Physlib.Electromagnetism.Dynamics.KineticTerm

Given an electromagnetic potential $A$ in $d$ spatial dimensions and a free space $\mathcal{F}$ with magnetic permeability $\mu_0$, the kinetic term at a spacetime point $x$ is defined as the scalar value: $$\mathcal{L}_{\text{kinetic}}(x) = -\frac{1}{4\mu_0} F_{\mu\nu}(x) F^{\mu\nu}(x)$$ where $F_{\mu\nu}$ is the electromagnetic field strength tensor (Faraday tensor) derived from the potential $A$, and the indices are contracted using the Minkowski metric $\eta$.

Let $d$ be the number of spatial dimensions and $\mathcal{F}$ be a free space with magnetic permeability $\mu_0$. For a differentiable electromagnetic potential $A$, the kinetic term at a spacetime point $x$ is defined as $\mathcal{L}_{\text{kinetic}}(A, x) = -\frac{1}{4\mu_0} F_{\mu\nu}(x) F^{\mu\nu}(x)$. For any Lorentz transformation $\Lambda$, the kinetic term is equivariant such that: $$\mathcal{L}_{\text{kinetic}}(\Lambda \cdot A(\Lambda^{-1} \cdot x), x) = \mathcal{L}_{\text{kinetic}}(A, \Lambda^{-1} \cdot x)$$ where $\Lambda \cdot A$ denotes the Lorentz action on the potential vector field.

For an electromagnetic potential $A$ in $d$ spatial dimensions and a free space $\mathcal{F}$ with magnetic permeability $\mu_0$, the kinetic term at a spacetime point $x$ is given by the following summation over spacetime indices $\mu, \nu, \mu', \nu' \in \{0, 1, \dots, d\}$: $$\mathcal{L}_{\text{kinetic}}(x) = -\frac{1}{4\mu_0} \sum_{\mu, \nu, \mu', \nu'} \eta_{\mu \mu'} \eta_{\nu \nu'} F_{\mu\nu}(x) F_{\mu'\nu'}(x)$$ where $F_{\mu\nu}(x)$ are the components of the electromagnetic field strength tensor (Faraday tensor) at point $x$ with respect to the standard basis, and $\eta$ represents the components of the Minkowski metric.

For an electromagnetic potential $A$ in $d$ spatial dimensions and a free space $\mathcal{F}$ with magnetic permeability $\mu_0$, the kinetic term at a spacetime point $x$ is given by the following summation over spacetime indices $\mu, \nu, \mu', \nu' \in \{0, 1, \dots, d\}$: $$\mathcal{L}_{\text{kinetic}}(x) = -\frac{1}{4\mu_0} \sum_{\mu, \nu, \mu', \nu'} \eta_{\mu \mu'} \eta_{\nu \nu'} F_{\mu\nu}(x) F_{\mu'\nu'}(x)$$ where $F_{\mu\nu}(x)$ are the components of the field strength matrix (Faraday tensor) at point $x$, and $\eta$ represents the components of the Minkowski metric.

For an electromagnetic potential $A$ in $d$ spatial dimensions and a free space $\mathcal{F}$ with magnetic permeability $\mu_0$, the kinetic term at a spacetime point $x$ is given by the following summation over the diagonal components of the Minkowski metric and the squared components of the field strength matrix: $$\mathcal{L}_{\text{kinetic}}(x) = -\frac{1}{4\mu_0} \sum_{\mu, \nu} \eta_{\mu\mu} \eta_{\nu\nu} (F_{\mu\nu}(x))^2$$ where $F_{\mu\nu}(x)$ are the components of the field strength matrix (Faraday tensor) at point $x$, and $\eta_{\mu\mu}$ are the diagonal components of the Minkowski metric.

For an electromagnetic potential $A$ in a spacetime with $d$ spatial dimensions and a free space $\mathcal{F}$ with magnetic permeability $\mu_0$, the kinetic term at a spacetime point $x$ is given by the following summation over spacetime indices $\mu, \nu \in \{0, 1, \dots, d\}$: $$\mathcal{L}_{\text{kinetic}}(x) = -\frac{1}{2\mu_0} \sum_{\mu, \nu} \left( \eta_{\mu\mu} \eta_{\nu\nu} (\partial_\mu A_\nu(x))^2 - \partial_\mu A_\nu(x) \partial_\nu A_\mu(x) \right)$$ where $\eta_{\mu\mu}$ are the diagonal components of the Minkowski metric, and $\partial_\mu A_\nu(x)$ denotes the partial derivative of the $\nu$-th component of the potential $A$ with respect to the $\mu$-th coordinate evaluated at point $x$.

