Exercise Sheet 7

Let $H=\tfrac12(p^{2}+\freq^{2}q^{2})+\varepsilon q^{4}$ with $\freq>0$. (i) Show that $H=\freq I+\varepsilon H_1$ with $H_1=\frac{I^{2}}{\freq^{2}}\big(\tfrac32-2\cos2\varphi+\tfrac12\cos4\varphi\big)$, and find $h_0$ and the other $h_k$. (ii) Solve the homological equation. Which denominators $k\freq$ appear, and can any of them vanish? (iii) Deduce that, to first order in $\varepsilon$, the frequency is $\freq+3\varepsilon I/\freq^{2}$. (iv) Take $\freq=1$, $\varepsilon=0.01$ and the energy $E=\tfrac12$. Show that the first-order frequency at energy $E$ is $1+3\varepsilon E+O(\varepsilon^{2})$, and compare it with $2\pi/T$, the period $T=\oint\dd q/p$ computed numerically.

Let $H=\tfrac12(p_1^{2}+\freq_1^{2}q_1^{2})+\tfrac12(p_2^{2}+\freq_2^{2}q_2^{2})+\varepsilon q_1^{2}q_2$ with $\freq_1,\freq_2>0$, and action–angle variables $(\varphi_i,I_i)$ for each oscillator, $I_1,I_2>0$. (i) Show that $H_0=\freq_1I_1+\freq_2I_2$ and $$H_1=A\Big(\tfrac12\sin\varphi_2-\tfrac14\sin(2\varphi_1+\varphi_2)+\tfrac14\sin(2\varphi_1-\varphi_2)\Big), \qquad A=\frac{2I_1}{\freq_1}\sqrt{\frac{2I_2}{\freq_2}} .$$ Which Fourier modes $k$ occur, and what is $h_0$? (ii) Solve the homological equation. Which denominator can vanish, and for which frequencies? For those frequencies, which tori are resonant? Is $H_0$ non-degenerate? (iii) At $\freq_1=\freq$, $\freq_2=2\freq$ and $\varepsilon\ne0$, keep only $H_0$ and the resonant term: $H_{\mathrm{res}}=\freq(I_1+2I_2)+\tfrac14\varepsilon A\sin(2\varphi_1-\varphi_2)$. Show that $I_1+2I_2$ is a first integral of $H_{\mathrm{res}}$ but $I_1$ is not.

Let $g=(\sqrt5-1)/2$, and recall from Lecture 7 that $\freq\in\R^{2}$ is $(\gamma,\tau)$-Diophantine if $\abs{k\cdot\freq}\ge\gamma/\abs{k}^{\tau}$ for every $k\in\Z^{2}\setminus\{0\}$, where $\abs k=\abs{k_1}+\abs{k_2}$. (i) Show that $g$ and $g'=-(\sqrt5+1)/2$ are the roots of $x^{2}+x-1$, and that for integers $p,q$, $(qg-p)(qg'-p)=p^{2}+pq-q^{2}$. (ii) Deduce that $\abs{qg-p}\ge1/\big((1+\sqrt5)\,q\big)$ for all integers $p$ and $q\ge1$. (iii) Deduce that $\freq=(1,g)$ is $(\gamma,1)$-Diophantine with $\gamma=1/(1+\sqrt5)$. Why is no $\freq$ with a rational ratio $\freq_2/\freq_1$ Diophantine, for any $\gamma>0$? (iv) (Computational.) Compute $m(N)=\min_{0<\abs k\le N}\abs k\,\abs{k\cdot\freq}$ for $\freq=(1,g)$ and $\freq=(1,\pi)$, at $N=10$, $100$ and $1000$. What do the numbers say?

