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\begin{document}

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\section*{Hamiltonian vector field}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{symplectic_geometry}{}\paragraph*{{Symplectic geometry}}\label{symplectic_geometry}

[[!include symplectic geometry - contents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{OnSymplecticManifolds}{On symplectic manifolds}\dotfill \pageref*{OnSymplecticManifolds} \linebreak
\noindent\hyperlink{on_plectic_manifolds}{On $n$-plectic manifolds}\dotfill \pageref*{on_plectic_manifolds} \linebreak
\noindent\hyperlink{OnNPlecticInfinityGroupoids}{On $n$-plectic smooth $\infty$-groupoids}\dotfill \pageref*{OnNPlecticInfinityGroupoids} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{hamiltonian_actions_and_moment_maps}{Hamiltonian actions and moment maps}\dotfill \pageref*{hamiltonian_actions_and_moment_maps} \linebreak
\noindent\hyperlink{RelationToSymplectic}{Relation to symplectic vector fields}\dotfill \pageref*{RelationToSymplectic} \linebreak
\noindent\hyperlink{RelationToFunctions}{Relation to functions and Poisson brackets}\dotfill \pageref*{RelationToFunctions} \linebreak
\noindent\hyperlink{group_of_hamiltonian_symplectomorphisms}{Group of Hamiltonian symplectomorphisms}\dotfill \pageref*{group_of_hamiltonian_symplectomorphisms} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{definition}{}\subsection*{{Definition}}\label{definition}

\hypertarget{OnSymplecticManifolds}{}\subsubsection*{{On symplectic manifolds}}\label{OnSymplecticManifolds}

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

\begin{displaymath}
\iota_v \omega = d \alpha
  \,.
\end{displaymath}
In this case $\alpha$ is called a [[Hamiltonian]] for $v$.

\end{defn}
\hypertarget{on_plectic_manifolds}{}\subsubsection*{{On $n$-plectic manifolds}}\label{on_plectic_manifolds}

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

\begin{displaymath}
\iota_v \omega = d \alpha
  \,.
\end{displaymath}
In this case $\alpha$ is called a [[Hamiltonian form|Hamiltonian (n-1)-form]] for $v$.

\end{defn}
\hypertarget{OnNPlecticInfinityGroupoids}{}\subsubsection*{{On $n$-plectic smooth $\infty$-groupoids}}\label{OnNPlecticInfinityGroupoids}

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 [[n-plectic geometry|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

\begin{displaymath}
\itexarray{
    && \mathbf{B}^n U(1)_{conn}
    \\
    & {}^{\mathllap{\nabla}}\nearrow & \downarrow
    \\
    X &\stackrel{\omega}{\to}& \Omega^{n+1}_{cl}
  }
  \,.
\end{displaymath}
The [[quantomorphism n-group]] of this prequantization is

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

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


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



\end{enumerate}
There is a canonical homomorphism of [[∞-groups]]

\begin{displaymath}
p \colon \mathbf{QuantMorph}(X,\omega) \to \mathbf{Aut}(X)
\end{displaymath}
to the [[automorphism ∞-group]] of $X$ (the [[diffeomorphism group]] of $X$), given as the restriction to invertible endomorphisms of the canonical morphism

\begin{displaymath}
\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]
\end{displaymath}
which is discussed at \emph{\href{internal%20hom#ExampleInSliceCategories}{internal hom -- Examples -- In slice categories}}.

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

\begin{displaymath}
p \colon \mathbf{QuantMorph}(X,\omega) \to \mathbf{HamSymp}(X,\omega)
  \hookrightarrow \mathbf{Aut}(X)
\end{displaymath}
of $p$ by an [[effective epimorphism in an (∞,1)-category|effective epimorphism]] followed by a [[monomorphism in an (∞,1)-category|monomorphism]].

The corresponding [[∞-Lie algebra]]

\begin{displaymath}
HamVect(X,\omega) \coloneqq Lie(\mathbf{HamSymp}(X,\omega))
\end{displaymath}
we call the $\infty$-Lie algebra of \textbf{Hamiltonian vector fields} on $(X,\omega)$.

\end{defn}
More explicitly:

\begin{defn}
\label{HamiltonianDiffeomorphismsOnInfinityGroupoid}\hypertarget{HamiltonianDiffeomorphismsOnInfinityGroupoid}{}
A \textbf{Hamiltonian diffeomorphism} $\phi$ on\newline 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

\begin{displaymath}
\itexarray{
    X &&\underoverset{\simeq}{\phi}{\to}&& X
    \\
    & {}_{\mathllap{\nabla}}\searrow 
    &
      \swArrow_{\alpha}
    & 
    \swarrow_{\mathrlap{\nabla}}
    \\
    && \mathbf{B}^n U(1)_{conn}
  }
\end{displaymath}
in $\mathbf{H}$.

