nLab Hamiltonian vector field

Definition

On symplectic manifolds

Definition

For $(X,\omega)$ a symplectic manifold, a vector field $v \in \Gamma(T X)$ is called a Hamiltonian vector field if its contraction with the differential 2-form $\omega$ is exact: if there exists $\alpha \in C^\infty(X)$ such that

$$\iota_v \omega = d \alpha \,.$$

In this case $\alpha$ is called a Hamiltonian for $v$.

On $n$-plectic manifolds

Definition

For $(X,\omega)$ an n-plectic manifold, a vector field $v \in \Gamma(T X)$ is called a Hamiltonian vector field if is contraction with the $(n+1)$-form $\omega$ is exact: there is $\alpha \in \Omega^{n-1}(X)$ such that

$$\iota_v \omega = d \alpha \,.$$

In this case $\alpha$ is called a Hamiltonian (n-1)-form for $v$.

On $n$-plectic smooth $\infty$-groupoids

We discuss now the notion of Hamiltonian vector fields in the full generality internal to a cohesive (∞,1)-topos $\mathbf{H}$. We write out the discussion for the case $\mathbf{H} =$ Smooth∞Grpd for convenience, but any other choice of cohesive $(\infty,1)$-topos works as well.

Consider the circle n-group $\mathbf{B}^{n-1}U(1)$ and the corresponding coefficient object $\mathbf{B}^n U(1)_{conn} \in \mathbf{H}$ for $U(1)$-differential cohomology in degree $(n+1)$, the smooth moduli stack of circle n-bundles with connection.

For any $X \in \mathbf{H}$, a morphism $\omega \colon X \to \Omega^{n+1}_{cl}$ is a pre-n-plectic structure on $X$. For instance $(X,\omega)$ might be a symplectic ∞-groupoid.

A higher geometric prequantization of $(X,\omega)$ is a lift $\nabla$ in

$$\array{ && \mathbf{B}^n U(1)_{conn} \\ & {}^{\mathllap{\nabla}}\nearrow & \downarrow \\ X &\stackrel{\omega}{\to}& \Omega^{n+1}_{cl} } \,.$$

The quantomorphism n-group of this prequantization is

$$\mathbf{QuantMorph}(X,\omega) \coloneqq \prod_{\mathbf{B}^n U(1)_{conn}} \mathbf{Aut}(\nabla) \,,$$

where

1. $\mathbf{Aut}(\nabla)$ is the automorphism ∞-group of $\nabla$ formed in the slice (∞,1)-topos $\mathbf{H}_{/\mathbf{B}^n U(1)_{conn}}$

2. $\prod_{\mathbf{B}^n U(1)_{conn}} \colon \mathbf{H}_{/\mathbf{B}^n U(1)_{conn}} \to \mathbf{H}$ is the dependent product (∞,1)-functor.

There is a canonical homomorphism of ∞-groups

$$p \colon \mathbf{QuantMorph}(X,\omega) \to \mathbf{Aut}(X)$$

to the automorphism ∞-group of $X$ (the diffeomorphism group of $X$), given as the restriction to invertible endomorphisms of the canonical morphism

$$\prod_{\mathbf{B}^n U(1)_{conn}} \left[ \nabla,\nabla \right] \to \left[ \sum_{\mathbf{B}^n U(1)_{conn}} \nabla, \sum_{\mathbf{B}^n U(1)_{conn}} \nabla \right] \simeq [X,X]$$

which is discussed at internal hom – Examples – In slice categories.

Definition

The Hamiltonian symplectomorphism n-group $\mathbf{HamSymp}(X,\omega)$ of $(X,\omega)$ is the ∞-image of this morphism $p$, hence the factorization

$$p \colon \mathbf{QuantMorph}(X,\omega) \to \mathbf{HamSymp}(X,\omega) \hookrightarrow \mathbf{Aut}(X)$$

of $p$ by an effective epimorphism followed by a monomorphism.

The corresponding ∞-Lie algebra

$$HamVect(X,\omega) \coloneqq Lie(\mathbf{HamSymp}(X,\omega))$$

we call the $\infty$-Lie algebra of Hamiltonian vector fields on $(X,\omega)$.

More explicitly:

Definition

A Hamiltonian diffeomorphism $\phi$ on
on $(X, \omega)$ is an element $\phi \colon X \stackrel{\simeq}{\to} X$ in the automorphism ∞-group $\phi \in \mathbf{Aut}(X)$ such that it fits into a diagram of the form

$$\array{ X &&\underoverset{\simeq}{\phi}{\to}&& X \\ & {}_{\mathllap{\nabla}}\searrow & \swArrow_{\alpha} & \swarrow_{\mathrlap{\nabla}} \\ && \mathbf{B}^n U(1)_{conn} }$$

in $\mathbf{H}$.