(i) For the rotors $H_0=\tfrac12(I_1^{2}+I_2^{2})$ of Lecture 7, on the level $H_0=E$ with the section $\varphi_2=0$ and $I_2>0$, find the rotation number $\rho(I_1)$ of the unperturbed section map, and show that it is a twist map. Show that for the isotropic oscillator, $H_0=I_1+I_2$, which Hénon–Heiles perturbs, the unperturbed section map is the identity. (ii) Show that the standard map $p'=p+K\sin x$, $x'=x+p'$ preserves area. (iii) For $K>0$, find its fixed points on the cylinder ($x$ modulo $2\pi$) and classify them. With $M$ the Jacobian, $\det M=1$, a fixed point is hyperbolic if $\abs{\tr M}>2$ (real eigenvalues $\mu$, $1/\mu$ with $\abs\mu>1$), elliptic if $\abs{\tr M}<2$ (eigenvalues $e^{\pm\ii\alpha}$), and parabolic if $\abs{\tr M}=2$. (iv) For a map, the Lyapunov exponent is $\lambda=\lim_{m\to\infty}\frac1m\ln\big(\abs{DP^{m}u}/\abs u\big)$. At $K=1$, find $\lambda$ at the hyperbolic fixed point, for $u$ not along the contracting eigenvector. Is this positive exponent a sign of chaos?

Let $V=\tfrac12(x^{2}+y^{2})+x^{2}y-\tfrac13y^{3}$, the Hénon–Heiles potential. (i) Show that in polar coordinates $V=\tfrac12r^{2}+\tfrac13r^{3}\sin3\theta$. Which symmetries does this show? (ii) Find all critical points of $V$, their type and their heights. (iii) Show that the Toda ring’s potential of Lecture 7, $\tfrac1{24}\big(e^{2y+2\sqrt3x}+e^{2y-2\sqrt3x}+e^{-4y}\big)-\tfrac18$, is $V+\tfrac12(x^{2}+y^{2})^{2}+O(r^{5})$. (iv) In the family $V_\lambda=\tfrac12(x^{2}+y^{2})+x^{2}y+\tfrac\lambda3y^{3}$ (Hénon–Heiles is $\lambda=-1$), show that $\lambda=1$ separates in $u=(x+y)/\sqrt2$, $v=(y-x)/\sqrt2$, and write down a second integral.

(i) Write the section code of Lecture 7’s tutorial and reproduce the Hénon–Heiles sections $x=0$, $p_x>0$ at $E=\tfrac1{12}$, $\tfrac18$ and $\tfrac16$, checking that $H$ drifts by less than $10^{-8}$ along each orbit. (ii) Do the same for the Toda ring of Exercise 5 at $E=\tfrac16$. What do you see? (iii) Estimate the largest Lyapunov exponent of the Hénon–Heiles orbits launched on the section at $(y,p_y)=(0,0)$, $(-0.1,0)$ and $(0.3,0)$ at $E=\tfrac16$, and at $(0,0)$ at $E=\tfrac1{12}$: integrate the linearised equations along the orbit from a generic unit tangent vector (try two), renormalise it every $10$ time units, and average the logarithms of the growth factors, up to $t=5\times10^{4}$. How does the estimate depend on the launch point, the starting vector and the length $t$ of the run? What goes wrong if the starting vector is $(1,0,0,0)$?

Poincaré’s mechanism, tested. Take Lecture 7’s rotors with $V=\cos q_1\cos q_2$: $H=\tfrac12\abs p^{2}+\varepsilon\cos q_1\cos q_2$ on $T^{2}\times\R^{2}$, with $q$ the angles and $p=I$, so that $H_0=\tfrac12\abs I^{2}$ and $\freq=I$. (i) Find the Fourier coefficients $h_k$ of $V$ and the Poincaré set $\mathfrak P$. Is it dense? (ii) Show that $F=\tfrac14(p_1+p_2)^{2}+\tfrac12\varepsilon\cos(q_1+q_2)$ and $G=\tfrac14(p_1-p_2)^{2}+\tfrac12\varepsilon\cos(q_1-q_2)$ Poisson-commute and add up to $H$. Deduce that $F$ is a first integral, independent of $H$ on a dense open set, for every $\varepsilon$. (iii) Write $F=F_0+\varepsilon F_1$. Check both conclusions of Lecture 7’s proposition on Poincaré’s mechanism at first order for this $F$, and show that $\dd F_0\wedge\dd H_0\ne0$ off $\mathfrak P$. The lecture concludes, for a dense $\mathfrak P$, that $\dd F_0\wedge\dd H_0=0$ everywhere. Which step of that argument fails here?