nLab symplectic gradient

Idea

The notion of symplectic gradient is the analog in symplectic geometry of the gradient in Riemannian geometry.

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

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