

  Part II, 2013

  (i) Consider a rigid body with principal moments of inertia $I_{1}, I_{2}, I_{3}$. Derive Euler's equations of torque-free motion,

  $$\begin{aligned} &I_{1} \dot{\omega}_{1}=\left(I_{2}-I_{3}\right) \omega_{2} \omega_{3}, \\ &I_{2} \dot{\omega}_{2}=\left(I_{3}-I_{1}\right) \omega_{3} \omega_{1}, \\ &I_{3} \dot{\omega}_{3}=\left(I_{1}-I_{2}\right) \omega_{1} \omega_{2}, \end{aligned}$$

  with components of the angular velocity $\boldsymbol{\omega}=\left(\omega_{1}, \omega_{2}, \omega_{3}\right)$ given in the body frame.

  (ii) Use Euler's equations to show that the energy $E$ and the square of the total angular momentum $\mathbf{L}^{2}$ of the body are conserved.

  (iii) Consider a torque-free motion of a symmetric top with $I_{1}=I_{2}=\frac{1}{2} I_{3}$. Show that in the body frame the vector of angular velocity $\boldsymbol{\omega}$ precesses about the body-fixed $\mathbf{e}_{3}$ axis with constant angular frequency equal to $\omega_{3}$.

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• Paper 1, Section I, B

  Classical Dynamics

  Part II, 2013

  Consider an $n$-dimensional dynamical system with generalized coordinates and momenta $\left(q_{i}, p_{i}\right), i=1,2, \ldots, n$.

  (a) Define the Poisson bracket $\{f, g\}$ of two functions $f\left(q_{i}, p_{i}, t\right)$ and $g\left(q_{i}, p_{i}, t\right)$.

  (b) Assuming Hamilton's equations of motion, prove that if a function $G\left(q_{i}, p_{i}\right)$ Poisson commutes with the Hamiltonian, that is $\{G, H\}=0$, then $G$ is a constant of the motion.

  (c) Assume that $q_{j}$ is an ignorable coordinate, that is the Hamiltonian does not depend on it explicitly. Using the formalism of Poisson brackets prove that the conjugate momentum $p_{j}$ is conserved.

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• Paper 4, Section II, B

  Classical Dynamics

  Part II, 2013

  The motion of a particle of charge $q$ and mass $m$ in an electromagnetic field with scalar potential $\phi(\mathbf{r}, t)$ and vector potential $\mathbf{A}(\mathbf{r}, t)$ is characterized by the Lagrangian

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

  (i) Write down the Hamiltonian of the particle.

  (ii) Write down Hamilton's equations of motion for the particle.

  (iii) Show that Hamilton's equations are invariant under the gauge transformation

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

  for an arbitrary function $\Lambda(\mathbf{r}, t)$.

  (iv) The particle moves in the presence of a field such that $\phi=0$ and $\mathbf{A}=\left(-\frac{1}{2} y B, \frac{1}{2} x B, 0\right)$, where $(x, y, z)$ are Cartesian coordinates and $B$ is a constant.

  (a) Find a gauge transformation such that only one component of $\mathbf{A}(x, y, z)$ remains non-zero.

  (b) Determine the motion of the particle.

  (v) Now assume that $B$ varies very slowly with time on a time-scale much longer than $(q B / m)^{-1}$. Find the quantity which remains approximately constant throughout the motion.

  [You may use the expression for the action variable $I=\frac{1}{2 \pi} \oint p_{i} d q_{i}$.]

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• Paper 2, Section II, B

  Classical Dynamics

  Part II, 2013

  (i) The action for a system with a generalized coordinate $q$ is given by

  $$S=\int_{t_{1}}^{t_{2}} L(q, \dot{q}, t) d t$$

  (a) State the Principle of Least Action and derive the Euler-Lagrange equation.

  (b) Consider an arbitrary function $f(q, t)$. Show that $L^{\prime}=L+d f / d t$ leads to the same equation of motion.

  (ii) A wire frame $A B C$ in a shape of an equilateral triangle with side $a$ rotates in a horizontal plane with constant angular frequency $\omega$ about a vertical axis through $A$. A bead of mass $m$ is threaded on $B C$ and moves without friction. The bead is connected to $B$ and $C$ by two identical light springs of force constant $k$ and equilibrium length $a / 2$.

  (a) Introducing the displacement $\eta$ of the particle from the mid point of $B C$, determine the Lagrangian $L(\eta, \dot{\eta})$.

  (b) Derive the equation of motion. Identify the integral of the motion.

  (c) Describe the motion of the bead. Find the condition for there to be a stable equilibrium and find the frequency of small oscillations about it when it exists.

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• Paper 4, Section I, A

  Classical Dynamics

  Part II, 2014

  Consider a heavy symmetric top of mass $M$ with principal moments of inertia $I_{1}$, $I_{2}$ and $I_{3}$, where $I_{1}=I_{2} \neq I_{3}$. The top is pinned at point $P$, which is at a distance $l$ from the centre of mass, $C$, as shown in the figure.

  

  Its angular velocity in a body frame $\left(\mathbf{e}_{\mathbf{1}}, \mathbf{e}_{\mathbf{2}}, \mathbf{e}_{\mathbf{3}}\right)$ is given by

  $$\boldsymbol{\omega}=[\dot{\phi} \sin \theta \sin \psi+\dot{\theta} \cos \psi] \mathbf{e}_{1}+[\dot{\phi} \sin \theta \cos \psi-\dot{\theta} \sin \psi] \mathbf{e}_{2}+[\dot{\psi}+\dot{\phi} \cos \theta] \mathbf{e}_{3}$$

  where $\phi, \theta$ and $\psi$ are the Euler angles.

  (a) Assuming that $\left\{\mathbf{e}_{a}\right\}, a=1,2,3$, are chosen to be the principal axes, write down the Lagrangian of the top in terms of $\omega_{a}$ and the principal moments of inertia. Hence find the Lagrangian in terms of the Euler angles.

  (b) Find all conserved quantities. Show that $\omega_{3}$, the spin of the top, is constant.

  (c) By eliminating $\dot{\phi}$ and $\dot{\psi}$, derive a second-order differential equation for $\theta$.

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• Paper 3, Section I, A

  Classical Dynamics

  Part II, 2014

  (a) The action for a one-dimensional dynamical system with a generalized coordinate $q$ and Lagrangian $L$ is given by

  $$S=\int_{t_{1}}^{t_{2}} L(q, \dot{q}, t) d t$$

  State the principle of least action. Write the expression for the Hamiltonian in terms of the generalized velocity $\dot{q}$, the generalized momentum $p$ and the Lagrangian $L$. Use it to derive Hamilton's equations from the principle of least action.

  (b) The motion of a particle of charge $q$ and mass $m$ in an electromagnetic field with scalar potential $\phi(\mathbf{r}, t)$ and vector potential $\mathbf{A}(\mathbf{r}, t)$ is characterized by the Lagrangian

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

  (i) Write down the Hamiltonian of the particle.

  (ii) Consider a particle which moves in three dimensions in a magnetic field with $\mathbf{A}=(0, B x, 0)$, where $B$ is a constant. There is no electric field. Obtain Hamilton's equations for the particle.

