Knowlpedia

Definition

Hamiltonian vector field

The vector field obtained from the differential of a Hamiltonian function through a symplectic form.

Definition

Let (M,ω)(M,\omega) be a and let H:M→RH:M\to\mathbb R be a . The Hamiltonian vector field of HH is the unique smooth XHX_H satisfying

ιXHω=dH.\iota_{X_H}\omega=dH.

Existence and uniqueness follow pointwise from the nondegeneracy of ω\omega, which identifies tangent vectors with covectors. This knowl fixes the plus-sign convention in the displayed equation. The vector field XHX_H, rather than the scalar function HH, supplies the first-order differential equation for the associated dynamics.

Basic properties

Cartan's formula and dω=0d\omega=0 give

LXHω=d(ιXHω)=d2H=0,\mathcal L_{X_H}\omega=d(\iota_{X_H}\omega)=d^2H=0,

so XHX_H is a . Also dH(XH)=ω(XH,XH)=0dH(X_H)=\omega(X_H,X_H)=0, and hence HH is constant along integral curves of XHX_H. The assignment H↦XHH\mapsto X_H is real-linear, and its kernel consists of functions that are locally constant.

These properties and the defining convention appear in Cannas da Silva, Lecture 18.1.

Coordinate form and a near miss

In Darboux coordinates with ω=∑idqi∧dpi\omega=\sum_i dq_i\wedge dp_i,

XH=∑i(∂H∂pi∂∂qi−∂H∂qi∂∂pi).X_H=\sum_i\left( \frac{\partial H}{\partial p_i}\frac{\partial}{\partial q_i} -\frac{\partial H}{\partial q_i}\frac{\partial}{\partial p_i} \right).

Every Hamiltonian vector field preserves ω\omega, but the converse can fail globally. On the symplectic torus, contraction of a constant translation field with ω\omega can be a closed nonexact 11-form, so no globally defined HH produces that field.

Conventions
References
  1. Ana Cannas da Silva, Lectures on Symplectic Geometry, Springer, 2008. DOI record. Relevant: Lecture 18.1, Definition 18.1 and the preservation calculation.
  2. Ralph Abraham and Jerrold E. Marsden, Foundations of Mechanics, 2nd ed., AMS Chelsea, 2008. DOI record. Relevant: §3.3.