Let $\mathcal{F}$ be a free space characterized by vacuum permittivity $\epsilon_0$, magnetic permeability $\mu_0$, and speed of light $c$. Given a differentiable electromagnetic potential $A$, the kinetic term of the electromagnetic Lagrangian at a spacetime point corresponding to time $t$ and spatial position $x$ is given by: $$\mathcal{L}_{\text{kinetic}}(t, x) = \frac{1}{2} \left( \epsilon_0 \|\mathbf{E}(t, x)\|^2 - \frac{1}{\mu_0} \|\mathbf{B}(t, x)\|^2 \right)$$ where $\mathbf{E}(t, x)$ and $\mathbf{B}(t, x)$ are the electric and magnetic fields derived from the potential $A$ at time $t$ and position $x$, respectively.

Let $\mathcal{F}$ be a free space characterized by vacuum permittivity $\epsilon_0$, magnetic permeability $\mu_0$, and speed of light $c$. For any differentiable electromagnetic potential $A$ and any spacetime point $x$, the kinetic term of the electromagnetic Lagrangian is given by: $$\mathcal{L}_{\text{kinetic}}(x) = \frac{1}{2} \left( \epsilon_0 \|\mathbf{E}(t, \mathbf{r})\|^2 - \frac{1}{\mu_0} \|\mathbf{B}(t, \mathbf{r})\|^2 \right)$$ where $t$ and $\mathbf{r}$ are the temporal and spatial components of $x$ (defined relative to the speed of light $c$), and $\mathbf{E}$ and $\mathbf{B}$ are the electric and magnetic fields derived from the potential $A$ at that point.

Let $\mathcal{F}$ be a free space environment characterized by vacuum permittivity $\epsilon_0$, magnetic permeability $\mu_0$, and speed of light $c$. For any differentiable electromagnetic potential $A$ in $d$ spatial dimensions, the kinetic term of the electromagnetic Lagrangian at a spacetime point $(t, x)$ is given by: $$\mathcal{L}_{\text{kinetic}}(t, x) = \frac{1}{2} \left( \epsilon_0 \|\mathbf{E}(t, x)\|^2 - \frac{1}{2\mu_0} \sum_{i,j} (B_{ij}(t, x))^2 \right)$$ where $\mathbf{E}(t, x)$ is the electric field vector and $B_{ij}(t, x)$ are the components of the magnetic field matrix at time $t$ and spatial position $x$.

Let $\mathcal{F}$ be a free space environment characterized by vacuum permittivity $\epsilon_0$, magnetic permeability $\mu_0$, and speed of light $c$. For any differentiable electromagnetic potential $A$ in $d$ spatial dimensions, the kinetic term of the electromagnetic Lagrangian at a spacetime point $x$ is given by: $$\mathcal{L}_{\text{kinetic}}(x) = \frac{1}{2} \left( \epsilon_0 \|\mathbf{E}(t, \mathbf{r})\|^2 - \frac{1}{2\mu_0} \sum_{i,j} (B_{ij}(t, \mathbf{r}))^2 \right)$$ where $t$ and $\mathbf{r}$ are the temporal and spatial components of $x$, $\mathbf{E}$ is the electric field vector, and $B_{ij}$ are the components of the magnetic field matrix at that point.

For any constant electromagnetic potential $A: \text{SpaceTime}_d \to \text{Vector}_d$ defined by $A(x) = A_0$ for some fixed Lorentz vector $A_0$, the kinetic term $\mathcal{L}_{\text{kinetic}}(x)$ is zero for all spacetime points $x$ and any free space environment $\mathcal{F}$: $$\mathcal{L}_{\text{kinetic}}(x) = 0$$

For an electromagnetic potential $A$ in $d$ spatial dimensions and a constant Lorentz vector $A_0$, the kinetic term of the shifted potential $x \mapsto A(x) + A_0$ is equal to the kinetic term of the original potential $A$. That is, for a given free space $\mathcal{F}$, $$\mathcal{L}_{\text{kinetic}}[A + A_0] = \mathcal{L}_{\text{kinetic}}[A]$$ where $\mathcal{L}_{\text{kinetic}}$ is defined as $-\frac{1}{4\mu_0} F_{\mu\nu} F^{\mu\nu}$.

