Classical Dynamics

2021

• 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 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 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 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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2020

• 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 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 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 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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2019

• 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 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 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 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 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 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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2018

• 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 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 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 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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2017

• 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 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 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 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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2016

• 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 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 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 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 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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2015

• 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 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 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 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 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 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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2014

• 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 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 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 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 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 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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2013

• 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.

  comment

• 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 $\omega_{3}$.

  comment

• 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 $f(q, t)$