Proposition

For $n = 1$ and $(X, \omega)$ an ordinary prequantizable symplectic manifold regarded as a smooth $\infty$-groupoid, this definition reproduces the ordinary definition of Hamiltonian vector fields above.

In particular it is independent of the choice of prequantum line bundle.

Proof

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 $\nabla$ given by a globally defined $A \in \Omega^1(X)$ with $d A = \omega$. Then the existence of the diagram in def. is equivalent to the condition

$$(\phi(t)^* A - A) = d \alpha(t) \,,$$

where $\alpha(t) \in C^\infty(X)$. Differentiating this at 0 yields the Lie derivative

$$\mathcal{L}_v A = d \alpha' \,,$$

where $v$ is the vector field of which $t \mapsto \phi(t)$ is the flow and where $\alpha' := \frac{d}{dt} \alpha$.

By Cartan calculus this is equivalently

$$d_{\mathrm{dR}} \iota_v A + \iota_v d_{dR} A = d \alpha'$$

and using that $A$ is the connection on a prequantum circle bundle for $\omega$

$$\iota_v \omega = d (\alpha' - \iota_v A) \,.$$

This says that for $v$ to be Hamiltonian, its contraction with $\omega$ must be exact. This is precisely the definition of Hamiltonian vector fields. The corresponding Hamiltonian function here is

$$h := \alpha'-\iota_v A \,.$$

In the general case that the prequantum bundle is not trivial, we can present it by a Cech cocycle on the Cech nerve $C(P_* X \to X)$ of the based path space surjective submersion (regarding $P_* X$ as a diffeological space and choosing one base point per connected component, or else assuming without restriction that $X$ is connected).

Any diffeomorphism $\phi = \exp(v) : X \to X$ lifts to a diffeomorphism $P_*\phi : P_* X \to P_* X$ by setting $P_* \phi(\gamma) : (t \in [0,1]) \mapsto \exp(t v)(\gamma(t))$. This way the Hamiltonian diffeomorphism is presented in the model structure on simplicial presheaves by a diagram

$$\array{ C(P_*X \to X) &&\underoverset{\simeq}{\phi}{\to}&& C(P_*X \to X) \\ & {}_{\mathllap{\nabla}}\searrow & \swArrow_{\alpha} & \swarrow_{\mathrlap{\nabla}} \\ && \mathbf{B}^n U(1)_{conn} }$$

of simplicial presheaves.

Now the same argument as above applies for $P_* X$.

Properties

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.

Proposition

For $(X, \omega)$ a finite dimensional symplectic manifold, there is an exact sequence

$$0 \to HamVect(X, \omega) \to SympVect(X, \omega) \to H^1(X, \mathbb{R}) \to 0 \,.$$

This appears as (Brylinski, 2.3.3).

Relation to functions and Poisson brackets

Proposition

Let $(X, \omega)$ be a connected symplectic manifold. Then there is a central extension of Lie algebras

$$0 \to \mathbb{R} \to (C^\infty(X),\{-,-\}) \to HamVect(X,\omega) \to 0 \,.$$

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 $\mathbb{G}$-principal ∞-connection)

$$(\Omega \mathbb{G})\mathbf{FlatConn}(X) \to \mathbf{QuantMorph}(X,\nabla) \to \mathbf{HamSympl}(X,\nabla)$$

$n$	geometry	structure	unextended structure	extension by	quantum extension
$\infty$	higher prequantum geometry	cohesive ∞-group	Hamiltonian symplectomorphism ∞-group	moduli ∞-stack of $(\Omega \mathbb{G})$-flat ∞-connections on $X$	quantomorphism ∞-group
1	symplectic geometry	Lie algebra	Hamiltonian vector fields	real numbers	Hamiltonians under Poisson bracket
1		Lie group	Hamiltonian symplectomorphism group	circle group	quantomorphism group
2	2-plectic geometry	Lie 2-algebra	Hamiltonian vector fields	line Lie 2-algebra	Poisson Lie 2-algebra
2		Lie 2-group	Hamiltonian 2-plectomorphisms	circle 2-group	quantomorphism 2-group
$n$	n-plectic geometry	Lie n-algebra	Hamiltonian vector fields	line Lie n-algebra	Poisson Lie n-algebra
$n$		smooth n-group	Hamiltonian n-plectomorphisms	circle n-group	quantomorphism n-group

(extension are listed for sufficiently connected $X$)

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.

(…)

Related concepts

• Hamiltonian, Hamiltonian action, Hamiltonian symplectomorphism

• Reeb vector field

• 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

References

Named after William Rowan Hamilton.

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

• Chris Rogers, Higher symplectic geometry PhD thesis (2011) (arXiv:1106.4068)

The notion of Hamiltonian vector field for $n$-plectic cohesive $\infty$-groupoids is discussed in section 4.8.1 of

• Urs Schreiber, differential cohomology in a cohesive topos

See also at higher geometric quantization.