For an electromagnetic potential $A$ in $d$ spatial dimensions and a free space configuration $\mathcal{F}$ with magnetic permeability $\mu_0$, if $A$ is of class $C^{n+1}$ (continuously differentiable $n+1$ times) for some $n \in \mathbb{N} \cup \{\infty\}$, then the associated electromagnetic kinetic term $\mathcal{L}_{\text{kinetic}}(x) = -\frac{1}{4\mu_0} F_{\mu\nu}(x) F^{\mu\nu}(x)$ is of class $C^n$.

For an electromagnetic potential $A$ in $d$ spatial dimensions and a free space $\mathcal{F}$ with magnetic permeability $\mu_0$, let $c$ be a constant Lorentz vector. If $A$ is differentiable, then at any spacetime point $x$ with time coordinate $x^0$, the kinetic term $\mathcal{L}_{\text{kinetic}}$ of the shifted potential $A'(x) = A(x) + x^0 c$ is given by: $$\mathcal{L}_{\text{kinetic}}(A + x^0 c)(x) = \mathcal{L}_{\text{kinetic}}(A)(x) - \frac{1}{2\mu_0} \left[ \sum_{\nu} \left( 2 c_\nu \eta_{\nu\nu} \partial_0 A_\nu(x) + \eta_{\nu\nu} c_\nu^2 - 2 c_\nu \partial_\nu A_0(x) \right) - c_0^2 \right]$$ where $\eta_{\nu\nu}$ are the diagonal components of the Minkowski metric, $c_\nu$ and $A_\nu$ are the components of $c$ and $A$ respectively, $\partial_\mu$ denotes the partial derivative with respect to the $\mu$-th coordinate, and the index $0$ refers to the temporal component.

For an electromagnetic potential $A$ in $d$ spatial dimensions and a free space configuration $\mathcal{F}$ (characterized by magnetic permeability $\mu_0$), the variational gradient of the kinetic term is the functional derivative of the action integral $S[A] = \int \mathcal{L}_{\text{kinetic}}(x) \, dx$ with respect to the potential $A$. Here, the kinetic term is defined as $\mathcal{L}_{\text{kinetic}} = -\frac{1}{4\mu_0} F_{\mu\nu} F^{\mu\nu}$, where $F_{\mu\nu}$ is the electromagnetic field strength tensor. The resulting value at each spacetime point $x$ is a Lorentz vector, which corresponds to the expression $\frac{1}{\mu_0} \partial_\mu F^{\mu\nu}$.

For an infinitely differentiable ($C^\infty$) electromagnetic potential $A$ in $d$ spatial dimensions and a free space $\mathcal{F}$ with magnetic permeability $\mu_0$, the variational gradient of the kinetic term at a spacetime point $x$ can be expressed as a sum over spacetime indices $\mu, \nu \in \{0, 1, \dots, d\}$: $$\text{gradKineticTerm}(x) = \sum_{\mu, \nu} F'_{\mu\nu}(\psi, x)$$ where $\psi$ is the constant function $\psi(x) = -\frac{1}{2\mu_0}$ and $F'_{\mu\nu}(\psi, x)$ is defined by the following expression involving Fréchet derivatives (which correspond to partial derivatives $\partial_\mu$ when evaluated along the standard basis $e_\mu$): $$F'_{\mu\nu}(\psi, x) = -2 \partial_\mu \left( \eta_{\mu\mu} \eta_{\nu\nu} \psi \partial_\mu A_\nu \right) e_\nu + \partial_\mu \left( \psi \partial_\nu A_\mu \right) e_\nu + \partial_\nu \left( \psi \partial_\mu A_\nu \right) e_\mu$$ Here, $\eta_{\mu\mu}$ denotes the diagonal components of the Minkowski metric, and $e_\mu, e_\nu$ are the standard basis vectors of the Lorentz vector space.

For an infinitely differentiable ($C^\infty$) electromagnetic potential $A$ in $d$ spatial dimensions and a free space configuration $\mathcal{F}$ with magnetic permeability $\mu_0$, the variational gradient of the kinetic term at a spacetime point $x$ is given by the following double sum over spacetime indices $\mu, \nu \in \{0, 1, \dots, d\}$: $$\text{gradKineticTerm}(x) = \sum_{\nu} \sum_{\mu} \frac{1}{\mu_0} \left( \eta_{\mu\mu} \eta_{\nu\nu} \frac{\partial^2 A_\nu(x)}{\partial (x^\mu)^2} - \frac{\partial^2 A_\mu(x)}{\partial x^\mu \partial x^\nu} \right) e_\nu$$ where $\eta_{\mu\mu}$ and $\eta_{\nu\nu}$ are the diagonal components of the Minkowski metric, $\partial_\mu$ denotes the partial derivative with respect to the $\mu$-th spacetime coordinate, and $e_\nu$ are the standard basis vectors of the Lorentz vector space.

