Universal operatorial Noether current associated with a quasi-complete locally convex linear space valued Lagrangian defined on an open set of a product of normed spaces

Abstract

In the context of locally convex vector valued integrable tensors on open sets of a $n$-dimensional manifold $M$ developped in our previous work \cite{sil09261621}, in this paper we construct a framework in which to address both a variational problem associated with a Lagrangian $\mathscr{L}$, and a conservation law on-shell associated with actions $ϕ$ and $ω$. $\mathscr{L}$ is a $C^{p}$-map valued in a quasi-complete Hausdorff lcs $\mathcal{Z}$ and defined on an open set $O$ of a normed space, $ϕ$ is a $C^{p}$-action parametrized by an open $A$ of a normed space $T$, acting over the space $\mathfrak{Z}^{O}$ of fields valued in a normed space $Y_{0}$ and defined on $W$ domain of a chart $ξ$ of $M$; $ω$ is a $C^{p}$-action on $W$ parametrized by $A$. Via a result in \cite{sil09261621}, we construct $\mathfrak{A}:\mathfrak{Z}^{O}\times A\to\mathcal{M}(W,\mathcal{Z})$ where $\mathfrak{A}(u,η)$ is the action functional associated with $\mathscr{L}$, $u$ and $η$. Here $\mathcal{M}(W,\mathcal{Z})=\mathcal{L}_{s}(\mathcal{H}(W),\mathcal{Z})$ is the lcs of measures on $W$ and valued in $\mathcal{Z}$. By letting $\mathcal{A}(u):A\to\mathcal{M}(W,\mathcal{Z})$ be such that $\mathcal{A}(u)(η)=\mathfrak{A}(u,η)$, we state that $\mathcal{A}(u)$ is differentiable off-shell, find its differential map off and on-shell. We establish that if the differential map of $\mathcal{A}(u)$ on $0$ is $0$, then $\mathscr{N}[u]$ is conserved on-shell, and if $p=\infty$, then the divergence of $\mathscr{N}[u]$ equals $0$, where the divergence of a lcs valued vector field is defined and analyzed in \cite{sil09261621}. Here $\mathscr{N}[u]$ is a $\mathcal{L}_{b}(T,\mathcal{Z})$-valued $C^{(p-1)}$-vector field of $M$ on $U=ξ(W)$.