nLab
symplectic gradient

Idea

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

Definition

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

The symplectic gradient of $H$ is the vector field

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

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

This is the unique vector field $X_H$ such that

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

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

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

Examples

If $(M,g)$ 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}.