# Hamiltonian vector field

Let $(M,\omega )$ be a symplectic manifold^{}, and $\stackrel{~}{\omega}:TM\to {T}^{*}M$ be the isomorphism from the tangent
bundle^{} to the cotangent bundle^{}

$$X\mapsto \omega (\cdot ,X)$$ |

and let $f:M\to \mathbb{R}$ is a smooth function. Then ${H}_{f}={\stackrel{~}{\omega}}^{-1}(df)$ is the Hamiltonian vector field of $f$. The vector field ${H}_{f}$ is symplectic (http://planetmath.org/SymplecticVectorField), and a symplectic vector field $X$ is http://planetmath.org/node/6410Hamiltonian if and only if the 1-form $\stackrel{~}{\omega}(X)=\omega (\cdot ,X)$ is exact.

If ${T}^{*}Q$ is the cotangent bundle of a manifold^{} $Q$, which is naturally identified with the phase
space of one particle on $Q$, and $f$ is the Hamiltonian, then the flow of the Hamiltonian
vector field ${H}_{f}$ is the time flow of the physical system.

Title | Hamiltonian vector field |
---|---|

Canonical name | HamiltonianVectorField |

Date of creation | 2013-03-22 13:14:07 |

Last modified on | 2013-03-22 13:14:07 |

Owner | rspuzio (6075) |

Last modified by | rspuzio (6075) |

Numerical id | 7 |

Author | rspuzio (6075) |

Entry type | Definition |

Classification | msc 53D05 |