For an infinitely differentiable ($C^\infty$) electromagnetic potential $A$ in $d$ spatial dimensions and a free space configuration $\mathcal{F}$ with magnetic permeability $\mu_0$, the variational gradient of the kinetic term at a spacetime point $x$ is given by: $$\text{gradKineticTerm}(x) = \sum_{\nu} \left( \frac{\eta_{\nu\nu}}{\mu_0} \sum_{\mu} \partial_\mu F_{\mu\nu}(x) \right) e_\nu$$ where $F_{\mu\nu}$ are the components of the field strength matrix (the electromagnetic field strength tensor), $\eta_{\nu\nu}$ are the diagonal entries of the Minkowski metric, $\partial_\mu$ denotes the partial derivative with respect to the $\mu$-th spacetime coordinate, and $e_\nu$ are the standard basis vectors of the Lorentz vector space.

For an infinitely differentiable ($C^\infty$) electromagnetic potential $A$ in $d$ spatial dimensions and a free space configuration $\mathcal{F}$ (with magnetic permeability $\mu_0$ and speed of light $c$), the variational gradient of the kinetic term at a spacetime point $x$ (with time component $t$ and spatial component $\mathbf{x}$) is given by: $$\text{gradKineticTerm}(x) = \left( \frac{1}{\mu_0 c} \nabla \cdot \mathbf{E}(t, \mathbf{x}) \right) e_0 + \sum_i \frac{1}{\mu_0} \left( \frac{1}{c^2} \frac{\partial E_i(t, \mathbf{x})}{\partial t} - \sum_j \frac{\partial M_{ji}(t, \mathbf{x})}{\partial x_j} \right) e_i$$ where $\mathbf{E}$ is the electric field, $M$ is the magnetic field matrix, $\nabla \cdot \mathbf{E}$ is the spatial divergence of the electric field, $\frac{\partial}{\partial t}$ is the partial derivative with respect to time, $\frac{\partial}{\partial x_j}$ is the partial derivative with respect to the $j$-th spatial coordinate, and $\{e_0, e_i\}$ are the standard basis vectors of the Lorentz vector space.

For an infinitely differentiable ($C^\infty$) electromagnetic potential $A$ in 3 spatial dimensions and a free space configuration $\mathcal{F}$ (with magnetic permeability $\mu_0$ and speed of light $c$), the variational gradient of the kinetic term at a spacetime point $x$ (with time component $t$ and spatial component $\mathbf{x}$) is given by: $$\text{gradKineticTerm}(x) = \left( \frac{1}{\mu_0 c} \nabla \cdot \mathbf{E}(t, \mathbf{x}) \right) e_0 + \sum_i \frac{1}{\mu_0} \left( \frac{1}{c^2} \frac{\partial E_i(t, \mathbf{x})}{\partial t} - (\nabla \times \mathbf{B}(t, \mathbf{x}))_i \right) e_i$$ where $\mathbf{E}$ is the electric field, $\mathbf{B}$ is the magnetic field, $\nabla \cdot \mathbf{E}$ is the spatial divergence of the electric field, $\nabla \times \mathbf{B}$ is the spatial curl of the magnetic field, $\frac{\partial}{\partial t}$ is the partial derivative with respect to time, and $\{e_0, e_i\}$ are the standard basis vectors of the Lorentz vector space.

For any spatial dimension $d$ and free space configuration $\mathcal{F}$ (characterized by magnetic permeability $\mu_0$), let $A_1$ and $A_2$ be infinitely differentiable ($C^\infty$) electromagnetic potentials. The variational gradient of the electromagnetic kinetic term satisfies the additivity property: $$\text{gradKineticTerm}(A_1 + A_2) = \text{gradKineticTerm}(A_1) + \text{gradKineticTerm}(A_2)$$ where $\text{gradKineticTerm}(A)$ corresponds to the expression $\frac{1}{\mu_0} \partial_\mu F^{\mu\nu}$ derived from the kinetic Lagrangian $\mathcal{L}_{\text{kinetic}} = -\frac{1}{4\mu_0} F_{\mu\nu} F^{\mu\nu}$.

