\( \def\CC{{\mathbb C}} \def\RR{{\mathbb R}} \def\NN{{\mathbb N}} \def\ZZ{{\mathbb Z}} \def\TT{{\mathbb T}} \def\CF{{\operatorname{CF}^\bullet}} \def\HF{{\operatorname{HF}^\bullet}} \def\SH{{\operatorname{SH}^\bullet}} \def\ot{{\leftarrow}} \def\st{\;:\;} \def\Fuk{{\operatorname{Fuk}}} \def\emprod{m} \def\cone{\operatorname{Cone}} \def\Flux{\operatorname{Flux}} \def\li{i} \def\ev{\operatorname{ev}} \def\id{\operatorname{id}} \def\grad{\operatorname{grad}} \def\ind{\operatorname{ind}} \def\weight{\operatorname{wt}} \def\Sym{\operatorname{Sym}} \def\HeF{\widehat{CHF}^\bullet} \def\HHeF{\widehat{HHF}^\bullet} \def\Spinc{\operatorname{Spin}^c} \def\min{\operatorname{min}} \def\div{\operatorname{div}} \def\SH{{\operatorname{SH}^\bullet}} \def\CF{{\operatorname{CF}^\bullet}} \def\Tw{{\operatorname{Tw}}} \def\Log{{\operatorname{Log}}} \def\TropB{{\operatorname{TropB}}} \def\wt{{\operatorname{wt}}} \def\Span{{\operatorname{span}}} \def\Crit{\operatorname{Crit}} \def\into{\hookrightarrow} \def\tensor{\otimes} \def\CP{\mathbb{CP}} \def\eps{\varepsilon} \) SympSnip:

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example 0.0.1

Consider \(S^2=\{(x_0, x_1, x_2)\;|\;x_0^2+x_1^2+x_2^2=1\}\) equipped with the symplectic form agreeing with the standard metric induced from \(\RR^3\). Take the Hamiltonian \begin{align*} H:S^2\to&\RR\\ (x_0,x_1,x_2)\mapsto& x_2 \end{align*} as drawn in figure 0.0.2. Since Hamiltonian flow preserves the level sets of \(H\), we know that the latitudinal slices are orbits under the action of the Hamiltonian flow. To show that the Hamiltonian flow uniformly rotates the sphere, consider the map \(\phi:S^2\setminus\{(0,0,1), (0,0,-1)\}\into S^1\times \RR\subset \RR^3\), where \(S^1\times \RR=\{(x_0, x_1, x_2)\;|\; x_0^2+x_1^2=1\}\), and the embedding is given by the latitudinal projection. This projection (the Gall-Peters map projection) is area-preserving, and so \(\phi\) is a symplectic embedding. In these new coordinates, \(\omega=d\theta\wedge dx_2\) and \(H=x_2\). In the Gall-Peters' coordinates, \(V_H=\partial_\theta\).
figure 0.0.2:The Hamiltonian flow of the standard height function rotates the sphere counterclockwise relative to the north pole