- some diffeomorphic spaces (i.e. \(T^*T^n\) and \((\CC^*)^n\)) which should be considered the same as symplectic manifolds.
- some spaces which are diffeomorphic (i.e. \(S^2\) and itself) which can be equipped with clearly different symplectic structures (i.e. \(\text{vol}_g\) and \(\text{vol}_{2g}\)).
definition 0.0.1
Let \((X, \omega)\) and \((X', \omega')\) be symplectic manifolds, and \(\phi: X\to X' \) be a diffeomorphism. We call \(\phi\) a symplectomorphism if \[\phi^*\omega'=\omega.\] Given an embedding \(\psi: X\to X'\), we say that \(\psi\) is a symplectic embedding if \[\psi^*\omega'=\omega\]definition 0.0.2
Given a time-dependent vector field \(V_t\), the flow associated to \(V\) is the function \(\phi_t: X\times \RR\to X\) which- for all \(t\), \(\phi_t:X\to X\) is a diffeomorphisms and;
- is the identity at time 0 so that \(\phi_0=\operatorname{id}_M\) and;
- generates the vector field \(V_t\) in the sense that \[(V_t)_p:=\frac{d}{ds}\phi_s(\phi^{-1}_t(p))|_{s=t}.\]
theorem 0.0.3
On compact manifolds, every time-dependent vector field has a well defined flow.proposition 0.0.4
\(\phi_t\) is a symplectic isotopy if and only if its infinitesimal generator \(V_t\) satisfies \[d(\iota_{V_t}\omega)=0.\]definition 0.0.5
Let \((X, \omega)\) be a symplectic manifold, and \(H: X\to \RR\) is a smooth function. The Hamiltonian vector field of \(H\) is the unique vector field \(V_H\) such that \(dH=\iota_{V_H}\omega\).example 0.0.6
Suppose that we are working in \(\RR^{2n}=(q_1, \ldots , q_n,p_1, \ldots p_n) \), equipped with the standard symplectic form \(\omega= \sum_{i=1}^n dp_i \wedge dq_i\). We can compute the Hamiltonian vector field for \(H: \RR^{2n}\to \RR\), in local coordinates as \[V_H=\sum_{k=1}^n \left(a_k \partial_{q_k}+b_k \partial_{p_k}\right)\] where the functions \(a_k\) and \(b_k\) are given by the formulas \begin{align*} \frac{\partial p_k}{\partial H}=-b_k & & \frac{\partial q_k}{\partial H}=a_k. \end{align*}example 0.0.7
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 0.0.8. 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, \(V_H=\partial_\theta\).lemma 0.0.9
Let \((X, \omega, g, J)\) be a compatible triple. Let \(H:X\to \RR\) be a Hamiltonian. \[\grad H = J V_H\]References
[dS01] | Ana Cannas da Silva. Lectures on symplectic geometry. Lecture Notes in Mathematics, 1764, 2001. |