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Definition

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

The symplectic gradient of $H$ is the vector field

$X_H := \omega^{-1} d_{dR} H \in \Gamma(T X) \,,$

where $d_{dR} : C^\infty(X) \to \Omega^1(X)$ is the de Rham differential.

This is the unique vector field $X_H$ such that

$d_{dR} H = \omega(-,X_H)$

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

$X_H(f)=\{H,f\}$

for any $f \in C^\infty(X)$, where $\{\,,\,\}$ is the Poisson bracket on $(M,\omega)$.

Examples

If $(M,g)$ is $\mathbb{R}^{2n}$ endowed with the standard symplectic form $\omega=dp_i\wedge dq^i$, then

$X_H= \sum_{i=1}^n\frac{\partial f}{\partial p_i}\frac{\partial}{\partial q^i}-\frac{\partial f}{\partial q^i}\frac{\partial}{\partial p_i}.$

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