\end{defn}
\begin{prop}
\label{}\hypertarget{}{}
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 \hyperlink{OnSymplecticManifolds}{above}.

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

\end{prop}
\begin{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. \ref{HamiltonianDiffeomorphismsOnInfinityGroupoid} is equivalent to the condition

\begin{displaymath}
(\phi(t)^* A - A) = d \alpha(t)
  \,,
\end{displaymath}
where $\alpha(t) \in C^\infty(X)$. Differentiating this at 0 yields the [[Lie derivative]]

\begin{displaymath}
\mathcal{L}_v A = d \alpha'
  \,,
\end{displaymath}
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

\begin{displaymath}
d_{\mathrm{dR}} \iota_v A + \iota_v d_{dR} A  = d \alpha'
\end{displaymath}
and using that $A$ is the connection on a prequantum circle bundle for $\omega$

\begin{displaymath}
\iota_v \omega = d (\alpha' - \iota_v A)
  \,.
\end{displaymath}
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

\begin{displaymath}
h := \alpha'-\iota_v A
  \,.
\end{displaymath}
In the general case that the prequantum bundle is not trivial, we can present it by a [[Cech cohomology|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

\begin{displaymath}
\itexarray{
    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}
  }
\end{displaymath}
of [[simplicial presheaves]].

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

\end{proof}
\hypertarget{properties}{}\subsection*{{Properties}}\label{properties}

\hypertarget{hamiltonian_actions_and_moment_maps}{}\subsubsection*{{Hamiltonian actions and moment maps}}\label{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 \emph{[[moment map]]}. See there for details.

\hypertarget{RelationToSymplectic}{}\subsubsection*{{Relation to symplectic vector fields}}\label{RelationToSymplectic}

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]].

\begin{prop}
\label{}\hypertarget{}{}
For $(X, \omega)$ a finite dimensional [[symplectic manifold]], there is an [[exact sequence]]

\begin{displaymath}
0 \to HamVect(X, \omega) \to SympVect(X, \omega) \to H^1(X, \mathbb{R}) 
  \to 0
  \,.
\end{displaymath}
\end{prop}
This appears as (\hyperlink{Brylinski}{Brylinski, 2.3.3}).

\hypertarget{RelationToFunctions}{}\subsubsection*{{Relation to functions and Poisson brackets}}\label{RelationToFunctions}

\begin{prop}
\label{}\hypertarget{}{}
Let $(X, \omega)$ be a connected [[symplectic manifold]]. Then there is a [[central extension]] of [[Lie algebras]]

\begin{displaymath}
0 \to \mathbb{R} \to (C^\infty(X),\{-,-\}) \to HamVect(X,\omega) \to 0
  \,.
\end{displaymath}
\end{prop}
This is a special case of what is called the \textbf{[[Kostant-Souriau central extension]]}. See around (\hyperlink{Brylinski}{Brylinski, prop. 2.3.9}).

[[!include geometric quantization extensions - table]]

\hypertarget{group_of_hamiltonian_symplectomorphisms}{}\subsubsection*{{Group of Hamiltonian symplectomorphisms}}\label{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]].

(\ldots{})

\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

The Hamiltonian vector field of a given function may also be called its \emph{[[symplectic gradient]]}.

The generalization to [[multisymplectic geometry]]/[[n-plectic geometry]]: [[Hamiltonian n-vector fields]]

\begin{itemize}%
\item [[Hamiltonian]], [[Hamiltonian action]], [[Hamiltonian symplectomorphism]]

\end{itemize}
\hypertarget{references}{}\subsection*{{References}}\label{references}

Named after [[William Rowan Hamilton]].

A textbook reference is section II.3 in

\begin{itemize}%
\item [[Jean-Luc Brylinski]], \emph{Loop spaces, characteristic classes and geometric quantization}, Birkh\"a{}user.

\end{itemize}
For more references on the ordinary notion of Hamiltonian vector fields see the references at \emph{[[symplectic geometry]]} and \emph{[[geometric quantization]]}.

The notion of Hamiltonian vector field in [[n-plectic geometry]] is discussed in

\begin{itemize}%
\item [[Chris Rogers]], \emph{Higher symplectic geometry} PhD thesis (2011) (\href{http://arxiv.org/abs/1106.4068}{arXiv:1106.4068})

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

\begin{itemize}%
\item [[Urs Schreiber]], \emph{[[schreiber:differential cohomology in a cohesive topos]]}

\end{itemize}
See also at \emph{[[higher geometric quantization]]}.

[[!redirects Hamiltonian vector fields]]

[[!redirects hamiltonian vector field]] [[!redirects hamiltonian vector fields]]



\end{document}