PlanetMath: Hamiltonian vector field

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Hamiltonian vector field

(Definition)

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

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

and let $f:M\to\mathbb{R}$ is a smooth function. Then $H_f=\tilde{\omega}^{-1}(df)$ is the Hamiltonian vector field of

. The vector field

is symplectic, and a symplectic vector field

is Hamiltonian if and only if the 1-form $\tilde{\omega}(X)= \omega (\cdot,X)$ is exact.

If is the cotangent bundle of a manifold , which is naturally identified with the phase space of one particle on , and is the Hamiltonian, then the flow of the Hamiltonian vector field is the time flow of the physical system.

"Hamiltonian vector field" is owned by rspuzio. [ full author list (2) | owner history (1) ]

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Cross-references: flow, Hamiltonian, manifold, 1-form, symplectic vector field, vector field, smooth function, cotangent bundle, tangent bundle, isomorphism, symplectic manifold
There are 2 references to this entry.

This is version 4 of Hamiltonian vector field, born on 2002-12-09, modified 2007-10-30.
Object id is 3706, canonical name is HamiltonianVectorField.
Accessed 2835 times total.

Classification:

AMS MSC:

53D05 (Differential geometry :: Symplectic geometry, contact geometry :: Symplectic manifolds, general)

Pending Errata and Addenda

None.[ View all 3 ]

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