Exercise Sheet 8

Let $H=\tfrac12(p^{2}+q^{2})$, so that $\dot q=p$ and $\dot p=-q$. (i) Lecture 8’s first step: show that $L=\begin{pmatrix}p&q\\q&-p\end{pmatrix}$ has $L^{2}=2H\,\Id$. What are its eigenvalues? (ii) The second step: show that $\dot L=\comm ML$ is a linear system for the entries of $M$; that if $M$ solves it, so does $M+\alpha\Id+\beta L$ for any functions $\alpha,\beta$; and that the only antisymmetric solution continuous on the phase space is $M=\tfrac12\begin{pmatrix}0&-1\\1&0\end{pmatrix}$. (iii) Show that $e^{tM}$ is the rotation $R(t/2)$ through $t/2$, and that $R(\theta)L(p,q)R(\theta)^{T}=L(p',q')$, where $(p',q')$ is the vector $(p,q)$ turned through $2\theta$. Deduce the motion from $L(t)=U(t)L(0)U(t)^{-1}$.

Let $H=\tfrac12(p_1^{2}+p_2^{2}+p_3^{2})+e^{q_1-q_2}+e^{q_2-q_3}$. (i) Show that Hamilton’s equations give $\dot a_n=\tfrac12a_n(b_n-b_{n+1})$ and $\dot b_n=a_{n-1}^{2}-a_n^{2}$, with $a_0=a_3=0$. With Lecture 8’s $3\times3$ matrix $L$ (symmetric tridiagonal, diagonal $b_n$, off-diagonal $a_n$), take $M$ antisymmetric and tridiagonal with unknown entries, $M_{12}=-m_1=-M_{21}$ and $M_{23}=-m_2=-M_{32}$, and derive $m_1$ and $m_2$ from $\dot L=\comm ML$, entry by entry. Which entries determine them, and which only test the result? (ii) Compute $\tr L$, $\tr L^{2}$ and $\tr L^{3}$ in terms of $q$ and $p$. Which of them did you know already? (iii) Use the Cayley–Hamilton theorem to write $\tr L^{4}$ as a polynomial in $s_k=\tr L^{k}$, $k=1,2,3$.

Take the $L$ and $M$ of Exercise 2 and $D=\operatorname{diag}(1,a_1,a_1a_2)$. (i) Show that $\tilde L=DLD^{-1}$ has $b_n$ on the diagonal, $1$ above it and $a_n^{2}=e^{q_n-q_{n+1}}$ below it. (ii) Compute $\tilde M=DMD^{-1}+\dot DD^{-1}$, which by Lecture 8’s proposition on gauge transformations satisfies $\dot{\tilde L}=\comm{\tilde M}{\tilde L}$. (iii) Show that $\tilde M=\tfrac12b_1\Id-\tfrac12\tilde L+\tilde L_-$, where $\tilde L_-$ is the strictly lower triangular part of $\tilde L$, and deduce $\dot{\tilde L}=\comm{\tilde L_-}{\tilde L}$. Which first integrals survive the gauge transformation, and what is lost?

Euler’s equations for the free top are $\dot\ell=\ell\times\omega$ with $\omega_i=\ell_i/I_i$, $I_1,I_2,I_3>0$ (Lecture 8’s reading). For $a\in\R^{3}$ let $\hat a$ be the antisymmetric matrix with $\hat a\,v=a\times v$. (i) Show that $\comm{\hat a}{\hat b}=\widehat{a\times b}$, and deduce that $L=\hat\ell$ satisfies $\dot L=\comm ML$ with $M=-\hat\omega$. (ii) Show that $\det(\eta-L)=\eta^{3}+\abs\ell^{2}\eta$. What does the spectrum of $L$ conserve? (iii) Show that $H=\sum_i\ell_i^{2}/(2I_i)$ is conserved, and that unless $I_1=I_2=I_3$ it is not a function of $\abs\ell^{2}$.

(i) For the Toda ring with $N=3$, $H=\tfrac12\sum_np_n^{2}+e^{q_1-q_2}+e^{q_2-q_3}+e^{q_3-q_1}$, show that $a_1a_2a_3=1$ with $a_3=e^{(q_3-q_1)/2}$. Write Lecture 8’s $L$ (with $a_3$ in the corners) and $M$ (antisymmetric, $M_{n,n+1}=-\tfrac12a_n$, indices modulo $3$), and verify $\dot L=\comm ML$. (ii) Why is $N=2$ degenerate? Compute $\tr L^{2}$ for the $N=2$ ring. (iii) Show that the site matrix $L_n(\lambda)=\begin{pmatrix}\lambda-p_n&e^{-q_n}\\-e^{q_n}&0\end{pmatrix}$ has determinant $1$, that $\psi_n=e^{q_n/2}\phi_n$ turns the eigenvalue recursion $a_{n-1}\psi_{n-1}+b_n\psi_n+a_n\psi_{n+1}=\lambda\psi_n$ into $\phi_{n+1}=(\lambda-p_n)\phi_n-e^{q_{n-1}-q_n}\phi_{n-1}$, and that then $\Phi_{n+1}=L_n(\lambda)\Phi_n$ for $\Phi_n=(\phi_n,\,-e^{q_{n-1}}\phi_{n-1})^{T}$. (iv) With $V_n(\lambda)=\begin{pmatrix}-\lambda/2&-e^{-q_n}\\e^{q_{n-1}}&\lambda/2\end{pmatrix}$, compute $V_{n+1}L_n-L_nV_n$, and show that $\dot L_n=V_{n+1}L_n-L_nV_n$ for all $n$ and every $\lambda$ exactly when the ring’s equations of motion hold.

(i) Show that every Hamiltonian has a Lax pair of size $1$. What does it conserve? (ii) For a free particle in $\R^{3}$, $H=\tfrac12\abs p^{2}$, with $\ell=q\times p$, show that $L=\operatorname{diag}(\ell_1,\ell_2,\ell_3)$, $M=0$ is a Lax pair, and compute $\pb{\tr L}{\tr L^{3}}$ using $\pb{\ell_i}{\ell_j}=\sum_k\epsilon_{ijk}\ell_k$ (Sheet 2, Exercise 1). Are the spectral invariants of a Lax pair in involution in general? (iii) In three sentences, say what the spectral parameter $\lambda$ of the site matrix buys, with the Toda ring as the example.

Manakov’s spectral parameter. For the top of Exercise 4, let $A=\operatorname{diag}(A_1,A_2,A_3)$ with $A_i=\tfrac12(I_1+I_2+I_3)-I_i$, and $\Omega=\hat\omega$. (i) Show that $A_2+A_3=I_1$ (and cyclically) and that $\hat\ell=A\Omega+\Omega A$. (ii) Show that $L(\lambda)=\hat\ell+\lambda A^{2}$ and $M(\lambda)=-(\Omega+\lambda A)$ satisfy $\dot L(\lambda)=\comm{M(\lambda)}{L(\lambda)}$ for every $\lambda$. (iii) Compute $\det\big(\eta-L(\lambda)\big)$ as a polynomial in $\eta$ and $\lambda$, and show that its coefficients determine $H$, for all $I_1,I_2,I_3>0$. (iv) For a symmetric top, $I_1=I_2\ne I_3$, show that $\ell_3$ is conserved and that $\ell_3^{2}$ is determined by the coefficients of $\det\big(\eta-L(\lambda)\big)$.