PhyslibAlpha.ClassicalFieldTheory.Local.FirstVariation.IntegrationByParts

For a Lagrangian $L$ of order $k$ on $\mathbb{R}^d$ and a field configuration $f: \mathbb{R}^d \to \mathbb{R}^m$, the property `HasIntegratedByPartsFormula` holds if for every admissible variation $\eta: \mathbb{R}^d \to \mathbb{R}^m$, the integral of the first-variation density is equal to the first-variation value. This is expressed as: $$\int_{\mathbb{R}^d} \sum_{|I| \le k} \sum_{a=1}^m \frac{\partial \mathcal{L}}{\partial u^a_I}(j^k f(x)) \cdot \partial^I \eta^a(x) \, dx = \int_{\mathbb{R}^d} \langle \mathcal{E}(L)[f](x), \eta(x) \rangle \, dx$$ where $\mathcal{E}(L)[f]$ is the Euler-Lagrange operator and $\langle \cdot, \cdot \rangle$ denotes the standard Euclidean inner product on $\mathbb{R}^m$. This formula represents the step in the first-variation formula where integration by parts is applied to relate the linearized Lagrangian density to the Euler-Lagrange equations.

For a Lagrangian $L$ of order $k$ on $\mathbb{R}^d$ and a field configuration $f: \mathbb{R}^d \to \mathbb{R}^m$, the property `HasTermwiseIntegratedByPartsFormula` holds if for every admissible variation $\eta: \mathbb{R}^d \to \mathbb{R}^m$, every multi-index $I$ with $|I| \leq k$, and every field component index $a \in \{0, \dots, m-1\}$, the following conditions are satisfied: 1. The term $\frac{\partial \mathcal{L}}{\partial u^a_I}(j^k f(x)) \cdot \partial^I \eta^a(x)$ is integrable over $\mathbb{R}^d$. 2. The term $(-1)^{|I|} D_I \left( \frac{\partial \mathcal{L}}{\partial u^a_I} \right)[f](x) \cdot \eta^a(x)$ is integrable over $\mathbb{R}^d$. 3. The integral of the first term over $\mathbb{R}^d$ is equal to the integral of the second term: $$\int_{\mathbb{R}^d} \frac{\partial \mathcal{L}}{\partial u^a_I}(j^k f(x)) \cdot \partial^I \eta^a(x) \, dx = \int_{\mathbb{R}^d} (-1)^{|I|} D_I \left( \frac{\partial \mathcal{L}}{\partial u^a_I} \right)[f](x) \cdot \eta^a(x) \, dx$$ where $j^k f(x)$ is the $k$-jet of the field $f$ at $x$, $\partial^I \eta^a$ is the iterated partial derivative of the $a$-th component of the variation, and $D_I$ is the iterated total derivative.

Let $d$ be a natural number and let $\text{Space } d$ be a $d$-dimensional real inner product space (isomorphic to $\mathbb{R}^d$). For any multi-index $I$ of dimension $d$ with order $|I|$, a smooth function $g: \text{Space } d \to \mathbb{R}$, and a test function $h: \text{Space } d \to \mathbb{R}$ (i.e., a smooth function with compact support), the following identity for repeated integration by parts holds: $$\int g(x) \partial^I h(x) \, dx = (-1)^{|I|} \int (\partial^I g(x)) h(x) \, dx$$ where $\partial^I$ denotes the iterated partial derivative $\frac{\partial^{|I|}}{\partial x_0^{I_0} \partial x_1^{I_1} \cdots \partial x_{d-1}^{I_{d-1}}}$.

