AI 中文总结
本文为局部凸向量值拉格朗日量建立变分框架，构造普适算子诺特流，证明其壳上守恒及无穷阶时的散度为零。
AI 中文摘要
在我们先前的工作\cite{sil09261621}中，针对$n$维流形$M$的开集上的局部凸向量值可积张量，本文构建了一个框架，用以处理与拉格朗日量$\mathscr{L}$相关的变分问题，以及与作用量$\phi$和$\omega$相关的壳上守恒律。$\mathscr{L}$是取值于拟完备Hausdorff局部凸空间$\mathcal{Z}$、定义在赋范空间的开集$O$上的$C^{p}$映射；$\phi$是由赋范空间$T$的开集$A$参数化的$C^{p}$作用，作用于取值于赋范空间$Y_{0}$、定义在$M$的图$\xi$的域$W$上的场空间$\mathfrak{Z}^{O}$；$\omega$是由$A$参数化的$W$上的$C^{p}$作用。通过\cite{sil09261621}中的结果，我们构造了$\mathfrak{A}:\mathfrak{Z}^{O}\times A\to\mathcal{M}(W,\mathcal{Z})$，其中$\mathfrak{A}(u,\eta)$是与$\mathscr{L}$、$u$和$\eta$相关的作用泛函。这里$\mathcal{M}(W,\mathcal{Z})=\mathcal{L}_{s}(\mathcal{H}(W),\mathcal{Z})$是$W$上取值于$\mathcal{Z}$的测度的局部凸空间。令$\mathcal{A}(u):A\to\mathcal{M}(W,\mathcal{Z})$使得$\mathcal{A}(u)(\eta)=\mathfrak{A}(u,\eta)$，我们断言$\mathcal{A}(u)$在壳外可微，并求出其微分映射在壳外和壳上的表达式。我们证明：若$\mathcal{A}(u)$在$0$处的微分映射为$0$，则$\mathscr{N}[u]$在壳上守恒；若$p=\infty$，则$\mathscr{N}[u]$的散度等于$0$，其中局部凸向量值向量场的散度在\cite{sil09261621}中定义并分析。这里$\mathscr{N}[u]$是$M$在$U=\xi(W)$上取值于$\mathcal{L}_{b}(T,\mathcal{Z})$的$C^{(p-1)}$向量场。
英文摘要
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 $U=ξ(W)$.
