PhyslibAlpha.ClassicalFieldTheory.Local.FirstVariation.Regularity

For a local $k$-th order Lagrangian $L$ and a field $f : \text{Space } d \to \mathbb{R}^m$, the property **Euler-Lagrange regularity at $f$** is defined by the conjunction of two conditions: 1. The coordinate derivatives of the Lagrangian evaluated along the $k$-jet of $f$, given by $x \mapsto \frac{\partial L}{\partial u_I^a}(j^k f(x))$, are infinitely differentiable ($C^\infty$) for all multi-indices $|I| \le k$ and component indices $a$. 2. For every admissible variation $\eta$, the coordinate derivatives evaluated along the family of varied fields $f + s\eta$, given by $(s, x) \mapsto \frac{\partial L}{\partial u_I^a}(j^k (f + s\eta)(x))$, are continuous as functions of the variation parameter $s \in \mathbb{R}$ and the position $x \in \text{Space } d$.

A local $k$-th order Lagrangian $L$ is said to have **smooth Euler-Lagrange regularity** if, for every infinitely differentiable ($C^\infty$) field $f: \mathbb{R}^d \to \mathbb{R}^m$, the condition of **Euler-Lagrange regularity at $f$** is satisfied. Specifically, this means that for any such smooth field $f$: 1. The coordinate derivatives of the Lagrangian evaluated along the $k$-jet of $f$, given by $x \mapsto \frac{\partial L}{\partial u_I^a}(j^k f(x))$, are $C^\infty$ for all multi-indices $|I| \le k$ and components $a$. 2. For every admissible variation $\eta$, the coordinate derivatives evaluated along the family of varied fields $f + s\eta$, given by $(s, x) \mapsto \frac{\partial L}{\partial u_I^a}(j^k (f + s\eta)(x))$, are continuous as functions of the variation parameter $s \in \mathbb{R}$ and the position $x \in \mathbb{R}^d$.

Let $L$ be a local Lagrangian of order $k$ for fields $f: \mathbb{R}^d \to \mathbb{R}^m$. Suppose that for a given field $f$, the coordinate derivatives of the Lagrangian evaluated along the $k$-jet of $f$, given by $x \mapsto \frac{\partial L}{\partial u_I^a}(j^k f(x))$, are infinitely differentiable ($C^\infty$) for all multi-indices $I$ with $|I| \le k$ and all field components $a \in \{0, \dots, m-1\}$. Then, for any such multi-index $I$ and component $a$, the Euler-Lagrange term $$x \mapsto (-1)^{|I|} D_I \left( \frac{\partial L}{\partial u^a_I} \right) [f](x)$$ is a continuous function, where $D_I$ denotes the iterated total derivative.

Let $L$ be a local Lagrangian of order $k$ for fields $f: \mathbb{R}^d \to \mathbb{R}^m$. Suppose that for a given field $f$, the coordinate derivatives of the Lagrangian evaluated along the $k$-jet of $f$, given by $x \mapsto \frac{\partial L}{\partial u_I^a}(j^k f(x))$, are infinitely differentiable ($C^\infty$) for all multi-indices $|I| \le k$ and all field components $a \in \{0, \dots, m-1\}$. Then, the $a$-th component of the local Euler-Lagrange operator, defined by $$E_a(L)[f](x) = \sum_{|I| \le k} (-1)^{|I|} D_I \left( \frac{\partial L}{\partial u^a_I} \right) [f](x)$$ is a continuous function from $\mathbb{R}^d$ to $\mathbb{R}$, where $D_I$ denotes the iterated total derivative.

Let $L$ be a local Lagrangian of order $k$ for fields $f: \mathbb{R}^d \to \mathbb{R}^m$. Suppose that for a given field $f$, the coordinate derivatives of the Lagrangian evaluated along the $k$-jet of $f$, given by the mapping $x \mapsto \frac{\partial L}{\partial u_I^a}(j^k f(x))$, are infinitely differentiable ($C^\infty$) for all multi-indices $I$ with $|I| \le k$ and all field components $a \in \{0, \dots, m-1\}$. Then the local Euler-Lagrange operator applied to $f$, denoted $\mathcal{E}(L)[f]$, is a continuous function from $\mathbb{R}^d$ to $\mathbb{R}^m$.

Let $L$ be a local $k$-th order Lagrangian. If the coordinate derivatives $\frac{\partial L}{\partial u^a_I}$ (for all field components $a \in \{0, \dots, m-1\}$ and multi-indices $|I| \le k$) are infinitely differentiable ($C^\infty$) functions of the base space coordinates $x \in \mathbb{R}^d$ and the jet coordinates $u$, then $L$ possesses **smooth Euler-Lagrange regularity**. This regularity ensures that for any $C^\infty$ field $f: \mathbb{R}^d \to \mathbb{R}^m$: 1. The functions $x \mapsto \frac{\partial L}{\partial u_I^a}(j^k f(x))$ are $C^\infty$ for all $a$ and $I$. 2. For any admissible variation $\eta$, the functions $(s, x) \mapsto \frac{\partial L}{\partial u_I^a}(j^k (f + s\eta)(x))$ are continuous with respect to the variation parameter $s \in \mathbb{R}$ and the position $x \in \mathbb{R}^d$.