Local variational problems and conservation laws. (English) Zbl 1233.58002

Ferraris, Marco; Palese, Marcella; Winterroth, Ekkehart

Differ. Geom. Appl. 29, Suppl. 1, S80-S85 (2011).

Let \(\pi:\mathbb{Y}\to \mathbb{X}\) be a fibered manifold with \(\dim \mathbb{X}=n\) and \(\dim \mathbb{Y}=n+m\). For \(r\geq0\), let \(J_r\mathbb{Y}\) be the space of \(r\)-jet prolongations of sections of \(\pi\). The natural fiberings \(\pi^r_s:J_r\mathbb{Y}\to J_s\mathbb{Y}\), \(r\geq s\), are affine bundles that induce the natural fibered splitting \[ J_r\mathbb{Y}\times_{J_{r-1}\mathbb{Y}}T^*J_{r-1}\mathbb{Y}=J_r\mathbb{Y}\times_{J_{r-1}\mathbb{Y}}(T^*\mathbb{X}\oplus V^*J_{r-1}\mathbb{Y})\;, \] which in turn induces a decomposition of the exterior differential into horizontal and vertical components, \((\pi^{r+1}_r)^*\circ d=d_H+d_V\). For a projectable vector field \(\Xi\) on \(\mathbb{Y}\), let \(j_r\Xi\) denote its jet prolongation, whose horizontal and vertical components are denoted by \(j_r\Xi_H\) and \(j_r\Xi_V\).
The variational sequence of order \(r\) defined by \(\pi:\mathbb{X}\to \mathbb{Y}\) is an exact soft resolution \((\mathcal{V}_r^*,\mathcal{E}_*)\) of the constant sheaf \(\mathbb{R}_\mathbb{Y}\) over \(\mathbb{Y}\), which is defined by using \(k\)-forms on jet spaces of order \(r\). The cohomology \(H_{VS}^*(\mathbb{Y})\) of the corresponding complex of global sections, \(((\mathcal{V}_r^*)_\mathbb{Y},\mathcal{E}_*)\), is naturally isomorphic to the de Rham cohomology \(H_{dR}^*(\mathbb{Y})\). In this setup, each \(\lambda\in(\mathcal{V}_r^n)_\mathbb{Y}\) is a Lagrangian, \(\mathcal{E}_n(\lambda)\) is an Euler-Lagrange form (\(\mathcal{E}_n\) is the Euler-Lagrange morphism), each \(\eta\in(\mathcal{V}_r^{n+1})_\mathbb{Y}\) is a dynamical form, and \(\widetilde{H}_{d\eta}:=\mathcal{E}_{n+1}(\eta)\) is a Helmholtz form (\(\mathcal{E}_{n+1}\) is the Helmholtz morphism).
\(\mathcal{E}_n(\mathcal{V}_r^n)\) is called the sheaf of Euler-Lagrange morphisms. It is said that \(\eta\in(\mathcal{V}_r^{n+1})_\mathbb{Y}\) is locally variational if \(\mathcal{E}_{n+1}(\eta)=0\) (the Helmholtz condition), and globally variational if \(\eta\in(\mathcal{E}_n(\mathcal{V}_r^n))_\mathbb{Y}\). The following cohomological condition determines when a locally variational \(\eta\) is globally variational. Let \(\mathbb{K}_r=\ker\mathcal{E}_n\). The short exact sequence of sheaves \[ 0 \to \mathbb{K}_r \to \mathcal{V}_r^n \to \mathcal{E}_n(\mathcal{V}_r^n) \to 0 \] gives rise to a long exact sequence in cohomology \[ 0 \to (\mathbb{K}_r)_\mathbb{Y} \to (\mathcal{V}_r^n)_\mathbb{Y} \to (\mathcal{E}_n(\mathcal{V}_r^n))_\mathbb{Y} \to H^1(\mathbb{Y},\mathbb{K}_r)\to0\;, \] with \(H^1(\mathbb{Y},\mathbb{K}_r)\cong H_{VS}^{n+1}(\mathbb{Y})\cong H_{dR}^{n+1}(\mathbb{Y})\). Then \(\eta\) is globally variational if and only if \(\delta\eta=0\), where \(\delta\) denotes the map \( (\mathcal{E}_n(\mathcal{V}_r^n))_\mathbb{Y} \to H^1(\mathbb{Y},\mathbb{K}_r)\) of the above sequence.
Given a countable open covering \(\{U_i\}\) of \(\mathbb{Y}\), a system of local sections \(\lambda_i\in(\mathcal{V}_r^n)_{U_i}\) is called a local variational problem when \(\mathcal{E}_n((\lambda_i-\lambda_j)|_{U_i\cap U_j})=0\). Two local variational problems are equivalent if they give rise to the same Euler-Lagrange morphism. Thus every cohomology class in \(H_{VS}^{n+1}(\mathbb{Y})\cong H_{dR}^{n+1}(\mathbb{Y})\) gives rise to a local variational problem.
The authors consider the symmetries of a local variational problem given by a vector field \(\Xi\) on \(\mathbb{Y}\) projectable to a vector field \(\xi\) on \(\mathbb{X}\). Precisely, they consider the variational Lie derivative operator \(\mathcal{L}_{j_r\Xi}\), and prove the following proposition. For the Euler-Lagrange morphism \(\eta_\lambda\) of a local variational problem \(\lambda\equiv(\lambda_i)\), if \(\mathcal{L}_{j_r\Xi}\eta_\lambda=0\), then the following local conservation law holds: \[ 0=d_H\big(j_r\Xi_V\lrcorner\, p_{d_V\lambda_i}+\xi\lrcorner\, \lambda_i-\beta(\lambda_i,\Xi)\big) \] for some \(\beta(\lambda_i,\Xi)\). Here, \(\epsilon(\lambda_i,\Xi)=j_r\Xi_V\lrcorner\, p_{d_V\lambda_i}+\xi\lrcorner\, \lambda_i\) is the canonical or Noether current. The local conserved currents are \(\epsilon(\lambda_i,\Xi)-\beta(\lambda_i,\Xi)\); indeed, \(\epsilon(\lambda_i,\Xi)\) is conserved if and only if \(\Xi\) is also a symmetry of \(\lambda_i\). The authors also show that \[ d_H\big(\epsilon(\lambda_i,\Xi)-\beta(\lambda_i,\Xi)-\epsilon(\lambda_j,\Xi)+\beta(\lambda_j,\Xi)\big)=0\;, \] and the local currents \(\epsilon(\lambda_i,\Xi)-\beta(\lambda_i,\Xi)\) define a global preserved current if and only if \([\Xi_V\lrcorner\,\eta_\lambda]=0\) in \(H^n_{dR}(\mathbb{Y})\). Finally, they prove that this obstruction \([\Xi_V\lrcorner\,\eta_\lambda]\) is the difference of two conceptually independent cohomology classes, one coming from the symmetries of the Euler-Lagrange morphism and the other from the the system of local Noether currents.

Reviewer: Jesus A. Álvarez López (Santiago de Compostela)

MSC:

58A25	Currents in global analysis
55N30	Sheaf cohomology in algebraic topology
55R10	Fiber bundles in algebraic topology
58A12	de Rham theory in global analysis
58A20	Jets in global analysis
58E30	Variational principles in infinite-dimensional spaces
70S10	Symmetries and conservation laws in mechanics of particles and systems

Keywords:

fibered manifold; jet space; Lagrangian formalism; variational sequence; cohomology; symmetry; conservation law

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