Exercise Sheet 0

Take $N$ unit masses on a line, joined by unit springs, with both ends fixed: with $x_n$ the displacement of mass $n$ and $x_0=x_{N+1}=0$, $$H=\frac12\sum_{n=1}^{N}p_n^{2}+\frac12\sum_{n=0}^{N}(x_{n+1}-x_n)^{2}.$$ (i) Write Hamilton’s equations, and show that $x_n(t)=\sin\big(nk\pi/(N+1)\big)\cos\omega t$, for $k\in\{1,\dots,N\}$, solves them exactly when $\omega=\omega_k=2\sin\big(k\pi/(2(N+1))\big)$, the frequencies of the lecture. (ii) Show that the vectors $s_k=\big(\sin(nk\pi/(N+1))\big)_{n=1}^{N}$ are orthogonal, with $|s_k|^{2}=(N+1)/2$. With $Q_k=\sqrt{2/(N+1)}\sum_n\sin(nk\pi/(N+1))\,x_n$ and $P_k$ the same combination of the $p_n$, show that $H=\sum_k\tfrac12(P_k^{2}+\omega_k^{2}Q_k^{2})$ and that each mode energy $E_k=\tfrac12(P_k^{2}+\omega_k^{2}Q_k^{2})$ is conserved: $N$ conserved quantities. (iii) For $N=2$, write $E_1$ and $E_2$ in terms of $x_1,x_2,p_1,p_2$ and check directly from Hamilton’s equations that each is conserved. (iv) For $N=32$, the chain of the lecture’s figure, compute $\omega_1$, the period of mode $1$, and how many of its periods fit into $t=10^{4}$. Compute $\omega_2/\omega_1$, $\omega_3/\omega_1$ and $\omega_4/\omega_1$.

Two particles of masses $m_1,m_2>0$ collide elastically on a line, particle $1$ on the left, with velocities $v_1>v_2$ before and $w_1,w_2$ after. (i) Solve the conservation of momentum and of energy for $(w_1,w_2)$. Show that the solutions are $(w_1,w_2)=(v_1,v_2)$ and $$w_1=\frac{(m_1-m_2)v_1+2m_2v_2}{m_1+m_2},\qquad w_2=\frac{(m_2-m_1)v_2+2m_1v_1}{m_1+m_2}.$$ From now on suppose the particles cannot pass through each other, so that particle $1$ stays on the left. Show that this rules out the first solution and leaves the second. (ii) For the second solution, show that the centre-of-mass velocity is unchanged and the relative velocity reverses: $w_2-w_1=-(v_2-v_1)$. (iii) For the second solution, show that $\{w_1,w_2\}=\{v_1,v_2\}$ as sets only if $m_1=m_2$. (iv) A particle of mass $1$ moving at velocity $1$ hits a particle of mass $3$ at rest. Find the velocities after the collision, and compare with two equal masses. What does this say about a one-dimensional gas of two species?

(i) Two particles of masses $m_1,m_2$ collide elastically in $d$ dimensions. Count the unknown outgoing velocity components and the conservation laws. With $M=m_1+m_2$, the centre-of-mass velocity $V$ and the relative velocity $u=v_1-v_2$, show that the kinetic energy is $\tfrac12M|V|^{2}+\tfrac12\mu|u|^{2}$ with $\mu=m_1m_2/M$, and deduce that the possible outcomes are exactly the relative velocities $u'$ with $|u'|=|u|$. What is this set for $d=1$, $d=2$ and $d=3$ when the particles have a relative velocity, $u\ne0$, and what is it when $u=0$? (ii) Now $N$ unit masses on a line meet at once. Show that fixed total momentum $P$ and energy $E$ confine the outgoing momenta to the set with $\sum_i(p_i-P/N)^{2}=2E-P^{2}/N$ in the plane $\sum_ip_i=P$, and that this contains every rearrangement of the incoming momenta. Show that it is a sphere of dimension $N-2$ and radius $\sqrt{2E-P^{2}/N}>0$ unless all incoming momenta are equal, when it is a single point. What is it for $N=2$ and $N=3$? (iii) The lecture’s figure of three colliding particles has incoming momenta $(1.6,\,0.3,\,-1.2)$, and prints the generic outcome as $(-0.94,\,-0.14,\,1.78)$. Compute $P$, $2E=\sum p_i^{2}$ and the radius of the circle. Check that the printed outcome has the same $P$ and, to the rounding of its last digit, the same $\sum p_i^{2}$, and that it is not a rearrangement of the incoming momenta.

