Convex polytope: Difference between revisions

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In [[physics]] and [[mathematics]], a '''symplectic vector field''' is one whose flow preserves a [[symplectic form]]. That is, if <math>(M,\omega)</math> is a [[symplectic manifold]], then a vector field <math>X\in\mathfrak{X}(M)</math> is symplectic if its flow preserves the symplectic structure. In other words, the [[Lie derivative]] must vanish:
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:<math>\mathcal{L}_X\omega=0</math>.<ref name="CdS">{{Citation|title=Lectures on Symplectic Geometry|series=Lecture Notes in Mathematics|volume=1764|first=Ana|last=Cannas da Silva|publisher=Springer-Verlag|year=2001|isbn=978-3-540-42195-5|page=106}}.</ref>
 
Alternatively, a vector field is symplectic if its interior product with the symplectic form is closed.<ref name="CdS"/> (The interior product gives a map from vector fields to 1-forms, which is an isomorphism due to the nondegeneracy of a symplectic form.)  The equivalence of the definitions follows from the closedness of the symplectic form and [[Cartan's magic formula]] for the [[Lie derivative]] in terms of the [[exterior derivative]].
 
If the interior product of a vector field with the symplectic form is exact (and in particular, closed), it is called a [[Hamiltonian vector field]].
If the first [[De Rham cohomology]] group <math>H^1(M)</math>is trivial, all closed forms are exact, so all symplectic vector fields are Hamiltonian. That is, ''the obstruction to a symplectic vector field being Hamiltonian lives in <math>H^1(M)</math>.''  In particular, symplectic vector fields on  simply connected spaces are Hamiltonian.
 
The [[Lie_bracket_of_vector_fields|lie bracket]] of two symplectic vector fields is Hamiltonian, and thus the collection of symplectic vector fields and the collection of Hamiltonian vector fields both form [[lie algebra]]s.
 
==References==
{{Reflist}}
 
{{PlanetMath attribution|id=3705|title=Symplectic vector field}}
 
[[Category:Symplectic geometry]]

Latest revision as of 08:22, 4 December 2014

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