On symplectic manifolds
For a symplectic manifold, a vector field is called a Hamiltonian vector field if its contraction with the differential 2-form is exact: if there exists such that
In this case is called a Hamiltonian for .
On -plectic manifolds
For an n-plectic manifold, a vector field is called a Hamiltonian vector field if is contraction with the -form is exact: there is such that
In this case is called a Hamiltonian (n-1)-form for .
On -plectic smooth -groupoids
We discuss now the notion of Hamiltonian vector fields in the full generality internal to a cohesive (∞,1)-topos . We write out the discussion for the case Smooth∞Grpd for convenience, but any other choice of cohesive -topos works as well.
Consider the circle n-group and the corresponding coefficient object for -differential cohomology in degree , the smooth moduli stack of circle n-bundles with connection.
For any , a morphism is a pre-n-plectic structure on . For instance might be a symplectic ∞-groupoid.
A higher geometric prequantization of is a lift in
The quantomorphism n-group of this prequantization is
is the automorphism ∞-group of formed in the slice (∞,1)-topos
is the dependent product (∞,1)-functor.
There is a canonical homomorphism of ∞-groups
to the automorphism ∞-group of (the diffeomorphism group of ), given as the restriction to invertible endomorphisms of the canonical morphism
which is discussed at internal hom – Examples – In slice categories.
The Hamiltonian symplectomorphism n-group of is the ∞-image of this morphism , hence the factorization
of by an effective epimorphism followed by a monomorphism.
The corresponding ∞-Lie algebra
we call the -Lie algebra of Hamiltonian vector fields on .
A Hamiltonian diffeomorphism on
on is an element in the automorphism ∞-group such that it fits into a diagram of the form
For and an ordinary prequantizable symplectic manifold regarded as a smooth -groupoid, this definition reproduces the ordinary definition of Hamiltonian vector fields above.
In particular it is independent of the choice of prequantum line bundle.
To compute the Lie algebra of this, we need to consider smooth 1-parameter families of Hamiltonian diffeomorphisms and differentiate them.
Assume first that the prequantum line bundle is trivial as a bundle, with the connection 1-form of given by a globally defined with . Then the existence of the diagram in def. 4 is equivalent to the condition
where . Differentiating this at 0 yields the Lie derivative
where is the vector field of which is the flow and where .
By Cartan calculus this is equivalently
and using that is the connection on a prequantum circle bundle for
This says that for to be Hamiltonian, its contraction with must be exact. This is precisely the definition of Hamiltonian vector fields. The corresponding Hamiltonian function here is
In the general case that the prequantum bundle is not trivial, we can present it by a Cech cocycle on the Cech nerve of the based path space surjective submersion (regarding as a diffeological space and choosing one base point per connected component, or else assuming without restriction that is connected).
Any diffeomorphism lifts to a diffeomorphism by setting . This way the Hamiltonian diffeomorphism is presented in the model structure on simplicial presheaves by a diagram
of simplicial presheaves.
Now the same argument as above applies for .
Hamiltonian actions and moment maps
An action of a Lie algebra by (flows of) Hamiltonian vector fields that can be lifted to a Hamiltonian action is equivalently given by a moment map. See there for details.
Relation to symplectic vector fields
Every Hamiltonian vector field is in particular a symplectic vector field. Where a symplectic vector field only preserves the symplectic form, a Hamiltonian vector field also preserves the connection on its prequantum line bundle.
For a finite dimensional symplectic manifold, there is an exact sequence
This appears as (Brylinski, 2.3.3).
Relation to functions and Poisson brackets
This is a special case of what is called the Kostant-Souriau central extension. See around (Brylinski, prop. 2.3.9).
higher and integrated Kostant-Souriau extensions:
(∞-group extension of ∞-group of bisections of higher Atiyah groupoid for -principal ∞-connection)
(extension are listed for sufficiently connected )
Group of Hamiltonian symplectomorphisms
The auto-symplectomorphisms on a symplectic manifold form a group, of which the symplectic vector fields generate the connected component. The Hamiltonian vector fields among the symplectic ones generate the group of Hamiltonian symplectomorphisms.
The Hamiltonian vector field of a given function may also be called its symplectic gradient.
The generalization to multisymplectic geometry/n-plectic geometry: Hamiltonian n-vector fields
A textbook reference is section II.3 in
- Jean-Luc Brylinski, Loop spaces, characteristic classes and geometric quantization, Birkhäuser.
For more references on the ordinary notion of Hamiltonian vector fields see the references at symplectic geometry and geometric quantization.
The notion of Hamiltonian vector field in n-plectic geometry is discussed in
The notion of Hamiltonian vector field for -plectic cohesive -groupoids is discussed in section 4.8.1 of
See also at higher geometric quantization.