Exercise Sheet 2

On $T^{*}\R^{3}$ let $L=q\times p$, that is $L_1=q_2p_3-q_3p_2$ and cyclically. (i) Compute $\pb{q_i}{p_j}$, $\pb{q_i}{q_j}$ and $\pb{p_i}{p_j}$. (ii) Show that $\pb{L_i}{L_j}=\sum_k\epsilon_{ijk}L_k$; in particular $\pb{L_1}{L_2}=L_3$. (iii) Show that each $L_i$ Poisson-commutes with $|q|^{2}$, $|p|^{2}$, $q\cdot p$ and $|L|^{2}$. Compute $\pb{L_3}{q_1}$ and $\pb{L_3}{q_2}$, and say what the flow of $L_3$ does. (iv) Deduce that for every central force, $H=\tfrac12|p|^{2}+V(|q|)$ with $V$ smooth, on $q\ne0$, each $L_i$ is a first integral, and state the Noether reading of this.

On $T^{*}\R$ let $f=q^{2}p$ and $g=qp^{2}$. (i) Compute $\hvf{f}$ and $\hvf{g}$. (ii) Compute $\comm{\hvf{f}}{\hvf{g}}$ from the component formula. (iii) Compute $\pb{f}{g}$ and $\hvf{\pb{f}{g}}$, and check the engine $\hvf{\pb{f}{g}}=-\comm{\hvf{f}}{\hvf{g}}$. Which sign would the convention $\hvf{f}h=\pb{f}{h}$ produce?

On $T^{*}\R^{2}$ take the anisotropic oscillator $H=\tfrac12(p_1^{2}+p_2^{2})+\tfrac12(q_1^{2}+4q_2^{2})$, with frequencies $1$ and $2$, and let $H_1=\tfrac12(p_1^{2}+q_1^{2})$, $H_2=\tfrac12(p_2^{2}+4q_2^{2})$, $L=q_1p_2-q_2p_1$. (i) Which of $H_1$, $H_2$, $L$ are first integrals? Do $H_1$ and $H_2$ Poisson-commute? For the isotropic oscillator $\tfrac12(p_1^{2}+p_2^{2}+q_1^{2}+q_2^{2})$, is $L$ a first integral? (ii) Find the flows of $H_1$ and of $L$, and check the Noether reading: $H$ is invariant under the one and not under the other. Compute $\tfrac{\dd}{\dd s}H$ at $s=0$ along the flow of $L$.

(i) On $\R^{2}$ let $f=x^{2}+y^{2}$. Write $\dd f$ in the basis $\dd x,\dd y$, and evaluate $\dd f$ at the point $(1,2)$ on the vectors $v=(3,1)$ and $w=(-2,1)$. Interpret the zero in the lecture’s picture of a covector as a stack. (ii) In polar coordinates $x=r\cos\theta$, $y=r\sin\theta$, express $\dd x$ and $\dd y$ through $\dd r$ and $\dd\theta$, and deduce that $x\,\dd y-y\,\dd x=r^{2}\,\dd\theta$ and $x\,\dd x+y\,\dd y=r\,\dd r$. (iii) Show that the $1$-form $p\,\dd q$ on $T^{*}\R$ is not $\dd F$ for any smooth function $F(q,p)$.

(i) Evaluate $(\dd x\wedge\dd y)(v,w)$ for $v=(1,2)$, $w=(3,-1)$, and $(\dd x\wedge\dd y)(w,v)$. Which way round do $v,w$ run? (ii) On $\R^{2}$, compute $(x\,\dd x+y\,\dd y)\wedge(-y\,\dd x+x\,\dd y)$. (iii) On $\R^{3}$, write $(\dd x+\dd y)\wedge(\dd y+\dd z)$ in the basis $\dd x\wedge\dd y$, $\dd x\wedge\dd z$, $\dd y\wedge\dd z$. (iv) On $\R^{4}$ with $\symp=\dd q_1\wedge\dd p_1+\dd q_2\wedge\dd p_2$, show that $\symp\wedge\symp=2\,\dd q_1\wedge\dd p_1\wedge\dd q_2\wedge\dd p_2$, and that adding $c\,\dd q_1\wedge\dd q_2$ to $\symp$ ($c$ constant) does not change it. (v) Show that $\alpha\wedge\alpha=0$ for every $1$-form $\alpha$, and give a $2$-form $\beta$ with $\beta\wedge\beta\ne0$.

The Jacobi identity with variable coefficients. On $\R^{3}$ with coordinates $z=(x,y,z)$, let $J^{\alpha\beta}(z)=-J^{\beta\alpha}(z)$ be smooth, and define $\pb{u}{v}=\sum_{\alpha,\beta}\partial_\alpha u\,J^{\alpha\beta}\,\partial_\beta v$, so that $J^{\alpha\beta}=\pb{z^{\alpha}}{z^{\beta}}$. The lecture’s reading proves the Jacobi identity when $J$ is constant. (i) Show that for variable $J$ the terms with second derivatives still cancel, and that the cyclic sum $\mathcal J(f,g,h)=\pb{f}{\pb{g}{h}}+\text{cyclic}$ satisfies $\mathcal J(f,g,h)=\mathcal J(x,y,z)\,\det\big[\partial_\alpha f, \partial_\alpha g,\partial_\alpha h\big]$. So Jacobi holds for all functions if and only if it holds for the three coordinates. (ii) Take $\pb{x_i}{x_j}=\sum_k\epsilon_{ijk}x_k$, the bracket of Exercise 1 (ii) with the $L_i$ as coordinates. Show that it satisfies Jacobi, and that $|x|^{2}$ Poisson-commutes with every function. (iii) Show that $\pb{x}{y}=1$, $\pb{y}{z}=0$, $\pb{z}{x}=x$ violates Jacobi.