PhyslibAlpha.ClassicalFieldTheory.Local.EulerLagrange

Let $L$ be a local Lagrangian of order $k$ defined for fields $f: \mathbb{R}^d \to \mathbb{R}^m$. For a multi-index $I$ such that $|I| \leq k$ and a specific field component index $a \in \{0, \dots, m-1\}$, the Euler-Lagrange term is a function that maps a field $f$ and a point $x \in \mathbb{R}^d$ to the real value: $$(-1)^{|I|} D_I \left( \frac{\partial L}{\partial u^a_I} \right) [f](x)$$ where $|I|$ is the order of the multi-index $I$, $\frac{\partial L}{\partial u^a_I}$ is the partial derivative of the Lagrangian with respect to the jet coordinate $u^a_I$, and $D_I$ is the iterated total derivative with respect to the spatial coordinates.

Let $L$ be a local Lagrangian of order $k$ for fields $f: \mathbb{R}^d \to \mathbb{R}^m$. For a multi-index $I$ with order $|I| \leq k$ and a field component index $a \in \{0, \dots, m-1\}$, the Euler-Lagrange term associated with $L, I$, and $a$ evaluated at a field $f$ and point $x \in \mathbb{R}^d$ is given by: $$\text{eulerLagrangeTerm}(L, I, a, f, x) = (-1)^{|I|} D_I \left( \frac{\partial L}{\partial u^a_I} \right) [f](x)$$ where $|I|$ denotes the order of the multi-index $I$, $\frac{\partial L}{\partial u^a_I}$ is the partial derivative of the Lagrangian with respect to the jet coordinate $u^a_I$, and $D_I$ is the iterated total derivative with respect to the spatial coordinates.

Let $L$ be a local Lagrangian of order $k$ defined for fields $f: \mathbb{R}^d \to \mathbb{R}^m$. For a field component index $a \in \{0, \dots, m-1\}$, the $a$-th component of the local Euler-Lagrange operator, denoted $E_a(L)$, is a function that maps a field $f$ and a point $x \in \mathbb{R}^d$ to the real value: $$E_a(L)[f](x) = \sum_{|I| \le k} (-1)^{|I|} D_I \left( \frac{\partial L}{\partial u^a_I} \right) [f](x)$$ where the sum is over all multi-indices $I$ of dimension $d$ with total order $|I| \le k$, $\frac{\partial L}{\partial u^a_I}$ is the partial derivative of the Lagrangian with respect to the jet coordinate $u^a_I$, and $D_I$ is the iterated total derivative with respect to the spatial coordinates.

Let $L$ be a local Lagrangian of order $k$ for fields $f: \mathbb{R}^d \to \mathbb{R}^m$. For a field component index $a \in \{0, \dots, m-1\}$, a field $f$, and a point $x \in \mathbb{R}^d$, the $a$-th component of the local Euler-Lagrange operator $E_a(L)$ is the sum of the Euler-Lagrange terms over all multi-indices $I$ of dimension $d$ with total order $|I| \le k$: $$E_a(L)[f](x) = \sum_{|I| \le k} (-1)^{|I|} D_I \left( \frac{\partial L}{\partial u^a_I} \right) [f](x)$$ where $D_I$ denotes the iterated total derivative with respect to the spatial coordinates, and $\frac{\partial L}{\partial u^a_I}$ is the partial derivative of the Lagrangian with respect to the jet coordinate $u^a_I$.

Given a local Lagrangian $L$ of order $k$ for fields $f: \mathbb{R}^d \to \mathbb{R}^m$, the local Euler-Lagrange operator $\mathcal{E}(L)$ maps a field $f$ to a function $\mathbb{R}^d \to \mathbb{R}^m$. At each point $x \in \mathbb{R}^d$, the $a$-th component of the value $\mathcal{E}(L)[f](x)$ is defined by the $a$-th component of the local Euler-Lagrange operator $E_a(L)$: $$(\mathcal{E}(L)[f](x))_a = E_a(L)[f](x) = \sum_{|I| \le k} (-1)^{|I|} D_I \left( \frac{\partial L}{\partial u^a_I} \right) [f](x)$$ where the sum is over all multi-indices $I$ of dimension $d$ with total order $|I| \le k$, $\frac{\partial L}{\partial u^a_I}$ is the partial derivative of the Lagrangian with respect to the jet coordinate $u^a_I$, and $D_I$ is the iterated total derivative with respect to the spatial coordinates.

Let $L$ be a local Lagrangian of order $k$ for fields $f: \mathbb{R}^d \to \mathbb{R}^m$. For any field $f$, point $x \in \mathbb{R}^d$, and component index $a \in \{0, \dots, m-1\}$, the $a$-th component of the value of the local Euler-Lagrange operator $\mathcal{E}(L)$ at $x$ is equal to the value of the $a$-th component of the Euler-Lagrange operator $E_a(L)$ at $x$: $$(\mathcal{E}(L)[f](x))_a = E_a(L)[f](x)$$

Let $L$ be a local Lagrangian of order $k$ for fields $f: \mathbb{R}^d \to \mathbb{R}^m$. The local Euler-Lagrange operator $\mathcal{E}(L)$ applied to a field $f$ is the function from $\mathbb{R}^d$ to $\mathbb{R}^m$ that maps each point $x \in \mathbb{R}^d$ to the vector in $\mathbb{R}^m$ whose $a$-th component is given by the $a$-th Euler-Lagrange component $E_a(L)[f](x)$. Specifically, for all $x \in \mathbb{R}^d$: $$\mathcal{E}(L)[f](x) = (E_0(L)[f](x), \dots, E_{m-1}(L)[f](x))$$