Exercise Sheet 4

On the half-plane $U=\{(x,y):x>0\}$ let $\symp=x^{-1}\,\dd x\wedge\dd y$. (i) Show that $(U,\symp)$ is a symplectic manifold. (ii) Solve $\iota_{\hvf{f}}\symp=\dd f$ for $\hvf{f}$, and find $\pb{f}{g}$ and $\pb{x}{y}$. (iii) Conversely, suppose you are told only that $\pb{x}{y}=x$. Recover $\symp$, using the relation $J=-W^{-1}$ of Lecture 4, where $J^{ab}=\pb{z^a}{z^b}$ and $W_{ab}=\symp(\partial_a,\partial_b)$. (iv) Find Darboux coordinates $(q,p)$ on $U$, and describe the flow of $H=y$ in both coordinate systems.

(i) Let $\freq>0$. Write down the time-$t$ flow of $H=\tfrac12(p^{2}+\freq^{2}q^{2})$ and verify directly that it satisfies $\dd q(t)\wedge\dd p(t)=\dd q\wedge\dd p$. (ii) For an arbitrary smooth $H$ on $\R^{2}$, let $D(t)$ be the Jacobian of the flow at a point, so that $\dot D=AD$, $D(0)=\Id$, with $A=\Omega S$ and $S$ the Hessian of $H$ along the trajectory (Lecture 4, in the reading that proves Hamiltonian flows symplectic). Using $\frac{\dd}{\dd t}\det D=\tr(A)\det D$, show that $\det D=1$ for every $t$ at which the flow is defined at that point, and conclude that the flow is symplectic wherever it is defined. Why can one not say “for all $t\in\R$”? (iii) Show by example that for $n=2$ the condition $\det D=1$ is not enough for a linear map of $\R^{4}$ to be symplectic.

Let $\freq>0$ and $F(q,Q)=\tfrac{\freq}{2}\,q^{2}\cot Q$, a type-1 generating function (the convention of a reading of Lecture 4), with $p=\partial F/\partial q$ and $P=-\partial F/\partial Q$, defined for $0<Q<\pi$ and $q\ne0$. (i) Find $p$ and $P$, and solve for $(q,p)$ as functions of $(Q,P)$. There are two branches; take the one with $q>0$. (ii) Verify directly that the transformation preserves $\symp$. (iii) Express $H=\tfrac12(p^{2}+\freq^{2}q^{2})$ in terms of $(Q,P)$ and solve the motion. Compare with Sheet 3, Exercise 4. (iv) Where does this generating function fail, and why is that not a defect of the transformation?

(i) With $\symp=\dd q_1\wedge\dd p_1+\dd q_2\wedge\dd p_2$, recall $\symp\wedge\symp=2\,\dd q_1\wedge\dd p_1\wedge\dd q_2\wedge\dd p_2$ (Sheet 2, Exercise 5). Identify $\symp^{2}/2!$ with the phase-space volume element of Sheet 1, Exercise 6, and show that a map $F$ with $F^{*}\symp=\symp$ preserves it. What does that give for Hamiltonian flows? (ii) For $F_i=\tfrac12(q_i^{2}+p_i^{2})$ and $c_1,c_2>0$, compute the Liouville volume of $\{F_1\le c_1,\ F_2\le c_2\}$ in two ways: directly, and in the coordinates of Sheet 3, Exercise 4 (with $\freq=1$) applied in each plane.

In $\R^{4}$ with coordinates $(q_1,q_2,p_1,p_2)$, decide which of the following are Lagrangian, and for each that is a graph of $\dd S$ over $q$-space, find $S$. (a) The plane $p_1=q_1+q_2$, $p_2=q_1-q_2$. (b) The plane $p_1=q_2$, $p_2=0$. (c) The cylinder $\{q_1^{2}+p_1^{2}=1,\ q_2=0\}$.

(i) Let $\symp=f(x,y)\,\dd x\wedge\dd y$ on $\R^{2}$ with $f>0$ everywhere. Show that $q=x$, $p=\int_0^{y}f(x,s)\,\dd s$ is a Darboux chart, i.e. a diffeomorphism onto its image with $\dd q\wedge\dd p=\symp$. (ii) What changes if $f<0$ everywhere? What if $f$ vanishes on the line $y=0$? (iii) Explain why no such argument can make the round metric of a sphere look like $\dd x^{2}+\dd y^{2}$ near a point, and what that contrast says.

(i) Let $q'=\phi(q)$ be a change of coordinates on configuration space, with momenta transforming as $p_i=\sum_jp'_j\,\partial q'_j/\partial q_i$. Show that $\sum_jp'_j\,\dd q'_j=\sum_ip_i\,\dd q_i$, and deduce that $-\dd(p\cdot\dd q)$ is the same symplectic form in both charts. (ii) Check it in polar coordinates: with $p_r=(xp_x+yp_y)/r$ and $p_\theta=xp_y-yp_x$, show that $\dd x\wedge\dd p_x+\dd y\wedge\dd p_y=\dd r\wedge\dd p_r+\dd\theta\wedge \dd p_\theta$ without computing a single wedge product. (iii) By contrast, on $T^{*}\R$ take $q'=q^{3}$ (on $q>0$). Compute $q'\,\dd p'$ and show that it is not $q\,\dd p$. Why is this not a contradiction with $\dd(q\,\dd p)=\symp$?

On $\R^{4}$ with coordinates $(q_1,q_2,p_1,p_2)$ let $a$ be a smooth, everywhere positive function of all four coordinates, and $$\symp_a=a\,\dd q_1\wedge\dd p_1+\dd q_2\wedge\dd p_2 .$$ (i) Show that $\symp_a$ is non-degenerate, compute $\dd\symp_a$, and find exactly which functions $a$ make $\symp_a$ symplectic. (ii) Compute the four fields $\hvf{q_1},\hvf{q_2},\hvf{p_1},\hvf{p_2}$ and the brackets of the coordinates. (iii) Compute the Jacobi sums for the triples $(q_1,p_1,p_2)$ and $(q_1,p_1,q_2)$, and check each against $\dd\symp_a$ evaluated on the three fields, as the theorem that Jacobi is $\dd\symp=0$ (a reading of Lecture 4) requires. Explain, from the form of $\dd\symp_a$, why the Jacobi sums never involve $\partial_{q_1}a$ or $\partial_{p_1}a$. (iv) When $\symp_a$ is symplectic, find Darboux coordinates.