For an infinitely differentiable ($C^\infty$) electromagnetic potential $A$ in a spacetime with $d$ spatial dimensions and a free space configuration $\mathcal{F}$ (with magnetic permeability $\mu_0$), the variational gradient of the kinetic term is homogeneous with respect to scalar multiplication. That is, for any real scalar $c \in \mathbb{R}$, the variational gradient of the scaled potential $c \cdot A$ is given by: $$\text{gradKineticTerm}(c \cdot A) = c \cdot \text{gradKineticTerm}(A)$$ where $\text{gradKineticTerm}$ corresponds to the expression $\frac{1}{\mu_0} \partial_\mu F^{\mu\nu}$.

For an infinitely differentiable ($C^\infty$) electromagnetic potential $A$ in a spacetime with $d$ spatial dimensions and a free space configuration $\mathcal{F}$ (with magnetic permeability $\mu_0$), the functional defined by the electromagnetic kinetic term $$\mathcal{L}_{\text{kinetic}}(A) = -\frac{1}{4\mu_0} F_{\mu\nu} F^{\mu\nu}$$ possesses a variational gradient at $A$. This variational gradient is given by the term $\text{gradKineticTerm}(A)$, which corresponds to the expression $\frac{1}{\mu_0} \partial_\mu F^{\mu\nu}$.

For an infinitely differentiable ($C^\infty$) electromagnetic potential $A$ in $d$ spatial dimensions and a free space configuration $\mathcal{F}$ with vacuum permeability $\mu_0$, the $\nu$-th component of the variational gradient of the kinetic term at a spacetime point $x$ is given by: $$(\text{gradKineticTerm}(A, x))_\nu = \eta_{\nu\nu} \left( \frac{1}{\mu_0} \sum_{\mu} \partial_\mu F_{\mu\nu}(x) \right)$$ where $F_{\mu\nu}$ are the components of the electromagnetic field strength tensor, $\eta_{\nu\nu}$ are the diagonal entries of the Minkowski metric, and $\partial_\mu F_{\mu\nu}$ denotes the contraction of the tensor derivative of the field strength matrix with respect to the derivative index and the first tensor index.

For a given spatial dimension $d$ and a physical environment $\mathcal{F}$ with vacuum permeability $\mu_0$, this definition provides the variational gradient of the kinetic term for an electromagnetic potential $A$ that is a distribution. The result is a linear map from the space of distributional electromagnetic potentials to vector-valued distributions on spacetime. Specifically, for a distributional potential $A$ and a test function $\varepsilon$ (a Schwartz function on spacetime), the variational gradient is the Lorentz vector given by the formula: $$\sum_{\nu} \left( \sum_{\mu} \frac{1}{\mu_0} \left( \eta_{\mu\mu} \eta_{\nu\nu} \partial_\mu \partial_\mu A_\nu(\varepsilon) - \partial_\mu \partial_\nu A_\mu(\varepsilon) \right) \right) \mathbf{e}_\nu$$ where: - $\eta$ is the Minkowski matrix with diagonal entries $\eta_{\mu\mu}$. - $\partial_\mu$ denotes the distributional derivative with respect to the $\mu$-th spacetime coordinate. - $A_\nu$ is the $\nu$-th component of the potential $A$. - $\mathbf{e}_\nu$ is the standard basis vector for the space of Lorentz vectors $\text{Vector}_d$.

Let $d$ be the spatial dimension and $\mathcal{F}$ be a physical environment with vacuum permeability $\mu_0$. For a distributional electromagnetic potential $A$ and a Schwartz test function $\varepsilon \in \mathcal{S}(\mathbb{R}^{1+d}, \mathbb{R})$, the variational gradient of the kinetic term evaluated at $\varepsilon$ is given by the following expression: $$\text{gradKineticTerm}(A)(\varepsilon) = \sum_{\nu} \left( \sum_{\mu} \frac{1}{\mu_0} \left( \eta_{\mu\mu} \eta_{\nu\nu} (\partial_\mu \partial_\mu A_\nu)(\varepsilon) - (\partial_\mu \partial_\nu A_\mu)(\varepsilon) \right) \right) \mathbf{e}_\nu$$ where: - $\eta$ is the Minkowski matrix with diagonal entries $\eta_{\mu\mu} = \mathrm{diag}(1, -1, \dots, -1)$. - $\partial_\mu$ denotes the distributional derivative with respect to the $\mu$-th spacetime coordinate. - $A_\nu$ is the $\nu$-th component of the potential distribution $A$. - $\mathbf{e}_\nu$ is the $\nu$-th standard basis vector of the space of Lorentz vectors.

