PhyslibAlpha.ClassicalFieldTheory.Local.FirstVariation.Density

Let $L$ be a local Lagrangian of order $k$ and $f: \mathbb{R}^d \to \mathbb{R}^m$ be a field. The property `HasActionVariationDerivativeUnderIntegral` states that for any admissible variation $\eta: \mathbb{R}^d \to \mathbb{R}^m$ where the varied action density $x \mapsto L(j^k(f + s\eta)(x))$ is integrable for all $s \in \mathbb{R}$ (the `HasFiniteActionVariation` property), the derivative of the action $S(s)$ with respect to the variation parameter $s$ at $s = 0$ can be computed by differentiating under the integral sign. Mathematically, this is expressed as: $$\left. \frac{d}{ds} \right|_{s=0} \int_{\mathbb{R}^d} L(j^k(f + s\eta)(x)) \, dx = \int_{\mathbb{R}^d} \left( \left. \frac{d}{ds} \right|_{s=0} L(j^k(f + s\eta)(x)) \right) \, dx$$ where $j^k(f + s\eta)(x)$ denotes the $k$-jet of the varied field at point $x$.

For a Lagrangian $\mathcal{L}$ of order $k$ and a field $f: \mathbb{R}^d \to \mathbb{R}^m$, the property `HasPointwiseLinearizedDensityFormula` holds if for every admissible variation $\eta: \mathbb{R}^d \to \mathbb{R}^m$ and every point $x \in \mathbb{R}^d$, the derivative of the action density of the varied field $f_s = f + s\eta$ with respect to the parameter $s$ at $s=0$ is equal to the first-variation density at $x$. Mathematically, this is expressed as: $$\left. \frac{d}{ds} \right|_{s=0} \mathcal{L}(j^k(f + s\eta)(x)) = \sum_{|I| \le k} \sum_{a=1}^m \frac{\partial \mathcal{L}}{\partial u^a_I}(j^k f(x)) \cdot \partial^I \eta^a(x)$$ where $j^k f(x)$ denotes the $k$-jet of the field $f$ at $x$, and $\frac{\partial \mathcal{L}}{\partial u^a_I}$ is the partial derivative of the Lagrangian with respect to the coordinate corresponding to the $I$-th derivative of the $a$-th component of the field.

Let $L$ be a Lagrangian of order $k$, $f: \text{Space } d \to \mathbb{R}^m$ be a field, and $\eta$ be an admissible variation. Suppose there exists a radius $\varepsilon > 0$, a family of functions $F': \mathbb{R} \times \text{Space } d \to \mathbb{R}$, and a dominating integrable function $B: \text{Space } d \to \mathbb{R}$ such that the following conditions hold: 1. For $s$ in a neighborhood of $0$, the action density of the varied field $x \mapsto \mathcal{L}(j^k(f + s\eta)(x))$ is almost everywhere strongly measurable. 2. The action density of the original field $x \mapsto \mathcal{L}(j^k f(x))$ is integrable over $\text{Space } d$. 3. The function $x \mapsto F'(0, x)$ is almost everywhere strongly measurable. 4. For almost every $x \in \text{Space } d$, and for all $s$ in the ball $B(0, \varepsilon)$, the norm of the derivative density satisfies $|F'(s, x)| \le B(x)$. 5. For almost every $x \in \text{Space } d$, and for all $s \in B(0, \varepsilon)$, the derivative of the action density with respect to the variation parameter, $\frac{d}{dr} \mathcal{L}(j^k(f + r\eta)(x)) \big|_{r=s}$, exists and equals $F'(s, x)$. Then the action variation function $s \mapsto S(L, f + s\eta) = \int_{\text{Space } d} \mathcal{L}(j^k(f + s\eta)(x)) \, dx$ is differentiable at $s=0$, and its derivative is given by the integral of the pointwise derivative: $$\left. \frac{d}{ds} \right|_{s=0} S(L, f + s\eta) = \int_{\text{Space } d} F'(0, x) \, dx.$$

