PhyslibAlpha.ClassicalFieldTheory.Local.FirstVariation

Let $L$ be a $k$-th order local Lagrangian on $\mathbb{R}^d$ with $m$ field components. If $L$ is smooth in its coordinates—meaning that $L(x, u)$ is continuous and its partial derivatives $\frac{\partial L}{\partial u^a_I}$ with respect to the jet coordinates $u^a_I$ are infinitely differentiable ($C^\infty$) for all field components $a$ and multi-indices $|I| \le k$—then $L$ satisfies smooth Euler-Lagrange regularity. Smooth Euler-Lagrange regularity implies that for any smooth field $f: \mathbb{R}^d \to \mathbb{R}^m$, the coordinate derivatives of the Lagrangian evaluated along the $k$-jet of $f$ are smooth, and the derivatives evaluated along a variation $f + s\eta$ are continuous with respect to the variation parameter $s$ and position $x$.

Let $L$ be a local Lagrangian of order $k$ for fields $f: \mathbb{R}^d \to \mathbb{R}^m$. Suppose $L$ is smooth in its coordinates (meaning $L(x, u)$ is continuous and its partial derivatives $\frac{\partial L}{\partial u^a_I}$ with respect to jet coordinates are $C^\infty$) and $f$ is an infinitely differentiable ($C^\infty$) field with finite action (the action density $x \mapsto L(j^k f(x))$ is integrable over $\mathbb{R}^d$). Then $f$ is critical for the action functional if and only if the local Euler-Lagrange operator $\mathcal{E}(L)$ vanishes at $f$: $$\mathcal{E}(L)[f] = 0$$ where criticality is defined as the first variation of the action $\left. \frac{d}{ds} \right|_{s=0} S(L, f + s\eta)$ being zero for all admissible variations $\eta$.

Let $L$ be a local Lagrangian of order $k$ for fields $f: \mathbb{R}^d \to \mathbb{R}^m$. Suppose that $L$ is smooth in its coordinates (i.e., $L(x, u)$ is continuous and its partial derivatives $\frac{\partial L}{\partial u^a_I}$ with respect to the jet coordinates are $C^\infty$) and that the pair $(L, f)$ is admissible for the action functional (implying $f$ is smooth and has finite action). Then, the field $f$ is critical for the action functional if and only if it satisfies the Euler-Lagrange equations: $$\mathcal{E}(L)[f] = 0$$ where $\mathcal{E}(L)$ is the local Euler-Lagrange operator whose components are defined by $$(\mathcal{E}(L)[f](x))_a = \sum_{|I| \le k} (-1)^{|I|} D_I \left( \frac{\partial L}{\partial u^a_I} \right) [f](x)$$ for each field component $a \in \{1, \dots, m\}$.