Let $d$ be the spatial dimension and $\mathcal{F}$ be a free space environment with vacuum permeability $\mu_0$. For a distributional electromagnetic potential $A$ and a Schwartz test function $\varepsilon \in \mathcal{S}(\mathbb{R}^{1+d}, \mathbb{R})$, the variational gradient of the kinetic term evaluated at $\varepsilon$ is given by: $$\text{gradKineticTerm}(A)(\varepsilon) = \sum_{\nu} \left[ \frac{1}{\mu_0} \eta_{\nu\nu} \sum_{\mu} \left( \partial_\mu F \right)_{\mu\nu} (\varepsilon) \right] \mathbf{e}_\nu$$ where: - $F$ is the field strength tensor (Faraday tensor) associated with the potential $A$. - $\eta$ is the Minkowski matrix with diagonal entries $\eta_{\nu\nu}$. - $\partial_\mu$ denotes the distributional derivative with respect to the $\mu$-th spacetime coordinate. - $(\partial_\mu F)_{\mu\nu}(\varepsilon)$ denotes the $(\mu, \nu)$-th component of the distribution $\partial_\mu F$ evaluated at the test function $\varepsilon$. - $\mathbf{e}_\nu$ is the $\nu$-th standard basis vector of the space of Lorentz vectors.

Let $d$ be the spatial dimension and $\mathcal{F}$ be a free space environment with vacuum permeability $\mu_0$ and speed of light $c$. For a distributional electromagnetic potential $A$ and a Schwartz test function $\varepsilon \in \mathcal{S}(\mathbb{R}^{1+d}, \mathbb{R})$, the time component (the component indexed by $0$) of the variational gradient of the kinetic term evaluated at $\varepsilon$ is equal to the distributional spatial divergence of the associated electric field $\mathbf{E}$ evaluated at $\varepsilon$, scaled by a physical constant: $$(\text{gradKineticTerm}(A)(\varepsilon))_0 = \frac{1}{\mu_0 c} (\nabla \cdot \mathbf{E})(\varepsilon)$$ where $\mathbf{E}$ is the distributional electric field associated with the potential $A$, and $\nabla \cdot \mathbf{E}$ denotes its distributional spatial divergence.

Let $d$ be the spatial dimension and $\mathcal{F}$ be a free space environment characterized by the vacuum permeability $\mu_0$ and the speed of light $c$. For a distributional electromagnetic potential $A$, a Schwartz test function $\varepsilon \in \mathcal{S}(\mathbb{R}^{1+d}, \mathbb{R})$, and a spatial index $i \in \{0, \dots, d-1\}$, the $i$-th spatial component of the variational gradient of the kinetic term evaluated at $\varepsilon$ is given by: $$\text{gradKineticTerm}(A)(\varepsilon)_i = \frac{1}{\mu_0} \left( \frac{1}{c^2} \frac{\partial E_i}{\partial t}(\varepsilon) - \sum_{j=1}^d \frac{\partial B_{ji}}{\partial x_j}(\varepsilon) \right)$$ where: - $E_i$ is the $i$-th component of the distributional electric field associated with $A$. - $B_{ji}$ is the $(j, i)$-th component of the distributional magnetic field matrix associated with $A$. - $\frac{\partial}{\partial t}$ and $\frac{\partial}{\partial x_j}$ denote the distributional temporal and spatial derivatives, respectively. - The term $(\text{distTimeSlice } c)^{-1}$ is used to appropriately map the spatial distributions to the spacetime test function $\varepsilon$.

Let $d$ be the spatial dimension and $\mathcal{F}$ be a free space environment with vacuum permeability $\mu_0$. For a distributional electromagnetic potential $A$ and a Schwartz test function $\varepsilon \in \mathcal{S}(\mathbb{R}^{1+d}, \mathbb{R})$, the $\nu$-th component of the variational gradient of the kinetic term evaluated at $\varepsilon$ is given by: $$(\text{gradKineticTerm}(A)(\varepsilon))_\nu = \frac{\eta_{\nu\nu}}{\mu_0} \left( \sum_{\mu} (\partial_\mu F)_{\mu\nu} (\varepsilon) \right)$$ where $F$ is the field strength tensor (Faraday tensor) associated with the potential $A$, $\eta_{\nu\nu}$ is the $\nu$-th diagonal entry of the Minkowski metric matrix, and $(\partial_\mu F)_{\mu\nu}(\varepsilon)$ is the $(\mu, \nu)$-th component of the distributional derivative of the field strength tensor with respect to the $\mu$-th spacetime coordinate.