Exercise Sheet 5

(i) Let $F$ be any smooth function on a phase space and $g$ any smooth function of one variable. Show that $\pb{F}{g(F)}=0$ and that $\dd F$ and $\dd\big(g(F)\big)$ are linearly dependent at every point. What does this say about counting functions in involution? (ii) On $T^{*}\R^{2}$ with $q\ne0$, let $H=\tfrac12|p|^{2}+V(|q|)$ with $V'(r)>0$ for $r>0$, and $\ell=q_1p_2-q_2p_1$. Show that $\pb{H}{\ell}=0$, and that $\dd H$ and $\dd\ell$ are linearly dependent exactly at the points where $p$ is perpendicular to $q$ with $|p|^{2}=|q|\,V'(|q|)$: the circular orbits. (iii) Take the Kepler potential $V=-1/r$. For fixed $\ell\ne0$, show that the smallest energy is $-1/(2\ell^{2})$, reached on the circular orbit of radius $\ell^{2}$, and that the level set $\{H=-1/(2\ell^{2}),\ \ell=\ell\}$ is exactly that circle. Which hypothesis of the theorem fails there, and which conclusion?

On $\R^{4}$ let $F_1=\tfrac12(q_1^{2}+q_2^{2}+p_1^{2}+p_2^{2})$ and $F_2=q_1$, and fix $c>0$ and $a$ with $a^{2}<2c$. (i) Show that $F_1,F_2$ are not in involution. (ii) Show that the level set $\Sigma=\{F_1=c,\ F_2=a\}$ is a $2$-sphere, compact and connected, and that $\dd F_1,\dd F_2$ are independent at every point of it. (iii) Show that $\Sigma$ is not Lagrangian and not invariant under the flow of $F_1$. Which conclusion of the theorem fails, and why does this not contradict it?

(i) (Stage (iv) with $n=1$.) Let $\Sigma$ be a compact connected component of a level set of $H$ on which $\dd H\ne0$, and $x_0\in\Sigma$. The flow of $H$ acts on $\Sigma$, and the lecture’s stage (iii) shows that the orbit of $x_0$ is all of $\Sigma$. Show that the stabiliser $\Lambda=\{t:\varphi_t(x_0)=x_0\}$ is either $\{0\}$ or $\tau\Z$ for some $\tau>0$, and that compactness excludes $\{0\}$. What is $\Sigma$, and what is $\tau$? (ii) For $H=\tfrac12p^{2}+\tfrac14q^{4}-\tfrac12q^{2}$ at $E=-\tfrac18$ (Sheet 1, Exercise 4), how many components has the level set? What does the theorem say about the level set, and what about each component? (iii) The pendulum $H=\tfrac12p^{2}-\cos\theta$ at $E>1$, with $\theta$ taken in $\R$ (not modulo $2\pi$): what is each component of the level set, and what is its stabiliser? Which hypothesis fails? What changes on the cylinder? (iv) On $\R^{4}$ let $F_1=\tfrac12p_1^{2}$ and $F_2=\tfrac12(q_2^{2}+p_2^{2})$. Show that they are in involution, and describe a component $\Sigma$ of $\{F_1=e_1,\ F_2=e_2\}$ for $e_1,e_2>0$. Find the stabiliser of the action of $\R^{2}$ on it, and check the lecture’s $\Sigma\cong T^{k}\times\R^{n-k}$.

Let $H_i=\tfrac12(p_i^{2}+\freq_i^{2}q_i^{2})$ with $\freq_i>0$, $i=1,2$, and a point $x_0$ with $H_1(x_0),H_2(x_0)>0$. (i) Write the action of $\R^{2}$, $\Phi_{(t_1,t_2)}(x_0)=\varphi^{H_1}_{t_1} \circ\varphi^{H_2}_{t_2}(x_0)$, explicitly. Why does the order of the two flows not matter? (ii) Find the stabiliser $\Lambda$ of $x_0$, and deduce that the level set through $x_0$ is a $2$-torus. (iii) Take the basis $e_1=(2\pi/\freq_1,0)$, $e_2=(0,2\pi/\freq_2)$ of $\Lambda$. Find the angles $\varphi=2\pi A^{-1}t$ and the two basis cycles, and show that the flow of $H=H_1+H_2$ is $\varphi(t)=\varphi(0)+(\freq_1,\freq_2)t$.

The spherical pendulum’s energy–momentum map. Away from the poles, with the conventions of optional lecture A (unit mass, length and gravity, $\theta$ measured from the upward vertical, so that the height is $\cos\theta$), $$H=\tfrac12\Big(p_\theta^{2}+\frac{p_\phi^{2}}{\sin^{2}\theta}\Big)+\cos\theta , \qquad J=p_\phi ,\qquad \theta\in(0,\pi).$$ (i) Show that $\pb{H}{J}=0$. With $V_J(\theta)=J^{2}/(2\sin^{2}\theta)+\cos\theta$, show that for $J\ne0$ the function $V_J$ has exactly one critical point, a minimum at some $\theta_J\in(\pi/2,\pi)$ with $J^{2}\cos\theta_J=-\sin^{4}\theta_J$. (ii) Deduce that for $J\ne0$ the energies reached are exactly $E\ge E_{\min}(J)=V_J(\theta_J)$, and show that $E_{\min}(J)=-1+|J|+O(J^{2})$. (iii) For $J\ne0$, show that $\dd H$ and $\dd J$ are dependent exactly on the circles $p_\theta=0$, $\theta=\theta_J$ (the conical motions), and that for $E>E_{\min}(J)$ the level set $\{H=E,\ J=J\}$ is a $2$-torus. (iv) At $J=0$ the pendulum swings in a vertical plane, and its two equilibria — hanging down and balanced upright — lie on the poles, outside these coordinates. Their energies are $-1$ and $1$. Show that the value $(J,E)=(0,1)$ lies inside the image of the map $(J,H)$, above the curve $E=E_{\min}(J)$.