Author: Eiko

Time: 2025-02-04 11:27:51 - 2025-02-04 11:27:51 (UTC)

Hamiltonian Flows

Recall that given a symplectic manifold $(M,\omega )$, and a Hamiltonian function $H:M\to \mathbb{R}$, the Hamiltonian vector field $X_H$ is defined by

$$\omega (\cdot ,X_H)=dH\in Z^1({\mathrm{\Omega }}_M^{\bullet },M).$$

On ${\mathbb{C}}^2=\mathbb{C}{e}_x\oplus \mathbb{C}{e}_y$ with $\omega =dx\wedge dy$, the Hamiltonian vector field of $H$ is given by

$$\omega (\cdot ,X_H)=X_H^{(y)}dx-X_H^{(x)}dy=dH=\frac{\partial H}{\partial x}dx+\frac{\partial H}{\partial y}dy.$$

So

$$X_H^{(x)}=-\frac{\partial H}{\partial y},X_H^{(y)}=\frac{\partial H}{\partial x}.$$

The Poisson structure on a symplectic manifold is defined by

$$\{f,g\}=\omega (X_f,X_g)=X_{f}g=-X_{g}f.$$

On $({\mathbb{C}}^2,dx\wedge dy)$, the Poisson bracket is given by

$$\{,\}:\mathcal{O}\wedge \mathcal{O}\to \mathcal{O}:\{f,g\}=\frac{\partial f}{\partial x}\frac{\partial g}{\partial y}-\frac{\partial f}{\partial y}\frac{\partial g}{\partial x}.$$

Hamiltonian Flows on $({\mathbb{C}}^2{)}^{\mathrm{\Gamma }}$ For $\mathrm{\Gamma }=C_2$

$$\{x^2,\}=2x\frac{\partial }{\partial y},\{xy,\}=y\frac{\partial }{\partial y}-x\frac{\partial }{\partial x},\{y^2,\}=-2y\frac{\partial }{\partial x}.$$