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• Paper 2, Section I, A

  Classical Dynamics

  Part II, 2014

  The components of the angular velocity $\omega$ of a rigid body and of the position vector $\mathbf{r}$ are given in a body frame.

  (a) The kinetic energy of the rigid body is defined as

  $$T=\frac{1}{2} \int d^{3} \mathbf{r} \rho(\mathbf{r}) \dot{\mathbf{r}} \cdot \dot{\mathbf{r}}$$

  Given that the centre of mass is at rest, show that $T$ can be written in the form

  $$T=\frac{1}{2} I_{a b} \omega_{a} \omega_{b},$$

  where the explicit form of the tensor $I_{a b}$ should be determined.

  (b) Explain what is meant by the principal moments of inertia.

  (c) Consider a rigid body with principal moments of inertia $I_{1}, I_{2}$ and $I_{3}$, which are all unequal. Derive Euler's equations of torque-free motion

  $$\begin{aligned} &I_{1} \dot{\omega}_{1}=\left(I_{2}-I_{3}\right) \omega_{2} \omega_{3} \\ &I_{2} \dot{\omega}_{2}=\left(I_{3}-I_{1}\right) \omega_{3} \omega_{1} \\ &I_{3} \dot{\omega}_{3}=\left(I_{1}-I_{2}\right) \omega_{1} \omega_{2} \end{aligned}$$

  (d) The body rotates about the principal axis with moment of inertia $I_{1}$. Derive the condition for stable rotation.

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• Paper 1, Section I, A

  Classical Dynamics

  Part II, 2014

  Consider a one-dimensional dynamical system with generalized coordinate and momentum $(q, p)$.

  (a) Define the Poisson bracket $\{f, g\}$ of two functions $f(q, p, t)$ and $g(q, p, t)$.

  (b) Verify the Leibniz rule

  $$\{f g, h\}=f\{g, h\}+g\{f, h\}$$

  (c) Explain what is meant by a canonical transformation $(q, p) \rightarrow(Q, P)$.

  (d) State the condition for a transformation $(q, p) \rightarrow(Q, P)$ to be canonical in terms of the Poisson bracket $\{Q, P\}$. Use this to determine whether or not the following transformations are canonical:

  (i) $Q=\frac{q^{2}}{2}, P=\frac{p}{q}$,

  (ii) $Q=\tan q, P=p \cos q$,

  (iii) $Q=\sqrt{2 q} e^{t} \cos p, P=\sqrt{2 q} e^{-t} \sin p$.

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• Paper 4, Section II, A

  Classical Dynamics

  Part II, 2014

  (a) Consider a system with one degree of freedom, which undergoes periodic motion in the potential $V(q)$. The system's Hamiltonian is

  $$H(p, q)=\frac{p^{2}}{2 m}+V(q)$$

  (i) Explain what is meant by the angle and action variables, $\theta$ and $I$, of the system and write down the integral expression for the action variable $I$. Is $I$ conserved? Is $\theta$ conserved?

  (ii) Consider $V(q)=\lambda q^{6}$, where $\lambda$ is a positive constant. Find $I$ in terms of $\lambda$, the total energy $E$, the mass $M$, and a dimensionless constant factor (which you need not compute explicitly).

  (iii) Hence describe how $E$ changes with $\lambda$ if $\lambda$ varies slowly with time. Justify your answer.

  (b) Consider now a particle which moves in a plane subject to a central force-field $\mathbf{F}=-k r^{-2} \hat{\mathbf{r}}$.

  (i) Working in plane polar coordinates $(r, \phi)$, write down the Hamiltonian of the system. Hence deduce two conserved quantities. Prove that the system is integrable and state the number of action variables.

  (ii) For a particle which moves on an elliptic orbit find the action variables associated with radial and tangential motions. Can the relationship between the frequencies of the two motions be deduced from this result? Justify your answer.

  (iii) Describe how $E$ changes with $m$ and $k$ if one or both of them vary slowly with time.

  [You may use

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

  where $0<r_{1}<r_{2}$.]

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• Paper 2, Section II, A

  Classical Dynamics

  Part II, 2014

  A planar pendulum consists of a mass $m$ at the end of a light rod of length $l$. The pivot of the pendulum is attached to a bead of mass $M$, which slides along a horizontal rod without friction. The bead is connected to the ends of the horizontal rod by two identical springs of force constant $k$. The pivot constrains the pendulum to swing in the vertical plane through the horizontal rod. The horizontal rod is mounted on a bracket, so the system could rotate about the vertical axis which goes through its centre as shown in the figure.

  

  (a) Initially, the system is not allowed to rotate about the vertical axis.

  (i) Identify suitable generalized coordinates and write down the Lagrangian of the system.

  (ii) Write down expression(s) for any conserved quantities. Justify your answer.

  (iii) Derive the equations of motion.

  (iv) For $M=m / 2$ and $g m / k l=3$, find the frequencies of small oscillations around the stable equilibrium and the corresponding normal modes. Describe the respective motions of the system.

  (b) Assume now that the system is free to rotate about the vertical axis without friction. Write down the Lagrangian of the system. Identify and calculate the additional conserved quantity.

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• Paper 4, Section I, D

  Classical Dynamics

  Part II, 2015

  A triatomic molecule is modelled by three masses moving in a line while connected to each other by two identical springs of force constant $k$ as shown in the figure.

  

  (a) Write down the Lagrangian and derive the equations describing the motion of the atoms.

  (b) Find the normal modes and their frequencies. What motion does the lowest frequency represent?

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• Paper 3, Section I, $7 \mathrm{D}$

  Classical Dynamics

  Part II, 2015

  (a) Consider a particle of mass $m$ that undergoes periodic motion in a one-dimensional potential $V(q)$. Write down the Hamiltonian $H(p, q)$ for the system. Explain what is meant by the angle-action variables $(\theta, I)$ of the system and write down the integral expression for the action variable $I$.

  (b) For $V(q)=\frac{1}{2} m \omega^{2} q^{2}$ and fixed total energy $E$, describe the shape of the trajectories in phase-space. By using the expression for the area enclosed by the trajectory, or otherwise, find the action variable $I$ in terms of $\omega$ and $E$. Hence describe how $E$ changes with $\omega$ if $\omega$ varies slowly with time. Justify your answer.

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• Paper 2, Section I, D

  Classical Dynamics

  Part II, 2015

  The Lagrangian for a heavy symmetric top of mass $M$, pinned at a point that is a distance $l$ from the centre of mass, is

  $$L=\frac{1}{2} I_{1}\left(\dot{\theta}^{2}+\dot{\phi}^{2} \sin ^{2} \theta\right)+\frac{1}{2} I_{3}(\dot{\psi}+\dot{\phi} \cos \theta)^{2}-M g l \cos \theta$$

  (a) Find all conserved quantities. In particular, show that $\omega_{3}$, the spin of the top, is constant.

  (b) Show that $\theta$ obeys the equation of motion

  $$I_{1} \ddot{\theta}=-\frac{d V_{\text {eff }}}{d \theta},$$

  where the explicit form of $V_{\text {eff }}$ should be determined.

