Exercise Sheet 3

On $\R^{3}$ with coordinates $(x,y,z)$ let $$\alpha=yz\,\dd x+xz\,\dd y+xy\,\dd z,\qquad \beta=-z\,\dd x+x^{2}\,\dd y,\qquad \gamma=x\,\dd y\wedge\dd z+y\,\dd z\wedge\dd x+z\,\dd x\wedge\dd y .$$ (i) Compute $\dd\alpha$. Is $\alpha$ exact? If it is, find a primitive. (ii) Compute $\dd\beta$, and then $\dd(\dd\beta)$ directly. Using the table in the lecture’s reading on grad, curl and div, say which classical operator $\dd\beta$ computes, and check the answer against the classical formula. (iii) Compute $\dd\gamma$ and name the classical quantity its coefficient is. (iv) Compute $\alpha\wedge\beta$ and $\dd(\alpha\wedge\beta)$, and check the graded Leibniz rule $\dd(\alpha\wedge\beta)=\dd\alpha\wedge\beta -\alpha\wedge\dd\beta$. Would the check catch the rule written with $+$?

Let $\alpha=P\,\dd x+Q\,\dd y$ with $P=xy^{2}$ and $Q=x^{3}$, and let $S_h=[a,a+h]\times[b,b+h]$, its boundary $\partial S_h$ traversed anticlockwise. (i) Compute $\oint_{\partial S_h}\alpha$ exactly, edge by edge, as a polynomial in $h$. (ii) Show that $h^{-2}\oint_{\partial S_h}\alpha\to (\partial_xQ-\partial_yP)(a,b)$ as $h\to0$, and that this limit is $\dd\alpha(\partial_x,\partial_y)$ at $(a,b)$: the lecture’s reading of $\dd\alpha$ as circulation per unit area. (iii) Compute $\int_{S_h}\dd\alpha$ and compare it with (i). Which theorem have you verified? (iv) For a general smooth $\alpha$, pair opposite edges and expand to first order to show $\oint_{\partial S_h}\alpha=h^{2}(\partial_xQ-\partial_yP)(a,b) +O(h^{3})$, although each edge alone is of order $h$.

Let $\alpha=(x\,\dd y-y\,\dd x)/(x^{2}+y^{2})$ be the angle form on $\R^{2}\setminus\{0\}$, and let $t\in[0,2\pi]$ throughout. (i) Compute $\oint\alpha$ over $\gamma_a(t)=(3+\cos t,\sin t)$, $\gamma_b(t)=(\cos2t,\sin2t)$, $\gamma_c(t)=(2\cos t,-\sin t)$, and $\gamma_m(t)=(\cos mt,\sin mt)$ for an integer $m\ge1$. For $\gamma_a$, find a primitive of $\alpha$ on the half-plane $x>0$ instead of integrating. For $\gamma_c$ you may use $\int_0^{2\pi}\dd t/(A^{2}\cos^{2}t+B^{2}\sin^{2}t) =2\pi/(AB)$ for $A,B>0$. (ii) Obtain the answer for $\gamma_c$ again without integrating: write down a deformation from the unit circle traversed clockwise to $\gamma_c$, check that it never meets the origin, and apply the lecture’s corollary on deformation invariance. (iii) The circle $r=2$ traversed anticlockwise and the circle $r=1$ traversed clockwise together make up the boundary of the annulus $A=\{1\le r\le2\}$, with its induced orientation. Compute both integrals and check Stokes’ theorem on $A$. Why does the deformation corollary not say that the two integrals are equal? (iv) Now let $\gamma$ be any smooth closed curve in $\R^{2}\setminus\{0\}$, written in polar form $\gamma(t)=r(t)\big(\cos\theta(t),\sin\theta(t)\big)$ with smooth $r>0$ and $\theta$ on $[0,2\pi]$. (Every such curve can be written so; take that as given.) Closedness forces $\theta(2\pi)-\theta(0)=2\pi m$ for an integer $m$, the winding number of $\gamma$. Show that $\oint_\gamma\alpha=2\pi m$, however the loop wanders or crosses itself, and that $F(t,s)=\big((1-s)\,r(t)+s\big)\big(\cos\theta(t),\sin\theta(t)\big)$, $s\in[0,1]$, is a deformation from $\gamma$ to a loop on the unit circle that never meets the origin.

