Peter Teichner. Hamiltonian formalism. Data for Lagrangian field theory: Smooth bundle E → M, M is the space-time, Φ = Γ(E) are the fields

on M, local Lagrangian λ ∈ Ω^{n,0}(JE), S: Φ → R, S := ∫_{M} λ is the classical action.

M × Φ → JE is the obvious map. We can canonically lift vector fields from M to JE. So we have j^{∗}: Ω^{r,s}(JE) → Ω^{r}M ⊗ Ω^{s}Φ. J^{1}E is an affine bundle with associated vector bundle Hom(π^{∗}T M, T_{v}E).

We have d_{H} → d_{M} and d_{V} = δ → d_{Φ}. λ ∈ Ω^{n,0} → δ(λ) ∈ Ω^{n,1} = d_{H}(α) + E(λ), where E(λ) ∈ Ω_{H}^{n+1}(JE → E) ∩ Ω^{n,1} ⊂ Ω^{n+1}(JE) is the

Euler-Lagrange form.

Definition. The field ϕ satisfies the classical equation of motion (Euler-Lagrange equation) if E(λ)_{[x,ϕ]} = 0 for all x ∈ M.

Theorem. (Takens.) There is a spectral sequence such that E_{0}^{r,s} = Ω^{r,s}(JE), E_{1}^{r,s} = H^{∗}(Ω^{r,s}(JE)), E_{2}^{r,s} = H^{r}M ⊗H^{s}(F). The horizontal rows of this spectral sequence are exact except for the edges: Ω^{r,0}, Ω^{r,s}. Hence H^{∗}E can be computed from the chain complex ⋯ → Ω^{k,0} → ⋯ → Ω^{n,0} → E^{1} → E^{2} → ⋯ → E^{k}. Here Ω^{n,k} = im(d_{H}) ⊕ E^{k}.

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

If Y^{r} ⊂ M is a compact oriented submanifold, then we have the following diagrams Y × Φ → M × Φ → JE. The composition Ω^{r,s}JE → Ω^{r}Y ⊗ Ω^{s}Φ → Ω^{s}Φ is the integral ∫_{Y}.

Key diagram: d_{H}: Ω^{r−1,s} → Ω^{r,s}, ∫_{∂Y}: Ω^{r−1,s} → Ω^{s}Φ, ∫_{Y}: Ω^{r,s} → Ω^{s}Φ, δ = d_{V}: Ω^{r,s} → Ω^{r,s+1}, d_{Φ}: Ω^{s}Φ → Ω^{s+1}Φ, ∫_{Y}: Ω^{r,s+1} → Ω^{s+1}Φ.

Applications: (a) Y = M is closed: Ω^{0}Φ ∋ S = ∫_{M} λ, λ ∈ Ω^{n,0}. d_{Φ}S = ∫_{M}(δλ) = ∫_{M}(d_{H}(α) + E(λ)) = ∫_{M} E(λ). Hence the classical solutions are the extrema of S.

(b) Y^{n−1} ⊂ M, M is closed. (“Space.”) Ω^{1}Φ ∋ a(Y) := ∫_{Y} α (independent of α!). Ω^{2}Φ ∋ ω(Y) = da(Y) = ∫_{Y} δα gives in good cases a symplectic form on Φ(Y).

(c) If ∂Σ^{n} = Y^{n−1} then we get Ω^{1}(Φ(Σ) ∋ d(S_{Σ}) = ∫_{Σ} δλ_{Σ}.