nLab
symplectic gradient

Contents

Idea

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

Definition

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

The symplectic gradient of HH is the vector field

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

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

This is the unique vector field X HX_H such that

d dRH=ω(,X H) d_{dR} H = \omega(-,X_H)

The function HH in this context is called an Hamiltonian and the vector field H XH_X an Hamiltonian vector field.

Equivalently, the vector field X HX_H is defined by the condition

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

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

Examples

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

X H= i=1 nfp iq ifq ip 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)