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 ℒ, and a conservation law on-shell associated with actions ϕ and ω. ℒ is a Cp-map valued in a quasi-complete Hausdorff lcs 𝒵 and defined on an open set O of a normed space, ϕ is a Cp-action parametrized by an open A of a normed space T, acting over the space ZO of fields valued in a normed space Y0 and defined on W domain of a chart ξ of M; ω is a Cp-action on W parametrized by A. Via a result in \cite{sil09261621}, we construct A:ZO× A→ℳ(W,𝒵) where A(u,η) is the action functional associated with ℒ, u and η. Here ℳ(W,𝒵)=ℒs(ℋ(W),𝒵) is the lcs of measures on W and valued in 𝒵. By letting 𝒜(u):A→ℳ(W,𝒵) be such that 𝒜(u)(η)=A(u,η), we state that 𝒜(u) is differentiable off-shell, find its differential map off and on-shell. We establish that if the differential map of 𝒜(u) on $0$ is $0$, then 𝒩[u] is conserved on-shell, and if p=∞, then the divergence of 𝒩[u] equals $0$, where the divergence of a lcs valued vector field is defined and analyzed in \cite{sil09261621}. Here 𝒩[u] is a ℒb(T,𝒵)-valued C(p−1)-vector field of M on U=ξ(W).
Related
- A solution to the quantum Yang--Baxter equation associated with a universal two-valued algebraic group
- Noether's problem for A6 and A7
- The Segre invariant and Brill-Noether theory on Hirzebruch surfaces
- Noether symmetries and conservation laws of a class of time-dependent multidimensional nonlinear wave equations
- Derived categories of threefolds near the Noether line
BibTeX
@misc{silvestri2026universal,
title = {{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}},
author = {Benedetto Silvestri},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.09137},
eprint = {2610.09137},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.09137},
}
RIS
TY - UNPB
TI - 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
AU - Benedetto Silvestri
PY - 2026
DA - 2026/10/06
PB - arXiv
DO - 10.48550/arxiv.2610.09137
UR - https://arxiv.org/abs/2610.09137
AB - 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 ℒ, and a conservation law on-shell associated with actions ϕ and ω. ℒ is a C^p-map valued in a quasi-complete Hausdorff lcs 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 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 A:Z^O× A→ℳ(W,𝒵) where A(u,η) is the action functional associated with ℒ, u and η. Here ℳ(W,𝒵)=ℒ_s(ℋ(W),𝒵) is the lcs of measures on W and valued in 𝒵. By letting 𝒜(u):A→ℳ(W,𝒵) be such that 𝒜(u)(η)=A(u,η), we state that 𝒜(u) is differentiable off-shell, find its differential map off and on-shell. We establish that if the differential map of 𝒜(u) on $0$ is $0$, then 𝒩[u] is conserved on-shell, and if p=∞, then the divergence of 𝒩[u] equals $0$, where the divergence of a lcs valued vector field is defined and analyzed in \cite{sil09261621}. Here 𝒩[u] is a ℒ_b(T,𝒵)-valued C^(p−1)-vector field of M on U=ξ(W).
ER -