(i) Let $F(u,v)=\big(x(u,v),y(u,v)\big)$ be smooth. Show that $F^{*}(\dd x\wedge\dd y)=\det\big(\partial(x,y)/\partial(u,v)\big)\, \dd u\wedge\dd v$, and deduce $\dd x\wedge\dd y=r\,\dd r\wedge\dd\theta$ in polar coordinates. (ii) Let $\freq>0$, and on $\R^{2}$ minus the origin introduce $(\varphi,I)$, $I>0$, by $$q=\sqrt{2I/\freq}\,\sin\varphi,\qquad p=\sqrt{2I\freq}\,\cos\varphi .$$ Show that $\dd q\wedge\dd p=\dd\varphi\wedge\dd I$, and that the choice $q=\sqrt{2I/\freq}\,\cos\varphi$, $p=\sqrt{2I\freq}\,\sin\varphi$ gives $-\dd\varphi\wedge\dd I$ instead. (iii) With the first choice and $H=\tfrac12(p^{2}+\freq^{2}q^{2})$, express $H$ in $(\varphi,I)$ and solve $\iota_X\symp=\dd H$ there, using $\symp=\dd\varphi\wedge\dd I$ and the lecture’s sign test with $\varphi$ in the place of $q$. Check at $\varphi=0$ that increasing $\varphi$ is the direction of the motion. (iv) Compute $\oint p\,\dd q$ round the curve $I=\text{const}$, traversed with $\varphi$ increasing, and compare it with the area of the region the curve encloses.

The pendulum $H=\tfrac12p^{2}-\cos q$ has phase space the cylinder $S^{1}\times\R$, with $q$ an angle defined modulo $2\pi$. (i) $q$ is not a function on the cylinder. Explain why $\dd q$ and $p\,\dd q$ are nevertheless $1$-forms there, and compute $\dd(p\,\dd q)$. Is $p\,\dd q$ closed on the cylinder? (ii) For $E>1$ the level $H=E$ has a component $\gamma_E$ on which $p>0$: the pendulum rotates. Write $\oint_{\gamma_E}p\,\dd q$, traversed in the direction of the motion, as an integral over $q\in[0,2\pi]$, and show that it lies strictly between $2\pi\sqrt{2(E-1)}$ and $2\pi\sqrt{2E}$. Deduce that the pullback of $p\,\dd q$ to the circle $\gamma_E$ is closed but not exact. (iii) Now let $H=p$ on the same cylinder. Show that every level set $\{p=c\}$ is an invariant circle, and compute $\oint p\,\dd q$ round it in the direction of the motion. What does the level $c=0$ say about the claim that the loop integral of $p\,\dd q$ round an invariant circle cannot vanish?

(i) Show that an antisymmetric $m\times m$ real matrix is singular when $m$ is odd. Deduce that a manifold carrying a non-degenerate $2$-form has even dimension. (ii) On $\R^{3}$ find every vector $v$ with $\iota_v\beta=0$, where $\beta=\dd x\wedge\dd y+\dd y\wedge\dd z+\dd z\wedge\dd x$. (iii) On $\R^{4}$ with coordinates $(q_1,q_2,p_1,p_2)$ find every $v$ with $\iota_v\beta'=0$, where $\beta'=\dd q_1\wedge\dd p_1+\dd q_1\wedge\dd p_2$. Is $\beta'$ closed? Is it symplectic?

A magnetic symplectic form. On $\R^{4}$ with coordinates $(q_1,q_2,p_1,p_2)$ and a real constant $B$, let $$\symp_B=\dd q_1\wedge\dd p_1+\dd q_2\wedge\dd p_2+B\,\dd q_1\wedge\dd q_2 .$$ (i) Show that $\symp_B$ is symplectic. (ii) For $H=\tfrac12(p_1^{2}+p_2^{2})$, solve $\iota_X\symp_B=\dd H$ for $X$, and solve the equations of motion it gives. Describe the orbits in the $(q_1,q_2)$-plane, and show that $H$ is conserved — first by computing, then from the antisymmetry of $\symp_B$ alone. (iii) Let $B=b(q_1,q_2)$ be a smooth function instead. Is $\symp_B$ still symplectic? What if $b$ depends on $p_1$ as well?