### Context

#### Symplectic geometry

symplectic geometry

higher symplectic geometry

# Contents

## Definition

Let $\left(X,\omega \right)$ be a symplectic manifold and $H\in {C}^{\infty }\left(X\right)$ a function.

The symplectic gradient of $H$ is the vector field

${X}_{H}:={\omega }^{-1}{d}_{\mathrm{dR}}H\in \Gamma \left(TX\right)\phantom{\rule{thinmathspace}{0ex}},$X_H := \omega^{-1} d_{dR} H \in \Gamma(T X) \,,

where ${d}_{\mathrm{dR}}:{C}^{\infty }\left(X\right)\to {\Omega }^{1}\left(X\right)$ is the de Rham differential.

This is the unique vector field ${X}_{H}$ such that

${d}_{\mathrm{dR}}H=\omega \left(-,{X}_{H}\right)$d_{dR} H = \omega(-,X_H)

The function $H$ in this context is called an Hamiltonian and the vector field ${H}_{X}$ an Hamiltonian vector field.

Equivalently, the vector field ${X}_{H}$ is defined by the condition

${X}_{H}\left(f\right)=\left\{H,f\right\}$X_H(f)=\{H,f\}

for any $f\in {C}^{\infty }\left(X\right)$, where $\left\{\phantom{\rule{thinmathspace}{0ex}},\phantom{\rule{thinmathspace}{0ex}}\right\}$ is the Poisson bracket on $\left(M,\omega \right)$.

## Examples

If $\left(M,g\right)$ is ${ℝ}^{2n}$ endowed with the standard symplectic form $\omega ={\mathrm{dp}}_{i}\wedge {\mathrm{dq}}^{i}$, then

${X}_{H}=\sum _{i=1}^{n}\frac{\partial f}{\partial {p}_{i}}\frac{\partial }{\partial {q}^{i}}-\frac{\partial f}{\partial {q}^{i}}\frac{\partial }{\partial {p}_{i}}.$X_H= \sum_{i=1}^n\frac{\partial f}{\partial p_i}\frac{\partial}{\partial q^i}-\frac{\partial f}{\partial q^i}\frac{\partial}{\partial p_i}.

Revised on August 31, 2011 20:02:42 by Urs Schreiber (131.211.238.114)