  (c) Determine the condition for uniform precession with no nutation, that is $\dot{\theta}=0$ and $\dot{\phi}=$ const. For what values of $\omega_{3}$ does such uniform precession occur?

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• Paper 1, Section I, D

  Classical Dynamics

  Part II, 2015

  (a) The action for a one-dimensional dynamical system with a generalized coordinate $q$ and Lagrangian $L$ is given by

  $$S=\int_{t_{1}}^{t_{2}} L(q, \dot{q}, t) d t$$

  State the principle of least action and derive the Euler-Lagrange equation.

  (b) A planar spring-pendulum consists of a light rod of length $l$ and a bead of mass $m$, which is able to slide along the rod without friction and is attached to the ends of the rod by two identical springs of force constant $k$ as shown in the figure. The rod is pivoted at one end and is free to swing in a vertical plane under the influence of gravity.

  

  (i) Identify suitable generalized coordinates and write down the Lagrangian of the system.

  (ii) Derive the equations of motion.

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• Paper 4, Section II, C

  Classical Dynamics

  Part II, 2015

  Consider a rigid body with angular velocity $\boldsymbol{\omega}$, angular momentum $\mathbf{L}$ and position vector $\mathbf{r}$, in its body frame.

  (a) Use the expression for the kinetic energy of the body,

  $$\frac{1}{2} \int d^{3} \mathbf{r} \rho(\mathbf{r}) \dot{\mathbf{r}}^{2},$$

  to derive an expression for the tensor of inertia of the body, I. Write down the relationship between $\mathbf{L}, \mathbf{I}$ and $\boldsymbol{\omega}$.

  (b) Euler's equations of torque-free motion of a rigid body are

  $$\begin{aligned} &I_{1} \dot{\omega}_{1}=\left(I_{2}-I_{3}\right) \omega_{2} \omega_{3} \\ &I_{2} \dot{\omega}_{2}=\left(I_{3}-I_{1}\right) \omega_{3} \omega_{1} \\ &I_{3} \dot{\omega}_{3}=\left(I_{1}-I_{2}\right) \omega_{1} \omega_{2} \end{aligned}$$

  Working in the frame of the principal axes of inertia, use Euler's equations to show that the energy $E$ and the squared angular momentum $\mathbf{L}^{2}$ are conserved.

  (c) Consider a cuboid with sides $a, b$ and $c$, and with mass $M$ distributed uniformly.

  (i) Use the expression for the tensor of inertia derived in (a) to calculate the principal moments of inertia of the body.

  (ii) Assume $b=2 a$ and $c=4 a$, and suppose that the initial conditions are such that

  $$\mathbf{L}^{2}=2 I_{2} E$$

  with the initial angular velocity $\omega$ perpendicular to the intermediate principal axis $\mathbf{e}_{2}$. Derive the first order differential equation for $\omega_{2}$ in terms of $E, M$ and $a$ and hence determine the long-term behaviour of $\boldsymbol{\omega}$.

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• Paper 2, Section II, C

  Classical Dynamics

  Part II, 2015

  (a) Consider a Lagrangian dynamical system with one degree of freedom. Write down the expression for the Hamiltonian of the system in terms of the generalized velocity $\dot{q}$, momentum $p$, and the Lagrangian $L(q, \dot{q}, t)$. By considering the differential of the Hamiltonian, or otherwise, derive Hamilton's equations.

  Show that if $q$ is ignorable (cyclic) with respect to the Lagrangian, i.e. $\partial L / \partial q=0$, then it is also ignorable with respect to the Hamiltonian.

  (b) A particle of charge $q$ and mass $m$ moves in the presence of electric and magnetic fields such that the scalar and vector potentials are $\phi=y E$ and $\mathbf{A}=(0, x B, 0)$, where $(x, y, z)$ are Cartesian coordinates and $E, B$ are constants. The Lagrangian of the particle is

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

  Starting with the Lagrangian, derive an explicit expression for the Hamiltonian and use Hamilton's equations to determine the motion of the particle.

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• Paper 4, Section I, E

  Classical Dynamics

  Part II, 2016

  Using conservation of angular momentum $\mathbf{L}=L_{a} \mathbf{e}_{a}$ in the body frame, derive the Euler equations for a rigid body:

  $$I_{1} \dot{\omega}_{1}+\left(I_{3}-I_{2}\right) \omega_{2} \omega_{3}=0, \quad I_{2} \dot{\omega}_{2}+\left(I_{1}-I_{3}\right) \omega_{3} \omega_{1}=0, \quad I_{3} \dot{\omega}_{3}+\left(I_{2}-I_{1}\right) \omega_{1} \omega_{2}=0$$

  [You may use the formula $\dot{\mathbf{e}}_{a}=\boldsymbol{\omega} \wedge \mathbf{e}_{a}$ without proof.]

  Assume that the principal moments of inertia satisfy $I_{1}<I_{2}<I_{3}$. Determine whether a rotation about the principal 3-axis leads to stable or unstable perturbations.

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• Paper 1, Section I, E

  Classical Dynamics

  Part II, 2016

  Consider a one-parameter family of transformations $q_{i}(t) \mapsto Q_{i}(s, t)$ such that $Q_{i}(0, t)=q_{i}(t)$ for all time $t$, and

  $$\frac{\partial}{\partial s} L\left(Q_{i}, \dot{Q}_{i}, t\right)=0$$

  where $L$ is a Lagrangian and a dot denotes differentiation with respect to $t$. State and prove Noether's theorem.

  Consider the Lagrangian

  $$L=\frac{1}{2}\left(\dot{x}^{2}+\dot{y}^{2}+\dot{z}^{2}\right)-V(x+y, y+z),$$

  where the potential $V$ is a function of two variables. Find a continuous symmetry of this Lagrangian and construct the corresponding conserved quantity. Use the Euler-Lagrange equations to explicitly verify that the function you have constructed is independent of $t$.

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• Paper 2, Section I, E

  Classical Dynamics

  Part II, 2016

  Consider the Lagrangian

  $$L=A\left(\dot{\theta}^{2}+\dot{\phi}^{2} \sin ^{2} \theta\right)+B(\dot{\psi}+\dot{\phi} \cos \theta)^{2}-C(\cos \theta)^{k}$$

  where $A, B, C$ are positive constants and $k$ is a positive integer. Find three conserved quantities and show that $u=\cos \theta$ satisfies

  $$\dot{u}^{2}=f(u)$$

  where $f(u)$ is a polynomial of degree $k+2$ which should be determined.

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• Paper 3, Section I, E

  Classical Dynamics

  Part II, 2016

  Consider a six-dimensional phase space with coordinates $\left(q_{i}, p_{i}\right)$ for $i=1,2,3$. Compute the Poisson brackets $\left\{L_{i}, L_{j}\right\}$, where $L_{i}=\epsilon_{i j k} q_{j} p_{k}$.

  Consider the Hamiltonian

  $$H=\frac{1}{2}|\mathbf{p}|^{2}+V(|\mathbf{q}|)$$

  and show that the resulting Hamiltonian system admits three Poisson-commuting independent first integrals.