The lecture’s tutorial matches the Toda spring $V_{\mathrm T}(r)=\frac ab\,e^{-br}+ar-\frac ab$ to the FPUT spring $\kappa\big(\tfrac12r^{2}+\tfrac\alpha3r^{3}\big)$ through cubic order, with $b=-2\alpha$ and $a=-\kappa/(2\alpha)$, for $\alpha\ne0$ and $\kappa>0$. (i) Expand $V_{\mathrm T}$ through order $r^{4}$. (ii) Show that with the matched constants $V_{\mathrm T}(r)-\kappa\big(\tfrac12r^{2}+\tfrac\alpha3r^{3}\big) =\tfrac16\kappa\alpha^{2}r^{4}+O(r^{5})$. (iii) Show that the matched Toda spring satisfies $V_{\mathrm T}(r)\ge0$ for every $r$, with equality only at $r=0$. (iv) Take $\kappa=1$. Show that the FPUT spring has, besides its minimum at $r=0$, a local maximum at $r=-1/\alpha$, of height $1/(6\alpha^{2})$, and that it decreases without bound beyond it. For the lecture’s $\alpha=\tfrac14$, where is the barrier and how high is it? What happens to a single FPUT spring given more energy than that?

Let $H$ be a Hermitian $n\times n$ matrix with distinct eigenvalues $\lambda_1,\dots,\lambda_n$, and $P_k$ the orthogonal projector onto the eigenline of $\lambda_k$. (i) Show that $HP_k=P_kH$, that $P_jP_k=\delta_{jk}P_k$, and that $P_1,\dots,P_n$ are linearly independent: every such $H$ has $n$ commuting, linearly independent conserved quantities. (ii) Show that $P_1,\dots,P_n$ span the same space of matrices as $1,H,\dots,H^{n-1}$. (iii) For $H=\begin{pmatrix}a&b\\b&-a\end{pmatrix}$ with $a,b$ real and $b\ne0$, find $P_\pm$ explicitly. (iv) What do (i) and (ii) say about defining a quantum integrable system as one with as many commuting conserved quantities as the dimension, or as degrees of freedom?

The Toda chain keeps the eigenvalues of a matrix. The open Toda chain of $N$ unit masses has $$H=\frac12\sum_{n=1}^{N}p_n^{2}+\sum_{n=1}^{N-1}e^{q_n-q_{n+1}},$$ so $\dot q_n=p_n$ and $\dot p_n=e^{q_{n-1}-q_n}-e^{q_n-q_{n+1}}$, the terms with $q_0$ and $q_{N+1}$ absent. Its Lax matrix, which Lecture 8 builds, is the symmetric tridiagonal matrix $L$ with $p_1,\dots,p_N$ on the diagonal and $a_n=e^{(q_n-q_{n+1})/2}$ beside it. (i) For $N=2$, show directly from Hamilton’s equations that $\tr L$ and $\det L$ are conserved, and conclude that the eigenvalues of $L$ are. (ii) Two Toda particles that collide separate again: $q_2-q_1\to+\infty$ as $t\to\pm\infty$, and their momenta tend to limits (take this as given: Lecture 12 proves it for two particles by solving the motion, and cites it for $N$). Deduce that the outgoing momenta are the incoming ones, exchanged. (iii) For $N=3$, show that $\tr L^{3}=\sum_np_n^{3}+3\sum_{n=1}^{2}e^{q_n-q_{n+1}}(p_n+p_{n+1})$ and that it is conserved. Why does it follow, with $\tr L=\sum_np_n$ and $\tr L^{2}=2H$, that the eigenvalues of $L$ are conserved?