Hamiltonian vector field

Hamilton 矢量场

Geometry & Topology · Graduate · Mathematics Resources editorial compilation (AI-assisted)

Translation evidence: Literature usage

Meaning in context

For a smooth real function H on a symplectic manifold with form omega, the unique smooth vector field X_H satisfying contraction of omega with X_H equals dH; this entry uses the positive-dH sign convention.

对辛流形上的光滑实函数 $H$，以缩并方程 $\iota_{X_H}\omega=dH$ 唯一确定的光滑向量场；本条采用正 $dH$ 的符号约定。

Example

With $\omega=dq\wedge dp$ and $H=(q^2+p^2)/2$, $X_H=p\partial_q-q\partial_p$, giving $\dot q=p$ and $\dot p=-q$.

Usage and distinctions

Some authors use the opposite sign. A vector field preserving the symplectic form need not be Hamiltonian: contraction must be globally exact, rather than merely closed.

Recommended reading in this subject · Preparation & skip guidance

Definition sources

1. Ana Cannas da Silva — Lectures on Symplectic Geometry§18.1, Hamiltonian and Symplectic Vector Fields, Definition 18.1 · Checked 2026-10-14

Translation sources

1. Liu Shixing, Xing Yan, Liu Chang and Guo Yongxin — 对称约化对完整系统数值积分的影响§1 完整约束系统的辛约化，paragraph immediately after equation (1)：Hamilton矢量场 · Checked 2026-10-14

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