\def\R{{\bf R}}
\def\Hom{\mathop{\rm Hom}}
\def\im{\mathop{\rm im}}

Peter Teichner.

Hamiltonian formalism.

Data for Lagrangian field theory: Smooth bundle $E\to M$,
$M$ is the space-time,
$\Phi=\Gamma(E)$ are the fields on~$M$,
local Lagrangian $\lambda\in\Omega^{n,0}(JE)$,
$S\colon\Phi\to\R$, $S:=\int_M\lambda$ is the classical action.

$M\times\Phi\to JE$ is the obvious map.
We can canonically lift vector fields from~$M$ to~$JE$.
So we have $j^*\colon\Omega^{r,s}(JE)\to\Omega^rM\otimes\Omega^s\Phi$.
$J^1E$ is an affine bundle with associated vector bundle $\Hom(\pi^*TM,T_vE)$.

We have $d_H\to d_M$ and $d_V=\delta\to d_\Phi$.

$\lambda\in\Omega^{n,0}\to\delta(\lambda)\in\Omega^{n,1}=d_H(\alpha)+E(\lambda)$,
where $E(\lambda)\in\Omega_H^{n+1}(JE\to E)\cap\Omega^{n,1}\subset\Omega^{n+1}(JE)$
is the Euler-Lagrange form.

Definition.  The field $\phi$ satisfies the classical equation of motion
(Euler-Lagrange equation) if $E(\lambda)_{[x,\phi]}=0$ for all~$x\in M$.

Theorem.  (Takens.)
There is a spectral sequence such that $E_0^{r,s}=\Omega^{r,s}(JE)$,
$E_1^{r,s}=H^*(\Omega^{r,s}(JE))$, $E_2^{r,s}=H^rM\otimes H^s(F)$.
The horizontal rows of this spectral sequence
are exact except for the edges: $\Omega^{r,0}$, $\Omega^{r,s}$.
Hence $H^*E$ can be computed from the chain complex
$\cdots\to\Omega^{k,0}\to\cdots\to\Omega^{n,0}\to E^1\to E^2\to\cdots\to E^k$.
Here $\Omega^{n,k}=\im(d_H)\oplus E^k$.

$\delta E(\lambda)=0$ is the Helmholtz equation.
This is the first obstruction for $E(\lambda)$ being an Euler-Lagrange equation.
The second obstruction is $[E(\lambda)]\in H^{n+1}(E)$.
Uniqueness: LFT correspond to $H^n(E)$.

If $Y^r\subset M$ is a compact oriented submanifold,
then we have the following diagrams $Y\times\Phi\to M\times\Phi\to JE$.
The composition $\Omega^{r,s}JE\to\Omega^rY\otimes\Omega^s\Phi\to\Omega^s\Phi$
is the integral $\int_Y$.

Key diagram: $d_H\colon\Omega^{r-1,s}\to\Omega^{r,s}$,
$\int_{\partial Y}\colon\Omega^{r-1,s}\to\Omega^s\Phi$,
$\int_Y\colon\Omega^{r,s}\to\Omega^s\Phi$,
$\delta=d_V\colon\Omega^{r,s}\to\Omega^{r,s+1}$,
$d_\Phi\colon\Omega^s\Phi\to\Omega^{s+1}\Phi$,
$\int_Y\colon\Omega^{r,s+1}\to\Omega^{s+1}\Phi$.

Applications: (a)~$Y=M$ is closed: $\Omega^0\Phi\ni S=\int_M\lambda$, $\lambda\in\Omega^{n,0}$.
$d_\Phi S=\int_M(\delta\lambda)=\int_M(d_H(\alpha)+E(\lambda))=\int_ME(\lambda)$.
Hence the classical solutions are the extrema of~$S$.

(b)~$Y^{n-1}\subset M$, $M$ is closed.  (``Space.'')
$\Omega^1\Phi\ni a(Y):=\int_Y\alpha$ (independent of~$\alpha$!).
$\Omega^2\Phi\ni\omega(Y)=da(Y)=\int_Y\delta\alpha$ gives
in good cases a symplectic form on~$\Phi(Y)$.

(c)~If $\partial\Sigma^n=Y^{n-1}$ then we get $\Omega^1(\Phi(\Sigma)
\ni d(S_\Sigma)=\int_\Sigma\delta\lambda_\Sigma$.

\bye