\def\R{{\bf R}}
\def\vol{{\rm vol}}
\def\colim{\mathop{\rm colim}}

Peter Teichner.

Hamiltonian approach to classical field theory.

Data for a classical Lagrangian field theory:
(1)~Smooth (oriented) manifold~$M$ (spacetime);
(2)~A smooth fiber bundle $E\to M$ (fields on~$M$ are $\Gamma(M,E)$);
(3)~Lagrangian density~$\lambda\in\Omega^{n,0}(JE)$.

Note: Get an action $S\colon\Phi\to\R$:
$M\times\Phi\to JE$ is the jet map.
We have $\Omega^{n,0}(JE)\supset\Omega^n(JE)\to\Omega^n(M\times\Phi)\to C^\infty(\Phi)\ni S$.

Goal for today: Filtration on the space of jets, Euler-Lagrange equations,
derive a Hamiltonian field theory picture.

Example: (Riemannian sigma-model.)  $M$~and~$N$ are Riemannian,
$E=M\times N\to M$, $\Phi=C^\infty(M,N)$, $\lambda(\phi)=\|T\phi(x)\|^2d\vol_M$.

Definition:
$J^\infty E=\{(x,\phi)\mid x\in M\land\hbox{$\phi$ is a local section of~$E$ near~$x$}\}/\sim$.
Here $(x,\phi_1)\sim(x,\phi_2)$ iff $\phi_1$ has $k$-contact with~$\phi_2$ at~$x$ for all~$k$.

Lemma: In coordinates this means that derivatives of $\phi_i$ coincide at point~$x$.

So we have a sequence of bundles $M\gets J^0E=E\gets J^1E=TE\gets J^2E\gets\cdots$.
$JE=\lim_kJ^kE$.

Define $C^\infty(JE):=\colim C^\infty(J^kE)$
and $\Omega^*(JE):=\colim\Omega^*(J^kE)$.

For any bundle $J\to M$ we have sub-dga of $\Omega^*(J)$ constructed as follows.
$\Omega^*_H(J)=C^\infty(J)\otimes_{C^\infty M}\pi^*\Omega^*M$.
This leads to the Serre spectral sequence for de Rham cohomology.
Key geometric fact about JE: it is the differential ideal of ``contact form''.
Jet bundle has a canonical flat connection: $\Omega^1(JE)=\Omega_H(JE)\oplus C^1(JE)$.
Here
$C^*(JE)=\{\omega\in\Omega^*JE\mid j^*(\omega)(\phi_u)=0\ \hbox{for all local sections~$\phi_u$}\}$.
Note that this works only for infinite jets.

%Introduction to the variational bicomplex
%Anderson

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