Exercise Sheet 1

(i) Write points of $S^{1}$ as $(\cos a,\sin a)$. Let $U$ be $S^{1}$ without its top point $(0,1)$, with the angle coordinate $\theta_U\in(-3\pi/2,\pi/2)$, and $V$ be $S^{1}$ without its bottom point $(0,-1)$, with $\theta_V\in(-\pi/2,3\pi/2)$. Show that $U\cap V$ consists of two arcs, and that $\theta_V-\theta_U$ is constant on each, with different constants. (ii) Give the configuration space, the phase space and its dimension for the spherical pendulum (a bob on a rigid massless rod, free to swing in every direction about a fixed point), the double pendulum (in a plane), and a free rigid body. (iii) A classical spin of fixed length has as its states the directions in space, the sphere $S^{2}$. Take as given: $S^{2}$ is compact; the cotangent bundle $T^{*}Q$ of a manifold $Q$ of dimension $n\ge1$ is the set of pairs (a point $q$ of $Q$, a momentum at $q$), the map $T^{*}Q\to Q$ forgetting the momentum is continuous, and the momenta at one point $q$ form a closed subset of $T^{*}Q$ that is a copy of $\R^{n}$ (Lectures 2 and 4 build this properly); a closed subset of a compact space is compact; and continuous maps take compact sets to compact sets. Show that $S^{2}$ is not a cotangent bundle $T^{*}Q$ of any manifold $Q$ of positive dimension.

(i) On $T^{*}\R$ take $H=qp$. Find the flow of $\hvf{H}$, check that $\varphi_t\circ\varphi_s=\varphi_{t+s}$, and say whether $\hvf{H}$ is complete. (ii) Now take $H=q^{2}p$. Solve Hamilton’s equations from $(q_0,p_0)$, and find for which $t$ the solution exists. Check that $H$ is constant along it. (iii) Let $X$ be a vector field on a manifold that conserves a smooth function $F$ ($XF=0$) all of whose level sets are compact. Show that $X$ is complete. (iv) Show that the only smooth functions conserved by $\dot x=x^{2}$ on $\R$ are the constants, and that the conserved $H$ of (ii) has non-compact level sets. Show that, by contrast, the pendulum $H=\tfrac12p^{2}-\cos\theta$ has compact level sets on its cylinder. What does (iii) conclude in each case?

On $\R^{2}$ with coordinates $(q,p)$ let $X=\partial_q$ and $Y=q\,\partial_p$. (i) Compute $\comm{X}{Y}$. Show that $X=\hvf{H}$ for $H=p$ and $Y=\hvf{K}$ for $K=-\tfrac12q^{2}$. (ii) Find the two flows, and compute $\varphi^Y_s\circ\varphi^X_t$ and $\varphi^X_t\circ\varphi^Y_s$. Are they equal? Is this consistent with the lecture’s theorem on commuting flows? (iii) Compute $\varphi^Y_{-s}\circ\varphi^X_{-t}\circ\varphi^Y_s\circ \varphi^X_t(q,p)$: follow $X$, then $Y$, then $X$ backwards, then $Y$ backwards. Compare the result with $ts\,\comm{X}{Y}$.

For $H=\tfrac12p^{2}+V(q)$ on $T^{*}\R$, with $$\text{(a)}\ V=\tfrac14q^{4}-\tfrac12q^{2},\qquad \text{(b)}\ V=\tfrac13q^{3}-q,\qquad \text{(c)}\ V=-1/\cosh^{2}q ,$$ find the equilibria and their energies. For each range of energies $E$, say how many orbits the level set $H=E$ consists of and which are closed curves; name the levels that contain a separatrix; and say whether any orbit leaves every bounded set in finite time.

For the pendulum $H=\tfrac12p^{2}-\cos\theta$ on its cylinder and $E>1$, the lecture says the rotating orbit with $p>0$ is periodic. Let $K(k)=\int_0^{\pi/2}\dd\varphi/\sqrt{1-k^{2}\sin^{2}\varphi}$ for $0\le k<1$. (i) Show that the period is $T(E)=\int_0^{2\pi}\dd\theta/\sqrt{2(E+\cos\theta)}$, and that $T(E)=2k\,K(k)$ with $k^{2}=2/(E+1)$. (ii) Show that $2\pi/\sqrt{2(E+1)}<T(E)<2\pi/\sqrt{2(E-1)}$, and that $T(E)\sqrt{2E}\to2\pi$ as $E\to\infty$. What motion is that limit? (iii) Show that $T(E)\to\infty$ as $E\to1^{+}$. (iv) Evaluate $T(2)$ numerically.

Let $X$ be a vector field on $\R^{m}$ with flow $\varphi_t$, and $J(t,x)=\det D\varphi_t(x)$, the Jacobian determinant of $x\mapsto\varphi_t(x)$. (i) Show that $\partial_tD\varphi_t(x)=DX(\varphi_t(x))\,D\varphi_t(x)$, and deduce $\partial_tJ=(\operatorname{div}X)(\varphi_t(x))\,J$, so that $J(t,x)=\exp\int_0^{t}(\operatorname{div}X)(\varphi_s(x))\,\dd s$. (ii) Show that $\operatorname{div}\hvf{H}=0$ on $T^{*}\R^{n}$ for every smooth $H$, and deduce Liouville’s theorem: the flow of $\hvf{H}$ preserves phase-space volume, $\operatorname{vol}(\varphi_t(R))=\operatorname{vol}(R)$, for every region $R$ and every time $t$ for which $\varphi_t$ is defined on all of $R$. (iii) For $H=q^{2}p$ (Exercise 2), at which points is $\varphi_1$ defined? Why does the theorem need its qualification? (iv) For $H=qp$, find the image of the unit disc under $\varphi_t$. What is preserved, and what is not? (v) For the damped oscillator $\dot x=v$, $\dot v=-x-\gamma v$ with $\gamma>0$, which is not Hamiltonian, find $J$.

The bracket is the failure of two flows to commute. Let $X$ and $Y$ be smooth vector fields on $\R^{m}$, with flows $\varphi^X_t$ and $\varphi^Y_s$. (i) Show that $\varphi^X_t(x)=x+t\,X(x)+\tfrac12t^{2}\,DX(x)\,X(x)+O(t^{3})$. (ii) Show that $$\varphi^Y_{-s}\circ\varphi^X_{-t}\circ\varphi^Y_{s}\circ\varphi^X_{t}(x) =x+ts\,\comm{X}{Y}(x)+O\big((|t|+|s|)^{3}\big).$$ (iii) Check (ii) against Exercise 3 (iii). (iv) Deduce one half of the lecture’s theorem on commuting flows: if $\varphi^X_t\circ\varphi^Y_s=\varphi^Y_s\circ\varphi^X_t$ for all small $s,t$, then $\comm{X}{Y}=0$.