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• Paper 2, Section II, E

  Classical Dynamics

  Part II, 2016

  Define what it means for the transformation $\mathbb{R}^{2 n} \rightarrow \mathbb{R}^{2 n}$ given by

  $$\left(q_{i}, p_{i}\right) \mapsto\left(Q_{i}\left(q_{j}, p_{j}\right), P_{i}\left(q_{j}, p_{j}\right)\right), \quad i, j=1, \ldots, n$$

  to be canonical. Show that a transformation is canonical if and only if

  $$\left\{Q_{i}, Q_{j}\right\}=0, \quad\left\{P_{i}, P_{j}\right\}=0, \quad\left\{Q_{i}, P_{j}\right\}=\delta_{i j}$$

  Show that the transformation $\mathbb{R}^{2} \rightarrow \mathbb{R}^{2}$ given by

  $$Q=q \cos \epsilon-p \sin \epsilon, \quad P=q \sin \epsilon+p \cos \epsilon$$

  is canonical for any real constant $\epsilon$. Find the corresponding generating function.

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• Paper 4, Section II, $14 \mathrm{E}$

  Classical Dynamics

  Part II, 2016

  A particle of unit mass is attached to one end of a light, stiff rod of length $\ell$. The other end of the rod is held at a fixed position, such that the rod is free to swing in any direction. Write down the Lagrangian for the system giving a clear definition of any angular variables you introduce. [You should assume the acceleration $g$ is constant.]

  Find two independent constants of the motion.

  The particle is projected horizontally with speed $v$ from a point where the rod lies at an angle $\alpha$ to the downward vertical, with $0<\alpha<\pi / 2$. In terms of $\ell, g$ and $\alpha$, find the critical speed $v_{c}$ such that the particle always remains at its initial height.

  The particle is now projected horizontally with speed $v_{c}$ but from a point at angle $\alpha+\delta \alpha$ to the vertical, where $\delta \alpha / \alpha \ll 1$. Show that the height of the particle oscillates, and find the period of oscillation in terms of $\ell, g$ and $\alpha$.

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• Paper 1, Section I, E

  Classical Dynamics

  Part II, 2017

  Consider a Lagrangian system with Lagrangian $L\left(x_{A}, \dot{x}_{A}, t\right)$, where $A=1, \ldots, 3 N$, and constraints

  $$f_{\alpha}\left(x_{A}, t\right)=0, \quad \alpha=1, \ldots, 3 N-n .$$

  Use the method of Lagrange multipliers to show that this is equivalent to a system with Lagrangian $\mathcal{L}\left(q_{i}, \dot{q}_{i}, t\right) \equiv L\left(x_{A}\left(q_{i}, t\right), \dot{x}_{A}\left(q_{i}, \dot{q}_{i}, t\right), t\right)$, where $i=1, \ldots, n$, and $q_{i}$ are coordinates on the surface of constraints.

  Consider a bead of unit mass in $\mathbb{R}^{2}$ constrained to move (with no potential) on a wire given by an equation $y=f(x)$, where $(x, y)$ are Cartesian coordinates. Show that the Euler-Lagrange equations take the form

  $$\frac{d}{d t} \frac{\partial \mathcal{L}}{\partial \dot{x}}=\frac{\partial \mathcal{L}}{\partial x}$$

  for some $\mathcal{L}=\mathcal{L}(x, \dot{x})$ which should be specified. Find one first integral of the EulerLagrange equations, and thus show that

  $$t=F(x)$$

  where $F(x)$ should be given in the form of an integral.

  [Hint: You may assume that the Euler-Lagrange equations hold in all coordinate systems.]

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• Paper 2, Section I, E

  Classical Dynamics

  Part II, 2017

  Derive the Lagrange equations from the principle of stationary action

  $$S[q]=\int_{t_{0}}^{t_{1}} \mathcal{L}\left(q_{i}(t), \dot{q}_{i}(t), t\right) d t, \quad \delta S=0$$

  where the end points $q_{i}\left(t_{0}\right)$ and $q_{i}\left(t_{1}\right)$ are fixed.

  Let $\phi$ and $\mathbf{A}$ be a scalar and a vector, respectively, depending on $\mathbf{r}=(x, y, z)$. Consider the Lagrangian

  $$\mathcal{L}=\frac{m \dot{\mathbf{r}}^{2}}{2}-(\phi-\dot{\mathbf{r}} \cdot \mathbf{A})$$

  and show that the resulting Euler-Lagrange equations are invariant under the transformations

  $$\phi \rightarrow \phi+\alpha \frac{\partial F}{\partial t}, \quad \mathbf{A} \rightarrow \mathbf{A}+\nabla F,$$

  where $F=F(\mathbf{r}, t)$ is an arbitrary function, and $\alpha$ is a constant which should be determined.

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• Paper 3, Section I, E

  Classical Dynamics

  Part II, 2017

  Define an integrable system with $2 n$-dimensional phase space. Define angle-action variables.

  Consider a two-dimensional phase space with the Hamiltonian

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

  where $k$ is a positive integer and the mass $m=m(t)$ changes slowly in time. Use the fact that the action is an adiabatic invariant to show that the energy varies in time as $m^{c}$, where $c$ is a constant which should be found.

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• Paper 4, Section I, E

  Classical Dynamics

  Part II, 2017

  Consider the Poisson bracket structure on $\mathbb{R}^{3}$ given by

  $$\{x, y\}=z, \quad\{y, z\}=x, \quad\{z, x\}=y$$

  and show that $\left\{f, \rho^{2}\right\}=0$, where $\rho^{2}=x^{2}+y^{2}+z^{2}$ and $f: \mathbb{R}^{3} \rightarrow \mathbb{R}$ is any polynomial function on $\mathbb{R}^{3}$.

  Let $H=\left(A x^{2}+B y^{2}+C z^{2}\right) / 2$, where $A, B, C$ are positive constants. Find the explicit form of Hamilton's equations

  $$\dot{\mathbf{r}}=\{\mathbf{r}, H\}, \quad \text { where } \quad \mathbf{r}=(x, y, z)$$

  Find a condition on $A, B, C$ such that the oscillation described by

  $$x=1+\alpha(t), \quad y=\beta(t), \quad z=\gamma(t)$$

  is linearly unstable, where $\alpha(t), \beta(t), \gamma(t)$ are small.

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• Paper 2, Section II, E

  Classical Dynamics

  Part II, 2017

  Show that an object's inertia tensor about a point displaced from the centre of mass by a vector $\mathbf{c}$ is given by

  $$\left(I_{\mathbf{c}}\right)_{a b}=\left(I_{0}\right)_{a b}+M\left(|\mathbf{c}|^{2} \delta_{a b}-c_{a} c_{b}\right),$$

  where $M$ is the total mass of the object, and $\left(I_{0}\right)_{a b}$ is the inertia tensor about the centre of mass.

  Find the inertia tensor of a cube of uniform density, with edge of length $L$, about one of its vertices.

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• Paper 4, Section II, $14 \mathrm{E}$

  Classical Dynamics

  Part II, 2017

  Explain how geodesics of a Riemannian metric

  $$g=g_{a b}\left(x^{c}\right) d x^{a} d x^{b}$$

  arise from the kinetic Lagrangian

  $$\mathcal{L}=\frac{1}{2} g_{a b}\left(x^{c}\right) \dot{x}^{a} \dot{x}^{b}$$

  where $a, b=1, \ldots, n$.