For a Lagrangian $\mathcal{L}$ of order $k$, a field configuration $f: \text{Space}(d) \to \mathbb{R}^m$, and an admissible variation $\eta$, the property `HasDominatedVariationDerivativeNear` holds if there exists a radius $\epsilon > 0$ and an integrable function $B: \text{Space}(d) \to \mathbb{R}$ such that: 1. The first-variation density of $f$ along $\eta$, given by $$\delta \mathcal{L}(f, \eta)(x) = \sum_{|I| \le k} \sum_{a=1}^m \frac{\partial \mathcal{L}}{\partial u^a_I}(j^k f(x)) \cdot \partial^I \eta^a(x),$$ is almost everywhere strongly measurable. 2. For almost every $x \in \text{Space}(d)$ and for all $s$ in the interval $(-\epsilon, \epsilon)$, the norm of the first-variation density of the varied field $f_s = f + s\eta$ is bounded by $B(x)$: $$\|\delta \mathcal{L}(f + s\eta, \eta)(x)\| \le B(x).$$ This condition provides the analytic domination required to differentiate the action integral with respect to the variation parameter $s$ at $s=0$.

Let $\mathcal{L}$ be a Lagrangian of order $k$, $f: \text{Space } d \to \mathbb{R}^m$ be a field configuration, and $\eta$ be an admissible variation. Suppose that for every multi-index $I$ with $|I| \le k$ and every field component $a \in \{1, \dots, m\}$, the map $(s, x) \mapsto \frac{\partial \mathcal{L}}{\partial u^a_I}(j^k (f + s\eta)(x))$ is continuous on $\mathbb{R} \times \text{Space } d$. Then $\mathcal{L}$ satisfies the dominated variation derivative property near $s = 0$ for $f$ along $\eta$. Specifically, there exists an $\epsilon > 0$ and an integrable majorant $B: \text{Space } d \to \mathbb{R}$ such that for all $s \in (-\epsilon, \epsilon)$, the first-variation density is bounded as $\|\delta \mathcal{L}(f + s\eta, \eta)(x)\| \le B(x)$ for almost every $x$.

Let $\mathcal{L}$ be a local Lagrangian of order $k$, $f: \mathbb{R}^d \to \mathbb{R}^m$ be a smooth field configuration ($C^\infty$), and $\eta: \mathbb{R}^d \to \mathbb{R}^m$ be an admissible variation. For any point $x \in \mathbb{R}^d$, the derivative with respect to the scalar parameter $s$ of the action density of the varied field $f + s\eta$ at $s = 0$ is given by the first-variation density: $$\left. \frac{d}{ds} \right|_{s=0} \mathcal{L}(j^k(f + s\eta)(x)) = \sum_{|I| \le k} \sum_{a=1}^m \frac{\partial \mathcal{L}}{\partial u^a_I}(j^k f(x)) \cdot \partial^I \eta^a(x)$$ where $j^k f(x)$ denotes the $k$-jet of the field $f$ at $x$, $I$ ranges over multi-indices with $|I| \le k$, and $a$ ranges over the field components.

Let $\mathcal{L}$ be a Lagrangian of order $k$ and $f: \mathbb{R}^d \to \mathbb{R}^m$ be a smooth ($C^\infty$) field. For any admissible variation $\eta: \mathbb{R}^d \to \mathbb{R}^m$, any point $x \in \mathbb{R}^d$, and any scalar $s \in \mathbb{R}$, the derivative of the action density of the varied field $f + r\eta$ with respect to $r$ at $r = s$ is given by the first-variation density: $$\frac{d}{dr} \mathcal{L}(j^k(f + r\eta)(x)) \Big|_{r=s} = \sum_{|I| \le k} \sum_{a=1}^m \frac{\partial \mathcal{L}}{\partial u^a_I}(j^k(f + s\eta)(x)) \cdot \partial^I \eta^a(x)$$ where $j^k(f + s\eta)(x)$ denotes the $k$-jet of the varied field $f + s\eta$ at $x$, and $\partial^I \eta^a(x)$ is the $I$-th partial derivative of the $a$-th component of the variation $\eta$ at $x$.

