Exercise Sheet 12

(i) Expand $V(r)=e^{-r}+r-1$ to fourth order in $r$, and show that to cubic order it is the FPUT spring $\tfrac12r^{2}+\tfrac\alpha3r^{3}$ with $\alpha=-\tfrac12$. (ii) Describe the force $-V'(r)$ for large compression ($r\to-\infty$) and large stretching ($r\to+\infty$). (iii) Why can the linear term $r$ be dropped on a ring of springs, and what does dropping it do on the open chain?

Take $N=2$, $\bar p=\tfrac12(p_1+p_2)$ and $x=q_1-q_2$, so that $\ddot x=-2e^{x}$ and $\tfrac14\dot x^{2}+e^{x}=\tfrac14v^{2}$ with $v>0$ (Lecture 12). (i) Verify that $x(t)=\ln\frac{v^{2}}4-2\ln\cosh\frac{v(t-t_0)}2$ solves both equations. (ii) Show that the momenta $p_{1,2}=\bar p\pm\tfrac12\dot x$ are exchanged by the collision. (iii) Show that the eigenvalues of $L$ are $\bar p\mp\tfrac v2$ at every time. (iv) With $t_0=0$ and $q_{1,2}=\bar pt\pm\tfrac x2$, show that $q_1\simeq\lambda_2t+\ln v$ as $t\to-\infty$ and $q_2\simeq\lambda_2t-\ln v$ as $t\to+\infty$, with $\lambda_2=\bar p+\tfrac v2$: the line carrying $\lambda_2$ is displaced by $-2\ln v$. Find the time delay when $\lambda_2>0$. (v) What happens to the displacement for $v<1$?

For $N=3$ let $F_k=\tr L^{k}$, $k=1,2,3$. (i) Compute the Jacobian $\det\big(\partial F_k/\partial p_n\big)$, and show that it is $6\big[(p_2-p_1)(p_3-p_1)(p_3-p_2)+x_1(p_1-p_2)+x_2(p_2-p_3)\big]$. (ii) Deduce that $\dd F_1,\dd F_2,\dd F_3$ are independent on a dense open set.

For general $N$: (i) Show that $\tr L=\sum_np_n$, $\tr L^{2}=2H$ and $\tr L^{3}=\sum_np_n^{3}+3\sum_{n=1}^{N-1}x_n(p_n+p_{n+1})$. (ii) Find the flows of $\tr L$ and of $\tr L^{3}$: how does each move the positions? (iii) Why did nobody see $\tr L^{3}$ as the generator of a symmetry of the positions?

(i) Assuming Moser’s separation, $q_{n+1}-q_n\to+\infty$ as $t\to\pm\infty$, show that the asymptotic momenta are the eigenvalues of $L$, in increasing order along the chain as $t\to+\infty$ and decreasing as $t\to-\infty$. (ii) For three particles with $\lambda=(-3,0,3)$, compute $D_1,D_2,D_3$, and write each as a sum of two-body displacements of Exercise 2, one per pair. (iii) Show that $\sum_kD_k=0$ for every spectrum. What does this say about the centre of mass?

(Computational.) Take $N=5$, $q(0)=(-1,-\tfrac12,0,\tfrac12,1)$ and $p(0)=(2,1,0,-1,-2)$. (i) Integrate the equations of motion from $t=-40$ to $t=40$, and check that $\tr L^{k}$, $k=1,\dots,5$, stay constant, and that $H=\tfrac12\sum_k\lambda_k^{2}$. (ii) Compare $p(\pm40)$ with the eigenvalues of $L$. (iii) Fit the asymptotic lines, $\beta_n^{\pm}=q_n(\pm40)-p_n(\pm40)\cdot(\pm40)$, and compare the displacements $D_k$ with Moser’s formula.

For two particles the fibre is a plane. Lecture 12’s reading proves, in outline, that every connected component of a regular level set of the open chain is a copy of $\R^{N}$. Do it for $N=2$. Let $\Sigma=\{p_1+p_2=2\bar p,\ H=\bar p^{2}+\tfrac14v^{2}\}$ with $v>0$, and use $Q=\tfrac12(q_1+q_2)$, $x=q_1-q_2$ and $p_x=\tfrac12(p_1-p_2)$. (i) Show that $(Q,p_x)$, with $\abs{p_x}<v/2$, are coordinates on $\Sigma$, so that $\Sigma\cong\R^{2}$. (ii) Show that the flow of $p_1+p_2$ moves $Q$ at unit speed and fixes $x$ and $p_x$, and that along the flow of $H$, $p_x$ strictly decreases. (iii) Deduce that no combination $s(p_1+p_2)+tH$ with $(s,t)\ne(0,0)$ returns a point of $\Sigma$ to itself: the stabiliser of Lecture 5’s $\R^{2}$-action is trivial.