  Find geodesics of the metric on the upper half plane

  $$\Sigma=\left\{(x, y) \in \mathbb{R}^{2}, y>0\right\}$$

  with the metric

  $$g=\frac{d x^{2}+d y^{2}}{y^{2}}$$

  and sketch the geodesic containing the points $(2,3)$ and $(10,3)$.

  [Hint: Consider $d y / d x .]$

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• Paper 1, Section I, B

  Classical Dynamics

  Part II, 2018

  Derive Hamilton's equations from an action principle.

  Consider a two-dimensional phase space with the Hamiltonian $H=p^{2}+q^{-2}$. Show that $F=p q-c t H$ is the first integral for some constant $c$ which should be determined. By considering the surfaces of constant $F$ in the extended phase space, solve Hamilton's equations, and sketch the orbits in the phase space.

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• Paper 2, Section I, B

  Classical Dynamics

  Part II, 2018

  Let $\mathbf{x}=x \mathbf{i}+y \mathbf{j}+z \mathbf{k}$. Consider a Lagrangian

  $$\mathcal{L}=\frac{1}{2} \dot{\mathbf{x}}^{2}+y \dot{x}$$

  of a particle constrained to move on a sphere $|\mathbf{x}|=1 / c$ of radius $1 / c$. Use Lagrange multipliers to show that

  $$\ddot{\mathbf{x}}+\ddot{y} \mathbf{i}-\dot{x} \mathbf{j}+c^{2}\left(|\dot{\mathbf{x}}|^{2}+y \dot{x}-x \dot{y}\right) \mathbf{x}=0$$

  Now, consider the system $(*)$ with $c=0$, and find the particle trajectories.

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• Paper 3, Section I, B

  Classical Dynamics

  Part II, 2018

  Three particles of unit mass move along a line in a potential

  $$V=\frac{1}{2}\left(\left(x_{1}-x_{2}\right)^{2}+\left(x_{1}-x_{3}\right)^{2}+\left(x_{3}-x_{2}\right)^{2}+x_{1}^{2}+x_{2}^{2}+x_{3}^{2}\right)$$

  where $x_{i}$ is the coordinate of the $i$ th particle, $i=1,2,3$.

  Write the Lagrangian in the form

  $$\mathcal{L}=\frac{1}{2} T_{i j} \dot{x}_{i} \dot{x}_{j}-\frac{1}{2} V_{i j} x_{i} x_{j}$$

  and specify the matrices $T_{i j}$ and $V_{i j}$.

  Find the normal frequencies and normal modes for this system.

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• Paper 4, Section I, B

  Classical Dynamics

  Part II, 2018

  State and prove Noether's theorem in Lagrangian mechanics.

  Consider a Lagrangian

  $$\mathcal{L}=\frac{1}{2} \frac{\dot{x}^{2}+\dot{y}^{2}}{y^{2}}-V\left(\frac{x}{y}\right)$$

  for a particle moving in the upper half-plane $\left\{(x, y) \in \mathbb{R}^{2}, y>0\right\}$ in a potential $V$ which only depends on $x / y$. Find two independent first integrals.

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• Paper 2, Section II, B

  Classical Dynamics

  Part II, 2018

  Define a body frame $\mathbf{e}_{a}(t), a=1,2,3$ of a rotating rigid body, and show that there exists a vector $\boldsymbol{\omega}=\left(\omega_{1}, \omega_{2}, \omega_{3}\right)$ such that

  $$\dot{\mathbf{e}}_{a}=\boldsymbol{\omega} \times \mathbf{e}_{a}$$

  Let $\mathbf{L}=I_{1} \omega_{1}(t) \mathbf{e}_{1}+I_{2} \omega_{2}(t) \mathbf{e}_{2}+I_{3} \omega_{3}(t) \mathbf{e}_{3}$ be the angular momentum of a free rigid body expressed in the body frame. Derive the Euler equations from the conservation of angular momentum.

  Verify that the kinetic energy $E$, and the total angular momentum $L^{2}$ are conserved. Hence show that

  $$\dot{\omega}_{3}^{2}=f\left(\omega_{3}\right),$$

  where $f\left(\omega_{3}\right)$ is a quartic polynomial which should be explicitly determined in terms of $L^{2}$ and $E$.

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• Paper 4, Section II, B

  Classical Dynamics

  Part II, 2018

  Given a Lagrangian $\mathcal{L}\left(q_{i}, \dot{q}_{i}, t\right)$ with degrees of freedom $q_{i}$, define the Hamiltonian and show how Hamilton's equations arise from the Lagrange equations and the Legendre transform.

  Consider the Lagrangian for a symmetric top moving in constant gravity:

  $$\mathcal{L}=\frac{1}{2} A\left(\dot{\theta}^{2}+\dot{\phi}^{2} \sin ^{2} \theta\right)+\frac{1}{2} B(\dot{\psi}+\dot{\phi} \cos \theta)^{2}-M g l \cos \theta$$

  where $A, B, M, g$ and $l$ are constants. Construct the corresponding Hamiltonian, and find three independent Poisson-commuting first integrals of Hamilton's equations.

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• Paper 4, Section I, E

  Classical Dynamics

  Part II, 2019

  (a) The angular momentum of a rigid body about its centre of mass is conserved.

  Derive Euler's equations,

  $$\begin{aligned} &I_{1} \dot{\omega}_{1}=\left(I_{2}-I_{3}\right) \omega_{2} \omega_{3} \\ &I_{2} \dot{\omega}_{2}=\left(I_{3}-I_{1}\right) \omega_{3} \omega_{1} \\ &I_{3} \dot{\omega}_{3}=\left(I_{1}-I_{2}\right) \omega_{1} \omega_{2} \end{aligned}$$

  explaining the meaning of the quantities appearing in the equations.

  (b) Show that there are two independent conserved quantities that are quadratic functions of $\boldsymbol{\omega}=\left(\omega_{1}, \omega_{2}, \omega_{3}\right)$, and give a physical interpretation of them.

  (c) Derive a linear approximation to Euler's equations that applies when $\left|\omega_{1}\right| \ll\left|\omega_{3}\right|$ and $\left|\omega_{2}\right| \ll\left|\omega_{3}\right|$. Use this to determine the stability of rotation about each of the three principal axes of an asymmetric top.

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• Paper 3, Section I, E

  Classical Dynamics

  Part II, 2019

  A simple harmonic oscillator of mass $m$ and spring constant $k$ has the equation of motion

  $$m \ddot{x}=-k x .$$

  (a) Describe the orbits of the system in phase space. State how the action $I$ of the oscillator is related to a geometrical property of the orbits in phase space. Derive the action-angle variables $(\theta, I)$ and give the form of the Hamiltonian of the oscillator in action-angle variables.

  (b) Suppose now that the spring constant $k$ varies in time. Under what conditions does the theory of adiabatic invariance apply? Assuming that these conditions hold, identify an adiabatic invariant and determine how the energy and amplitude of the oscillator vary with $k$ in this approximation.