Let $\mathcal{L}$ be a Lagrangian of order $k$ and $f: \mathbb{R}^d \to \mathbb{R}^m$ be a smooth ($C^\infty$) field configuration. Then the pointwise linearized density formula holds for $f$. That is, for every admissible variation $\eta: \mathbb{R}^d \to \mathbb{R}^m$ and every point $x \in \mathbb{R}^d$, the derivative of the action density of the varied field $f + s\eta$ with respect to the parameter $s$ at $s=0$ is given by $$\left. \frac{d}{ds} \right|_{s=0} \mathcal{L}(j^k(f + s\eta)(x)) = \sum_{|I| \le k} \sum_{a=1}^m \frac{\partial \mathcal{L}}{\partial u^a_I}(j^k f(x)) \cdot \partial^I \eta^a(x)$$ where $j^k f(x)$ denotes the $k$-jet of the field $f$ at $x$, and $\frac{\partial \mathcal{L}}{\partial u^a_I}$ is the partial derivative of the Lagrangian with respect to the coordinate corresponding to the $I$-th derivative of the $a$-th component of the field.

Let $L$ be a Lagrangian of order $k$ and $f: \mathbb{R}^d \to \mathbb{R}^m$ be a smooth ($C^\infty$) field configuration. Suppose that for every admissible variation $\eta: \mathbb{R}^d \to \mathbb{R}^m$, if the varied action density $x \mapsto L(j^k(f + s\eta)(x))$ is integrable for all $s \in \mathbb{R}$, then the first-variation density satisfies a domination condition near $s=0$. Specifically, there exists a neighborhood $(-\epsilon, \epsilon)$ and an integrable function $B: \mathbb{R}^d \to \mathbb{R}$ such that for almost every $x$ and all $s \in (-\epsilon, \epsilon)$, the norm of the first-variation density of the varied field is bounded by $B(x)$. Under these conditions, the derivative of the action with respect to the variation parameter $s$ at $s=0$ can be computed by differentiating under the integral sign: $$\left. \frac{d}{ds} \right|_{s=0} \int_{\mathbb{R}^d} L(j^k(f + s\eta)(x)) \, dx = \int_{\mathbb{R}^d} \left( \left. \frac{d}{ds} \right|_{s=0} L(j^k(f + s\eta)(x)) \right) \, dx.$$

Let $\mathcal{L}$ be a Lagrangian of order $k$ and $f: \mathbb{R}^d \to \mathbb{R}^m$ be a smooth ($C^\infty$) field configuration. Suppose that for every admissible variation $\eta: \mathbb{R}^d \to \mathbb{R}^m$, every multi-index $I$ with $|I| \le k$, and every component $a \in \{1, \dots, m\}$, the map $$(s, x) \mapsto \frac{\partial \mathcal{L}}{\partial u^a_I}(j^k(f + s\eta)(x))$$ is continuous on $\mathbb{R} \times \mathbb{R}^d$, where $j^k(f + s\eta)(x)$ denotes the $k$-jet of the varied field $f + s\eta$ at the point $x$. Then, the derivative of the action with respect to the variation parameter $s$ at $s=0$ can be computed by differentiating under the integral sign: $$\left. \frac{d}{ds} \right|_{s=0} \int_{\mathbb{R}^d} \mathcal{L}(j^k(f + s\eta)(x)) \, dx = \int_{\mathbb{R}^d} \left( \left. \frac{d}{ds} \right|_{s=0} \mathcal{L}(j^k(f + s\eta)(x)) \right) \, dx.$$

Let $L$ be a local Lagrangian of order $k$ and $f: \mathbb{R}^d \to \mathbb{R}^m$ be a smooth ($C^\infty$) field. Suppose that for every admissible variation $\eta: \mathbb{R}^d \to \mathbb{R}^m$, the coordinate derivatives of the Lagrangian are continuous along the family of varied fields $f_s = f + s\eta$. Specifically, for every multi-index $I$ with $|I| \le k$ and every field component index $a \in \{0, \dots, m-1\}$, the mapping $$(s, x) \mapsto \frac{\partial L}{\partial u^a_I}(j^k(f + s\eta)(x))$$ is continuous as a function from $\mathbb{R} \times \mathbb{R}^d$ to $\mathbb{R}$, where $j^k$ denotes the $k$-jet of the field. Then, the derivative of the action with respect to the variation parameter $s$ at $s=0$ can be computed by differentiating under the integral sign: $$\left. \frac{d}{ds} \right|_{s=0} \int_{\mathbb{R}^d} L(j^k(f + s\eta)(x)) \, dx = \int_{\mathbb{R}^d} \left( \left. \frac{d}{ds} \right|_{s=0} L(j^k(f + s\eta)(x)) \right) \, dx.$$