PhyslibAlpha.ClassicalFieldTheory.Local.FirstVariation.Basic

Given a Lagrangian $\mathcal{L}$ of order $k$, a field configuration $f: \mathbb{R}^d \to \mathbb{R}^m$, and an admissible variation $\eta: \mathbb{R}^d \to \mathbb{R}^m$, for a specific multi-index $I$ (with $|I| \le k$) and field component $a \in \{0, \dots, m-1\}$, this function maps each point $x \in \mathbb{R}^d$ to the following term of the linearized first-variation density: $$\frac{\partial \mathcal{L}}{\partial u^a_I}(j^k f(x)) \cdot \partial^I \eta^a(x)$$ where: - $\frac{\partial \mathcal{L}}{\partial u^a_I}$ is the partial derivative of the Lagrangian with respect to the jet coordinate $u^a_I$ (representing the $I$-th derivative of the $a$-th field component). - $j^k f(x)$ is the $k$-jet of the field $f$ at the point $x$. - $\partial^I \eta^a(x)$ is the iterated partial derivative of the $a$-th component of the variation $\eta$ at $x$ corresponding to the multi-index $I$.

Given a Lagrangian $\mathcal{L}$ of order $k$, a field configuration $f: \mathbb{R}^d \to \mathbb{R}^m$, and an admissible variation $\eta: \mathbb{R}^d \to \mathbb{R}^m$, the first-variation density is a function that maps each point $x \in \mathbb{R}^d$ to the total sum of the linearized variations: $$\sum_{|I| \le k} \sum_{a=1}^m \frac{\partial \mathcal{L}}{\partial u^a_I}(j^k f(x)) \cdot \partial^I \eta^a(x)$$ where: - $I$ ranges over all multi-indices with total order $|I| \le k$. - $a$ ranges over the components of the field $a \in \{1, \dots, m\}$. - $\frac{\partial \mathcal{L}}{\partial u^a_I}$ is the partial derivative of the Lagrangian with respect to the jet coordinate $u^a_I$ (the coordinate corresponding to the $I$-th derivative of the $a$-th field component). - $j^k f(x)$ is the $k$-jet of the field $f$ at point $x$. - $\partial^I \eta^a(x)$ is the $I$-th partial derivative of the $a$-th component of the variation $\eta$ at point $x$.

Given a local Lagrangian $L$ of order $k$ for fields $f: \mathbb{R}^d \to \mathbb{R}^m$ and an admissible variation $\eta: \mathbb{R}^d \to \mathbb{R}^m$, the first-variation value is defined as the integral over $\mathbb{R}^d$ of the inner product between the Euler-Lagrange operator $\mathcal{E}(L)[f]$ and the variation $\eta$: $$\int_{\mathbb{R}^d} \langle \mathcal{E}(L)[f](x), \eta(x) \rangle \, dx$$ where $\langle \cdot, \cdot \rangle$ denotes the standard Euclidean inner product on $\mathbb{R}^m$. This represents the value predicted by the first-variation formula for an admissible variation after integration by parts.

For a Lagrangian $\mathcal{L}$ of order $k$, a field configuration $f: \mathbb{R}^d \to \mathbb{R}^m$, an admissible variation $\eta: \mathbb{R}^d \to \mathbb{R}^m$, a multi-index $I$ such that $|I| \le k$, and a field component index $a \in \{0, \dots, m-1\}$, the value of the $(I, a)$-th term of the first-variation density at a point $x \in \mathbb{R}^d$ is given by: $$\text{firstVariationDensityTerm}(\mathcal{L}, f, \eta, I, a, x) = \frac{\partial \mathcal{L}}{\partial u^a_I}(j^k f(x)) \cdot \partial^I \eta^a(x)$$ where $\frac{\partial \mathcal{L}}{\partial u^a_I}$ is the partial derivative of the Lagrangian with respect to the jet coordinate $u^a_I$, $j^k f(x)$ is the $k$-jet of $f$ at $x$, and $\partial^I \eta^a(x)$ is the iterated partial derivative of the $a$-th component of the variation $\eta$ at $x$ corresponding to the multi-index $I$.

For a Lagrangian $\mathcal{L}$ of order $k$, a field configuration $f: \mathbb{R}^d \to \mathbb{R}^m$, and an admissible variation $\eta: \mathbb{R}^d \to \mathbb{R}^m$, the first-variation density evaluated at a point $x \in \mathbb{R}^d$ is equal to the sum over all multi-indices $I$ (where $|I| \le k$) and all field components $a \in \{1, \dots, m\}$ of the individual first-variation density terms: $$\text{firstVariationDensity}(\mathcal{L}, f, \eta, x) = \sum_{|I| \le k} \sum_{a=1}^m \text{firstVariationDensityTerm}(\mathcal{L}, f, \eta, I, a, x)$$ where each term in the sum is given by $\frac{\partial \mathcal{L}}{\partial u^a_I}(j^k f(x)) \cdot \partial^I \eta^a(x)$.

For a local Lagrangian $L$ of order $k$, a field configuration $f: \mathbb{R}^d \to \mathbb{R}^m$, and an admissible variation $\eta: \mathbb{R}^d \to \mathbb{R}^m$, the first-variation value is equal to the integral over $\mathbb{R}^d$ of the inner product between the Euler-Lagrange operator $\mathcal{E}(L)[f]$ and the variation $\eta$: $$\text{firstVariationValue}(L, f, \eta) = \int_{\mathbb{R}^d} \langle \mathcal{E}(L)[f](x), \eta(x) \rangle \, dx$$ where $\langle \cdot, \cdot \rangle$ denotes the standard Euclidean inner product on $\mathbb{R}^m$.

Let $L$ be a local Lagrangian of order $k$ for fields $f: \mathbb{R}^d \to \mathbb{R}^m$, and let $\eta: \mathbb{R}^d \to \mathbb{R}^m$ be an admissible variation. If the field $f$ satisfies the Euler-Lagrange equations, such that the Euler-Lagrange operator vanishes ($\mathcal{E}(L)[f] = 0$), then the first-variation value is zero: $$\int_{\mathbb{R}^d} \langle \mathcal{E}(L)[f](x), \eta(x) \rangle \, dx = 0$$ where $\langle \cdot, \cdot \rangle$ denotes the standard Euclidean inner product on $\mathbb{R}^m$.