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• Paper 2, Section I, E

  Classical Dynamics

  Part II, 2019

  (a) State Hamilton's equations for a system with $n$ degrees of freedom and Hamilto$\operatorname{nian} H(\mathbf{q}, \mathbf{p}, t)$, where $(\mathbf{q}, \mathbf{p})=\left(q_{1}, \ldots, q_{n}, p_{1}, \ldots, p_{n}\right)$ are canonical phase-space variables.

  (b) Define the Poisson bracket $\{f, g\}$ of two functions $f(\mathbf{q}, \mathbf{p}, t)$ and $g(\mathbf{q}, \mathbf{p}, t)$.

  (c) State the canonical commutation relations of the variables $\mathbf{q}$ and $\mathbf{p}$.

  (d) Show that the time-evolution of any function $f(\mathbf{q}, \mathbf{p}, t)$ is given by

  $$\frac{d f}{d t}=\{f, H\}+\frac{\partial f}{\partial t}$$

  (e) Show further that the Poisson bracket of any two conserved quantities is also a conserved quantity.

  [You may assume the Jacobi identity,

  $$\{f,\{g, h\}\}+\{g,\{h, f\}\}+\{h,\{f, g\}\}=0 .]$$

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• Paper 1, Section I, E

  Classical Dynamics

  Part II, 2019

  (a) A mechanical system with $n$ degrees of freedom has the Lagrangian $L(\mathbf{q}, \dot{\mathbf{q}})$, where $\mathbf{q}=\left(q_{1}, \ldots, q_{n}\right)$ are the generalized coordinates and $\dot{\mathbf{q}}=d \mathbf{q} / d t$.

  Suppose that $L$ is invariant under the continuous symmetry transformation $\mathbf{q}(t) \mapsto$ $\mathbf{Q}(s, t)$, where $s$ is a real parameter and $\mathbf{Q}(0, t)=\mathbf{q}(t)$. State and prove Noether's theorem for this system.

  (b) A particle of mass $m$ moves in a conservative force field with potential energy $V(\mathbf{r})$, where $\mathbf{r}$ is the position vector in three-dimensional space.

  Let $(r, \phi, z)$ be cylindrical polar coordinates. $V(\mathbf{r})$ is said to have helical symmetry if it is of the form

  $$V(\mathbf{r})=f(r, \phi-k z),$$

  for some constant $k$. Show that a particle moving in a potential with helical symmetry has a conserved quantity that is a linear combination of angular and linear momenta.

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• Paper 2, Section II, E

  Classical Dynamics

  Part II, 2019

  The Lagrangian of a particle of mass $m$ and charge $q$ moving in an electromagnetic field described by scalar and vector potentials $\phi(\mathbf{r}, t)$ and $\mathbf{A}(\mathbf{r}, t)$ is

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

  where $\mathbf{r}(t)$ is the position vector of the particle and $\dot{\mathbf{r}}=d \mathbf{r} / d t$.

  (a) Show that Lagrange's equations are equivalent to the equation of motion

  $$m \ddot{\mathbf{r}}=q(\mathbf{E}+\mathbf{v} \times \mathbf{B}),$$

  where

  $$\mathbf{E}=-\nabla \phi-\frac{\partial \mathbf{A}}{\partial t}, \quad \mathbf{B}=\nabla \times \mathbf{A}$$

  are the electric and magnetic fields.

  (b) Show that the related Hamiltonian is

  $$H=\frac{|\mathbf{p}-q \mathbf{A}|^{2}}{2 m}+q \phi,$$

  where $\mathbf{p}=m \dot{\mathbf{r}}+q \mathbf{A}$. Obtain Hamilton's equations for this system.

  (c) Verify that the electric and magnetic fields remain unchanged if the scalar and vector potentials are transformed according to

  where $f(\mathbf{r}, t)$ is a scalar field. Show that the transformed Lagrangian $\tilde{L}$ differs from $L$ by the total time-derivative of a certain quantity. Why does this leave the form of Lagrange's equations invariant? Show that the transformed Hamiltonian $\tilde{H}$ and phase-space variables $(\mathbf{r}, \tilde{\mathbf{p}})$ are related to $H$ and $(\mathbf{r}, \mathbf{p})$ by a canonical transformation.

  [Hint: In standard notation, the canonical transformation associated with the type-2 generating function $F_{2}(\mathbf{q}, \mathbf{P}, t)$ is given by

  $$\left.\mathbf{p}=\frac{\partial F_{2}}{\partial \mathbf{q}}, \quad \mathbf{Q}=\frac{\partial F_{2}}{\partial \mathbf{P}}, \quad K=H+\frac{\partial F_{2}}{\partial t} .\right]$$

  $$\begin{aligned} & \phi \mapsto \tilde{\phi}=\phi-\frac{\partial f}{\partial t}, \\ & \mathbf{A} \mapsto \tilde{\mathbf{A}}=\mathbf{A}+\nabla f, \end{aligned}$$

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• Paper 4, Section II, E

  Classical Dynamics

  Part II, 2019

  (a) Explain what is meant by a Lagrange top. You may assume that such a top has the Lagrangian

  $$L=\frac{1}{2} I_{1}\left(\dot{\theta}^{2}+\dot{\phi}^{2} \sin ^{2} \theta\right)+\frac{1}{2} I_{3}(\dot{\psi}+\dot{\phi} \cos \theta)^{2}-M g l \cos \theta$$

  in terms of the Euler angles $(\theta, \phi, \psi)$. State the meaning of the quantities $I_{1}, I_{3}, M$ and $l$ appearing in this expression.

  Explain why the quantity

  $$p_{\psi}=\frac{\partial L}{\partial \dot{\psi}}$$

  is conserved, and give two other independent integrals of motion.

  Show that steady precession, with a constant value of $\theta \in\left(0, \frac{\pi}{2}\right)$, is possible if

  $$p_{\psi}^{2} \geqslant 4 M g l I_{1} \cos \theta .$$

  (b) A rigid body of mass $M$ is of uniform density and its surface is defined by

  $$x_{1}^{2}+x_{2}^{2}=x_{3}^{2}-\frac{x_{3}^{3}}{h}$$

  where $h$ is a positive constant and $\left(x_{1}, x_{2}, x_{3}\right)$ are Cartesian coordinates in the body frame.

  Calculate the values of $I_{1}, I_{3}$ and $l$ for this symmetric top, when it rotates about the sharp point at the origin of this coordinate system.

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• Paper 1, Section I, B

  Classical Dynamics

  Part II, 2020

  A linear molecule is modelled as four equal masses connected by three equal springs. Using the Cartesian coordinates $x_{1}, x_{2}, x_{3}, x_{4}$ of the centres of the four masses, and neglecting any forces other than those due to the springs, write down the Lagrangian of the system describing longitudinal motions of the molecule.

  Rewrite and simplify the Lagrangian in terms of the generalized coordinates

  $$q_{1}=\frac{x_{1}+x_{4}}{2}, \quad q_{2}=\frac{x_{2}+x_{3}}{2}, \quad q_{3}=\frac{x_{1}-x_{4}}{2}, \quad q_{4}=\frac{x_{2}-x_{3}}{2}$$

  Deduce Lagrange's equations for $q_{1}, q_{2}, q_{3}, q_{4}$. Hence find the normal modes of the system and their angular frequencies, treating separately the symmetric and antisymmetric modes of oscillation.

