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\begin{document}

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\section*{Hamiltonian mechanics}

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

\hypertarget{physics}{}\paragraph*{{Physics}}\label{physics}

[[!include physicscontents]]

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

[[!include symplectic geometry - contents]]

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

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definition_in_terms_of_symplectic_geometry}{Definition in terms of symplectic geometry}\dotfill \pageref*{definition_in_terms_of_symplectic_geometry} \linebreak
\noindent\hyperlink{comments_on_this_definition}{Comments on this definition}\dotfill \pageref*{comments_on_this_definition} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{vortices_on_the_sphere}{Vortices on the sphere}\dotfill \pageref*{vortices_on_the_sphere} \linebreak
\noindent\hyperlink{related_entries}{Related entries}\dotfill \pageref*{related_entries} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

Hamiltonian mechanics is a formulation of [[mechanics]] in which the basic datum in a mechanical system is a function $H$, the [[Hamiltonian]] of the system, which gives the total energy in the system in terms of the [[position|positions]] and [[momentum|momenta]] of the objects in the system.

More abstractly, the Hamiltonian is a function on [[phase space]], a [[manifold]] whose coordinates are [[generalized position|generalised]] positions $q^i$ and momenta $p_i$. (Compare this to [[Lagrangian mechanics]], in which the [[Lagrangian]] is a function on [[state space]], whose coordinates are generalised positions and [[velocity|velocities]].) So to do Hamiltonian mechanics properly, you must `mind your $p$s and $q$s' (blame [[John Baez]] for this pun).

To begin with, we often take phase space to be the [[cotangent bundle]] of [[configuration space]]. (Compare that state space is the [[tangent bundle]] of configuration space.) This comes equipped with a natural $2$-[[differential form|form]]

\begin{displaymath}
\omega = \sum_i \mathrm{d}p_i \wedge \mathrm{d}q^i ,
\end{displaymath}
or simply $\omega = \mathrm{d}p_i \wedge \mathrm{d}q^i$ using the [[Einstein summation convention]]. This $2$-form is closed, in fact exact, since it is the [[exterior derivative|differential]] of the [[action functional|action form]] $\bar{\mathrm{d}}S = p_i \wedge \mathrm{d}q^i$, and therefore it is a [[symplectic form]].

However, it is also possible to take phase space to be \emph{any} [[symplectic manifold]], or even any [[Poisson manifold]]. In any case, phase space itself gives only the [[kinematics]] (in a momentum-based rather than velocity-based sense); you need the Hamiltonian $H$ to get the [[dynamics]].

Hamiltonian mechanics was developed originally for [[classical mechanics]], but it is also the best known formulation of [[quantum mechanics]]; many students of [[physics]] (and even more so, students of [[chemistry]]) learn it only when they study the latter. This sometimes leads to confusion about the essential differences between classical and quantum physics.

\hypertarget{definition_in_terms_of_symplectic_geometry}{}\subsection*{{Definition in terms of symplectic geometry}}\label{definition_in_terms_of_symplectic_geometry}

Hamiltonian mechanics is best formalized in terms of [[symplectic geometry]] as described for instance in the monograoph

\begin{itemize}%
\item [[Vladimir Arnold]], \emph{Mathemtical Methods of Classical Mechanics} Springer.

\end{itemize}
A classical Hamiltonian [[mechanical system]] is a pair $((X,\omega), H)$ consisting of a

\begin{itemize}%
\item [[symplectic manifold]] $(X,\omega)$


\item and a [[Hamiltonian]] function $H \in C^\infty(X)$.



\end{itemize}
Here

\begin{itemize}%
\item $X$ is the \textbf{[[phase space]]} of the physical system;


\item a curve $\gamma : \mathbb{R} \to X$ is a \textbf{[[trajectory]]} of the physical system in time;


\item $(X,\omega)$ defines the \textbf{[[kinematics]]} of the system;


\item $H$ is the \textbf{[[Hamiltonian]]} that defined the \textbf{[[dynamics]]} of the system.



\end{itemize}
The \textbf{dynamics} is encoded by declaring that those trajectories $\gamma : \mathbb{R} \to X$ are the physically realized trajectories that satisfy the equation

\begin{displaymath}
d H = \omega(\gamma', -)
\end{displaymath}
The components of this are \textbf{[[Hamilton's equations]]}.

In more detail this equation means that for each $t \in \mathbb{R}$ the [[differential form|1-form]]

\begin{displaymath}
(d H)_{\gamma(t)}
\end{displaymath}
and the 1-form

\begin{displaymath}
\omega_{\gamma(t)}((\frac{d}{d t}\gamma(t), -)
\end{displaymath}
coincide.

\hypertarget{comments_on_this_definition}{}\subsubsection*{{Comments on this definition}}\label{comments_on_this_definition}

At first, this formulation of Hamilton's mechanics is just that, an equivalent reformulation. But as any reformulation in more abstract terms, it serves to

\begin{enumerate}%
\item clarify a structure


\item allow more powerful thinking about that structure


\item and eventually it bears in it the seed for further developments pointing beyond this structure



\end{enumerate}
Regarding the first point: this formulation of Hamiltonian mechanics makes clear what th meaning of Hamilton's equations is for systems topologically more interesting than the example $X = \mathbb{R}^{2 n}$ that many introductory physics texts concentrate on-

Regarding the second point: the differential calculus formulation lends itself much more to high-powered arguments than the traditional component-ridden presentation. Of course the latter may still be the preferred method for some concrete computations.

Regarding the second point: after Hamilton's times people started thinking about what [[quantization]] of a classical system should mean. One successful formalization is that of [[geometric quantization]] which takes a symplectic manifold with Hamiltonian function on it as input datum.

The impact that this idea of quantization from symplectic geometry eventually had is hard to underestimate. In the hands of [[Alan Weinstein]] and his school it led to [[symplectic groupoid]]s, [[Courant algebroid]]s and other higher [[Lie theory|Lie theoretic structures]]. In the hands of [[Maxim Kontsevich]] it led to the theorem on formal [[deformation quantization]] and the vast machinery nowadays associated with that.

\hypertarget{examples}{}\subsubsection*{{Examples}}\label{examples}

The symplectic-geometry description of Hamiltonian mechanics is especially well-suited to describe topologically nontrivial phase spaces that are not [[cotangent bundle]]s.

\hypertarget{vortices_on_the_sphere}{}\paragraph*{{Vortices on the sphere}}\label{vortices_on_the_sphere}

$n$ vortices on the sphere as finite dimensional limit of 2D [[equation of motion|Euler equations]]: the phase space of the system of $n$ vortices is not a [[cotangent bundle]] but is $(S^2)^n$ .

\hypertarget{related_entries}{}\subsection*{{Related entries}}\label{related_entries}

\begin{itemize}%
\item [[Hamilton's equations]]


\item [[Hamilton-Jacobi equation]]


\item traditional [[Lagrangian mechanics]] and [[Hamiltonian mechanics]] are naturally embedding into [[local prequantum field theory]] by the notion of [[prequantized Lagrangian correspondences]]



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

Named after [[William Rowan Hamilton]].

A motivation for formulating Hamiltonian mechanics in terms of symplectic manifolds can be found in

\begin{itemize}%
\item Henry Cohn, \emph{\href{http://math.mit.edu/~cohn/Thoughts/symplectic.html}{Why symplectic geometry is the natural setting for classical mechanics}}

\end{itemize}


\end{document}