PhyslibAlpha.ClassicalFieldTheory.Local.TotalDivergence

A Lagrangian density $L$ of order $k+1$ and a jet-dependent current $B = (B_1, \dots, B_d)$ of order $k$ satisfy this property if, for every field $f: \mathbb{R}^d \to \mathbb{R}^m$, the action density associated with $L$ is equal to the total divergence of the current $B$. Specifically, for all $x \in \mathbb{R}^d$: $$L(j^{k+1} f(x)) = \sum_{i=1}^d \frac{d}{dx_i} \left( B_i(j^k f(x)) \right)$$ where $j^k f(x)$ denotes the $k$-jet of the field $f$ at $x$, and $\frac{d}{dx_i}$ denotes the total derivative with respect to the $i$-th spatial coordinate.

A local Lagrangian $L$ of order $k$ is called Euler-Lagrange-trivial if, for every field $f: \mathbb{R}^d \to \mathbb{R}^m$, the resulting Euler-Lagrange operator $\mathcal{E}(L)$ vanishes identically. That is, for all $f$: $$\mathcal{E}(L)[f] = 0$$ where $\mathcal{E}(L)[f](x)$ is the vector in $\mathbb{R}^m$ whose components are given by the standard Euler-Lagrange equations: $$\sum_{|I| \le k} (-1)^{|I|} D_I \left( \frac{\partial L}{\partial u^a_I} \right) [f](x) = 0$$ for each component $a \in \{1, \dots, m\}$.

This definition provides a coercion that allows a `TotalDivergence` object $T$ of order $k$ (which packages a current of order $k$ and a Lagrangian density whose divergence matches that current) to be treated directly as its underlying Lagrangian density $\mathcal{L}$ of order $k+1$. Here $d$ represents the dimension of the space, $m$ the dimension of the field value space, and $k$ the jet order.

Let $T$ be a total divergence package of order $k$, consisting of a Lagrangian $\mathcal{L}$ of order $k+1$ and a current $B = (B_0, \dots, B_{d-1})$ of order $k$. For any field $f: \mathbb{R}^d \to \mathbb{R}^m$, the action density of the Lagrangian $\mathcal{L}$ evaluated on $f$ at a point $x \in \mathbb{R}^d$ is equal to the total divergence of the current $B$ evaluated on $f$. Mathematically: $$\mathcal{L}(j^{k+1} f(x)) = \sum_{i=0}^{d-1} \frac{d}{dx_i} \left( B_i(j^k f(x)) \right)$$ where $j^k f(x)$ denotes the $k$-jet of the field $f$ at $x$, and $\frac{d}{dx_i}$ denotes the total derivative with respect to the $i$-th spatial coordinate.

Let $T$ be a total divergence package of order $k$, consisting of a Lagrangian density $\mathcal{L}$ of order $k+1$ and a current $B$ such that the action density of $\mathcal{L}$ is the total divergence of $B$. For any field $f: \mathbb{R}^d \to \mathbb{R}^m$, the local Euler-Lagrange operator $\mathcal{E}(\mathcal{L})$ evaluated on $f$ is identically zero: $$\mathcal{E}(\mathcal{L})[f] = 0$$ This reflects the physical principle that Lagrangians that are total divergences are Euler-Lagrange trivial.

Let $T$ be a total divergence package of order $k$, which consists of a local Lagrangian $\mathcal{L}$ of order $k+1$ and a current $B$ such that the action density of $\mathcal{L}$ is the total divergence of $B$. Let $f: \mathbb{R}^d \to \mathbb{R}^m$ be a field. Suppose that the pair $(\mathcal{L}, f)$ is admissible for the action functional (meaning $f$ is smooth and has a finite action) and that the Lagrangian $\mathcal{L}$ is smooth in its jet coordinates. Then the field $f$ is critical for the action functional, meaning that for every admissible variation $\eta$, the directional derivative of the action functional $S$ at $f$ in the direction $\eta$ is zero: $$\left. \frac{d}{ds} \right|_{s=0} S(\mathcal{L}, f + s\eta) = 0$$