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• Paper 2, Section I, B

  Classical Dynamics

  Part II, 2020

  A particle of mass $m$ has position vector $\mathbf{r}(t)$ in a frame of reference that rotates with angular velocity $\boldsymbol{\omega}(t)$. The particle moves under the gravitational influence of masses that are fixed in the rotating frame. Explain why the Lagrangian of the particle is of the form

  $$L=\frac{1}{2} m(\dot{\mathbf{r}}+\boldsymbol{\omega} \times \mathbf{r})^{2}-V(\mathbf{r}) .$$

  Show that Lagrange's equations of motion are equivalent to

  $$m(\ddot{\mathbf{r}}+2 \boldsymbol{\omega} \times \dot{\mathbf{r}}+\dot{\boldsymbol{\omega}} \times \mathbf{r}+\boldsymbol{\omega} \times(\boldsymbol{\omega} \times \mathbf{r}))=-\boldsymbol{\nabla} V$$

  Identify the canonical momentum $\mathbf{p}$ conjugate to $\mathbf{r}$. Obtain the Hamiltonian $H(\mathbf{r}, \mathbf{p})$ and Hamilton's equations for this system.

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• Paper 3, Section I, B

  Classical Dynamics

  Part II, 2020

  A particle of mass $m$ experiences a repulsive central force of magnitude $k / r^{2}$, where $r=|\mathbf{r}|$ is its distance from the origin. Write down the Hamiltonian of the system.

  The Laplace-Runge-Lenz vector for this system is defined by

  $$\mathbf{A}=\mathbf{p} \times \mathbf{L}+m k \hat{\mathbf{r}}$$

  where $\mathbf{L}=\mathbf{r} \times \mathbf{p}$ is the angular momentum and $\hat{\mathbf{r}}=\mathbf{r} / r$ is the radial unit vector. Show that

  $$\{\mathbf{L}, H\}=\{\mathbf{A}, H\}=\mathbf{0},$$

  where $\{\cdot, \cdot\}$ is the Poisson bracket. What are the integrals of motion of the system? Show that the polar equation of the orbit can be written as

  $$r=\frac{\lambda}{e \cos \theta-1},$$

  where $\lambda$ and $e$ are non-negative constants.

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• Paper 4, Section I, B

  Classical Dynamics

  Part II, 2020

  Derive expressions for the angular momentum and kinetic energy of a rigid body in terms of its mass $M$, the position $\mathbf{X}(t)$ of its centre of mass, its inertia tensor $I$ (which should be defined) about its centre of mass, and its angular velocity $\boldsymbol{\omega}$.

  A spherical planet of mass $M$ and radius $R$ has density proportional to $r^{-1} \sin (\pi r / R)$. Given that $\int_{0}^{\pi} x \sin x d x=\pi$ and $\int_{0}^{\pi} x^{3} \sin x d x=\pi\left(\pi^{2}-6\right)$, evaluate the inertia tensor of the planet in terms of $M$ and $R$.

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• Paper 2, Section II, B

  Classical Dynamics

  Part II, 2020

  A symmetric top of mass $M$ rotates about a fixed point that is a distance $l$ from the centre of mass along the axis of symmetry; its principal moments of inertia about the fixed point are $I_{1}=I_{2}$ and $I_{3}$. The Lagrangian of the top is

  $$L=\frac{1}{2} I_{1}\left(\dot{\theta}^{2}+\dot{\phi}^{2} \sin ^{2} \theta\right)+\frac{1}{2} I_{3}(\dot{\psi}+\dot{\phi} \cos \theta)^{2}-M g l \cos \theta$$

  (i) Draw a diagram explaining the meaning of the Euler angles $\theta, \phi$ and $\psi$.

  (ii) Derive expressions for the three integrals of motion $E, L_{3}$ and $L_{z}$.

  (iii) Show that the nutational motion is governed by the equation

  $$\frac{1}{2} I_{1} \dot{\theta}^{2}+V_{\text {eff }}(\theta)=E^{\prime}$$

  and derive expressions for the effective potential $V_{\mathrm{eff}}(\theta)$ and the modified energy $E^{\prime}$ in terms of $E, L_{3}$ and $L_{z}$.

  (iv) Suppose that

  $$L_{z}=L_{3}\left(1-\frac{\epsilon^{2}}{2}\right)$$

  where $\epsilon$ is a small positive number. By expanding $V_{\text {eff }}$ to second order in $\epsilon$ and $\theta$, show that there is a stable equilibrium solution with $\theta=O(\epsilon)$, provided that $L_{3}^{2}>4 M g l I_{1}$. Determine the equilibrium value of $\theta$ and the precession rate $\dot{\phi}$, to the same level of approximation.

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• Paper 4, Section II, B

  Classical Dynamics

  Part II, 2020

  (a) Explain how the Hamiltonian $H(\mathbf{q}, \mathbf{p}, t)$ of a system can be obtained from its Lagrangian $L(\mathbf{q}, \dot{\mathbf{q}}, t)$. Deduce that the action can be written as

  $$S=\int(\mathbf{p} \cdot d \mathbf{q}-H d t)$$

  Show that Hamilton's equations are obtained if the action, computed between fixed initial and final configurations $\mathbf{q}\left(t_{1}\right)$ and $\mathbf{q}\left(t_{2}\right)$, is minimized with respect to independent variations of $\mathbf{q}$ and $\mathbf{p}$.

  (b) Let $(\mathbf{Q}, \mathbf{P})$ be a new set of coordinates on the same phase space. If the old and new coordinates are related by a type-2 generating function $F_{2}(\mathbf{q}, \mathbf{P}, t)$ such that

  $$\mathbf{p}=\frac{\partial F_{2}}{\partial \mathbf{q}}, \quad \mathbf{Q}=\frac{\partial F_{2}}{\partial \mathbf{P}}$$

  deduce that the canonical form of Hamilton's equations applies in the new coordinates, but with a new Hamiltonian given by

  $$K=H+\frac{\partial F_{2}}{\partial t}$$

  (c) For each of the Hamiltonians (i) $H=H(p)$, (ii) $H=\frac{1}{2}\left(q^{2}+p^{2}\right)$,

  express the general solution $(q(t), p(t))$ at time $t$ in terms of the initial values given by $(Q, P)=(q(0), p(0))$ at time $t=0$. In each case, show that the transformation from $(q, p)$ to $(Q, P)$ is canonical for all values of $t$, and find the corresponding generating function $F_{2}(q, P, t)$ explicitly.

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• Paper 1, Section I, D

  Classical Dynamics

  Part II, 2021

  Two equal masses $m$ move along a straight line between two stationary walls. The mass on the left is connected to the wall on its left by a spring of spring constant $k_{1}$, and the mass on the right is connected to the wall on its right by a spring of spring constant $k_{2}$. The two masses are connected by a third spring of spring constant $k_{3}$.

  (a) Show that the Lagrangian of the system can be written in the form

  $$L=\frac{1}{2} T_{i j} \dot{x}_{i} \dot{x}_{j}-\frac{1}{2} V_{i j} x_{i} x_{j}$$

  where $x_{i}(t)$, for $i=1,2$, are the displacements of the two masses from their equilibrium positions, and $T_{i j}$ and $V_{i j}$ are symmetric $2 \times 2$ matrices that should be determined.