Let $L$ be a Lagrangian of order $k$ on $\mathbb{R}^d$ with $m$ field components, and let $f: \mathbb{R}^d \to \mathbb{R}^m$ be a field configuration. Suppose that for every multi-index $I$ with $|I| \leq k$ and every field component index $a \in \{0, \dots, m-1\}$, the coefficient function defined by the partial derivative of the Lagrangian density $\mathcal{L}$ with respect to the jet coordinate $u^a_I$, evaluated along the field $f$, is smooth ($C^\infty$): $$x \mapsto \frac{\partial \mathcal{L}}{\partial u^a_I}(j^k f(x)) \in C^\infty(\mathbb{R}^d, \mathbb{R})$$ Then the termwise integration-by-parts formula holds for $L$ and $f$. Specifically, for any test function $\eta^a$ (a smooth function with compact support), the following identity holds: $$\int_{\mathbb{R}^d} \frac{\partial \mathcal{L}}{\partial u^a_I}(j^k f(x)) \partial^I \eta^a(x) \, dx = \int_{\mathbb{R}^d} (-1)^{|I|} \left[ \partial^I \left( \frac{\partial \mathcal{L}}{\partial u^a_I} \circ j^k f \right)(x) \right] \eta^a(x) \, dx$$ where $j^k f(x)$ is the $k$-jet of the field $f$ at $x$, and $\partial^I$ denotes the iterated partial derivative $\frac{\partial^{|I|}}{\partial x_1^{I_1} \dots \partial x_d^{I_d}}$.

Let $L$ be a $k$-th order Lagrangian on $\mathbb{R}^d$ and $f: \mathbb{R}^d \to \mathbb{R}^m$ be a field configuration. If the coordinate derivatives of the Lagrangian along the field $f$ are smooth—meaning that for every multi-index $I$ with $|I| \le k$ and every field component $a$, the mapping $x \mapsto \frac{\partial L}{\partial u_I^a}(j^k f(x))$ is $C^\infty$—then the termwise integrated-by-parts formula holds for $L$ and $f$. Specifically, for any admissible variation (test function) $\eta: \mathbb{R}^d \to \mathbb{R}^m$, the following identity is satisfied for each $I$ and $a$: $$\int_{\mathbb{R}^d} \frac{\partial L}{\partial u^a_I}(j^k f(x)) \cdot \partial^I \eta^a(x) \, dx = \int_{\mathbb{R}^d} (-1)^{|I|} D_I \left( \frac{\partial L}{\partial u^a_I} \right)[f](x) \cdot \eta^a(x) \, dx$$ where $j^k f(x)$ is the $k$-jet of $f$, $\partial^I$ is the iterated partial derivative, and $D_I$ is the iterated total derivative.

For a Lagrangian $L$ of order $k$ on $\mathbb{R}^d$ and a field configuration $f: \mathbb{R}^d \to \mathbb{R}^m$, if the termwise integration-by-parts formula holds for all multi-indices $I$ (with $|I| \le k$) and all field components $a$, then the integrated first-variation formula holds. Specifically, if for every admissible variation $\eta: \mathbb{R}^d \to \mathbb{R}^m$, every multi-index $I$ with $|I| \leq k$, and every field component index $a$, we have: $$\int_{\mathbb{R}^d} \frac{\partial \mathcal{L}}{\partial u^a_I}(j^k f(x)) \cdot \partial^I \eta^a(x) \, dx = \int_{\mathbb{R}^d} (-1)^{|I|} D_I \left( \frac{\partial \mathcal{L}}{\partial u^a_I} \right)[f](x) \cdot \eta^a(x) \, dx$$ then the sum over all terms satisfies the first-variation identity: $$\int_{\mathbb{R}^d} \sum_{|I| \le k} \sum_{a=1}^m \frac{\partial \mathcal{L}}{\partial u^a_I}(j^k f(x)) \cdot \partial^I \eta^a(x) \, dx = \int_{\mathbb{R}^d} \langle \mathcal{E}(L)[f](x), \eta(x) \rangle \, dx$$ where $\mathcal{E}(L)[f]$ is the Euler-Lagrange operator and $\langle \cdot, \cdot \rangle$ denotes the standard Euclidean inner product on $\mathbb{R}^m$.