In physics

Physics

Physics is a natural science that involves the study of matter and its motion through spacetime, along with related concepts such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves.Physics is one of the oldest academic...

 and mathematics

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, a symplectic vector field is one whose flow preserves a symplectic form. That is, if is a symplectic manifold

Symplectic manifold

In mathematics, a symplectic manifold is a smooth manifold, M, equipped with a closed nondegenerate differential 2-form, ω, called the symplectic form. The study of symplectic manifolds is called symplectic geometry or symplectic topology...

, then a vector field is symplectic if its flow preserves the symplectic structure. In other words, the Lie derivative

Lie derivative

In mathematics, the Lie derivative , named after Sophus Lie by Władysław Ślebodziński, evaluates the change of a vector field or more generally a tensor field, along the flow of another vector field...

 must vanish:
.

Alternatively, a vector field is symplectic if its interior product with the symplectic form is closed. (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

Lie derivative

In mathematics, the Lie derivative , named after Sophus Lie by Władysław Ślebodziński, evaluates the change of a vector field or more generally a tensor field, along the flow of another vector field...

 in terms of the exterior derivative

Exterior derivative

In differential geometry, the exterior derivative extends the concept of the differential of a function, which is a 1-form, to differential forms of higher degree. Its current form was invented by Élie Cartan....

.

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

Hamiltonian vector field

In mathematics and physics, a Hamiltonian vector field on a symplectic manifold is a vector field, defined for any energy function or Hamiltonian. Named after the physicist and mathematician Sir William Rowan Hamilton, a Hamiltonian vector field is a geometric manifestation of Hamilton's equations...

.
If the first De Rham cohomology

De Rham cohomology

In mathematics, de Rham cohomology is a tool belonging both to algebraic topology and to differential topology, capable of expressing basic topological information about smooth manifolds in a form particularly adapted to computation and the concrete representation of cohomology classes...

 group 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 . In particular, symplectic vector fields on simply connected spaces are Hamiltonian.

The lie bracket

Lie bracket

Lie bracket can refer to:*A bilinear binary operation defined on elements of a Lie algebra*Lie bracket of vector fields...

 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

Lie algebra

In mathematics, a Lie algebra is an algebraic structure whose main use is in studying geometric objects such as Lie groups and differentiable manifolds. Lie algebras were introduced to study the concept of infinitesimal transformations. The term "Lie algebra" was introduced by Hermann Weyl in the...

s.