  (b) Let

  $$k_{1}=k(1+\epsilon \delta), \quad k_{2}=k(1-\epsilon \delta), \quad k_{3}=k \epsilon,$$

  where $k>0, \epsilon>0$ and $|\epsilon \delta|<1$. Using Lagrange's equations of motion, show that the angular frequencies $\omega$ of the normal modes of the system are given by

  $$\omega^{2}=\lambda \frac{k}{m}$$

  where

  $$\lambda=1+\epsilon\left(1 \pm \sqrt{1+\delta^{2}}\right)$$

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• Paper 2, Section I, D

  Classical Dynamics

  Part II, 2021

  Show that, in a uniform gravitational field, the net gravitational torque on a system of particles, about its centre of mass, is zero.

  Let $S$ be an inertial frame of reference, and let $S^{\prime}$ be the frame of reference with the same origin and rotating with angular velocity $\boldsymbol{\omega}(t)$ with respect to $S$. You may assume that the rates of change of a vector $v$ observed in the two frames are related by

  $$\left(\frac{d \mathbf{v}}{d t}\right)_{S}=\left(\frac{d \mathbf{v}}{d t}\right)_{S^{\prime}}+\omega \times \mathbf{v} .$$

  Derive Euler's equations for the torque-free motion of a rigid body.

  Show that the general torque-free motion of a symmetric top involves precession of the angular-velocity vector about the symmetry axis of the body. Determine how the direction and rate of precession depend on the moments of inertia of the body and its angular velocity.

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• Paper 3 , Section I, D

  Classical Dynamics

  Part II, 2021

  The Lagrangian of a particle of mass $m$ and charge $q$ in an electromagnetic field takes the form

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

  Explain the meaning of $\phi$ and $\mathbf{A}$, and how they are related to the electric and magnetic fields.

  Obtain the canonical momentum $\mathbf{p}$ and the Hamiltonian $H(\mathbf{r}, \mathbf{p}, t)$.

  Suppose that the electric and magnetic fields have Cartesian components $(E, 0,0)$ and $(0,0, B)$, respectively, where $E$ and $B$ are positive constants. Explain why the Hamiltonian of the particle can be taken to be

  $$H=\frac{p_{x}^{2}}{2 m}+\frac{\left(p_{y}-q B x\right)^{2}}{2 m}+\frac{p_{z}^{2}}{2 m}-q E x$$

  State three independent integrals of motion in this case.

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• Paper 4, Section I, D

  Classical Dynamics

  Part II, 2021

  Briefly describe a physical object (a Lagrange top) whose Lagrangian is

  $$L=\frac{1}{2} I_{1}\left(\dot{\theta}^{2}+\dot{\phi}^{2} \sin ^{2} \theta\right)+\frac{1}{2} I_{3}(\dot{\psi}+\dot{\phi} \cos \theta)^{2}-M g l \cos \theta$$

  Explain the meaning of the symbols in this equation.

  Write down three independent integrals of motion for this system, and show that the nutation of the top is governed by the equation

  $$\dot{u}^{2}=f(u),$$

  where $u=\cos \theta$ and $f(u)$ is a certain cubic function that you need not determine.

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• Paper 2, Section II, D

  Classical Dynamics

  Part II, 2021

  (a) Show that the Hamiltonian

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

  where $\omega$ is a positive constant, describes a simple harmonic oscillator with angular frequency $\omega$. Show that the energy $E$ and the action $I$ of the oscillator are related by $E=\omega I$.

  (b) Let $0<\epsilon<2$ be a constant. Verify that the differential equation

  $$\ddot{x}+\frac{x}{(\epsilon t)^{2}}=0 \quad \text { subject to } \quad x(1)=0, \quad \dot{x}(1)=1$$

  is solved by

  $$x(t)=\frac{\sqrt{t}}{k} \sin (k \log t)$$

  when $t>1$, where $k$ is a constant you should determine in terms of $\epsilon$.

  (c) Show that the solution in part (b) obeys

  $$\frac{1}{2} \dot{x}^{2}+\frac{1}{2} \frac{x^{2}}{(\epsilon t)^{2}}=\frac{1-\cos (2 k \log t)+2 k \sin (2 k \log t)+4 k^{2}}{8 k^{2} t}$$

  Hence show that the fractional variation of the action in the limit $\epsilon \ll 1$ is $O(\epsilon)$, but that these variations do not accumulate. Comment on this behaviour in relation to the theory of adiabatic invariance.

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• Paper 4, Section II, 15D

  Classical Dynamics

  Part II, 2021

  (a) Let $(\mathbf{q}, \mathbf{p})$ be a set of canonical phase-space variables for a Hamiltonian system with $n$ degrees of freedom. Define the Poisson bracket $\{f, g\}$ of two functions $f(\mathbf{q}, \mathbf{p})$ and $g(\mathbf{q}, \mathbf{p})$. Write down the canonical commutation relations that imply that a second set $(\mathbf{Q}, \mathbf{P})$ of phase-space variables is also canonical.

  (b) Consider the near-identity transformation

  $$\mathbf{Q}=\mathbf{q}+\delta \mathbf{q}, \quad \mathbf{P}=\mathbf{p}+\delta \mathbf{p}$$

  where $\delta \mathbf{q}(\mathbf{q}, \mathbf{p})$ and $\delta \mathbf{p}(\mathbf{q}, \mathbf{p})$ are small. Determine the approximate forms of the canonical commutation relations, accurate to first order in $\delta \mathbf{q}$ and $\delta \mathbf{p}$. Show that these are satisfied when

  $$\delta \mathbf{q}=\epsilon \frac{\partial F}{\partial \mathbf{p}}, \quad \delta \mathbf{p}=-\epsilon \frac{\partial F}{\partial \mathbf{q}}$$

  where $\epsilon$ is a small parameter and $F(\mathbf{q}, \mathbf{p})$ is some function of the phase-space variables.

  (c) In the limit $\epsilon \rightarrow 0$ this near-identity transformation is called the infinitesimal canonical transformation generated by $F$. Let $H(\mathbf{q}, \mathbf{p})$ be an autonomous Hamiltonian. Show that the change in the Hamiltonian induced by the infinitesimal canonical transformation is

  $$\delta H=-\epsilon\{F, H\} .$$

  Explain why $F$ is an integral of motion if and only if the Hamiltonian is invariant under the infinitesimal canonical transformation generated by $F$.

  (d) The Hamiltonian of the gravitational $N$-body problem in three-dimensional space is

  $$H=\frac{1}{2} \sum_{i=1}^{N} \frac{\left|\mathbf{p}_{i}\right|^{2}}{2 m_{i}}-\sum_{i=1}^{N-1} \sum_{j=i+1}^{N} \frac{G m_{i} m_{j}}{\left|\mathbf{r}_{i}-\mathbf{r}_{j}\right|}$$

  where $m_{i}, \mathbf{r}_{i}$ and $\mathbf{p}_{i}$ are the mass, position and momentum of body $i$. Determine the form of $F$ and the infinitesimal canonical transformation that correspond to the translational symmetry of the system.

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