Physlib.Electromagnetism.Dynamics.Lagrangian

For a spatial dimension $d$, given an electromagnetic potential field $A$, a Lorentz current density field $J$, and a spacetime point $x$, the free current potential is defined as the Minkowski inner product of $A$ and $J$ at $x$: $$\langle A(x), J(x) \rangle_m = A^0(x) J^0(x) - \sum_{i=1}^d A^i(x) J^i(x)$$ where $A^0, J^0$ denote the temporal components and $A^i, J^i$ denote the spatial components. This quantity represents the local potential energy density resulting from the interaction between the electromagnetic potential and the free current density.

For a given spatial dimension $d$, let $A$ be an electromagnetic potential field, $J$ be a Lorentz current density field, $c$ be a constant Lorentz vector, and $x$ be a point in spacetime. The free current potential at $x$ for the potential shifted by $c$ (defined by $x \mapsto A(x) + c$) is equal to the sum of the free current potential of $A$ at $x$ and the Minkowski inner product of $c$ and $J(x)$: $$\text{freeCurrentPotential}(A + c, J, x) = \text{freeCurrentPotential}(A, J, x) + \langle c, J(x) \rangle_m$$ where $\langle \cdot, \cdot \rangle_m$ denotes the Minkowski inner product.

For a spatial dimension $d$, let $A$ be an electromagnetic potential field and $J$ be a Lorentz current density field, both of which are infinitely differentiable ($C^\infty$). The functional that maps the potential $A$ to the free current potential $\langle A, J \rangle_m$ possesses a variational gradient at $A$. This gradient is the vector field that maps each spacetime point $x$ to the Lorentz vector: $$\sum_{\mu} (\eta_{\mu\mu} J^\mu(x)) \mathbf{e}_\mu$$ where $\eta$ is the Minkowski matrix $\mathrm{diag}(1, -1, \dots, -1)$, $J^\mu(x)$ are the components of the current density, and $\mathbf{e}_\mu$ are the standard basis vectors of the Lorentz vector space.

For a given spatial dimension $d$ and a free space configuration $\mathcal{F}$ with speed of light $c$, let $A$ be an electromagnetic potential field, $J$ be a Lorentz current density field, and $x$ be a point in spacetime with time $t$ and spatial position $\mathbf{x}$. The free current potential at $x$, defined by the Minkowski product $\langle A(x), J(x) \rangle_m$, is equal to the product of the scalar potential $\Phi$ and the charge density $\rho$ minus the dot product of the vector potential $\mathbf{A}$ and the current density $\mathbf{J}$: $$\langle A(x), J(x) \rangle_m = \Phi(t, \mathbf{x}) \rho(t, \mathbf{x}) - \sum_{i=1}^d A^i(t, \mathbf{x}) J^i(t, \mathbf{x})$$ where the scalar potential $\Phi$ and charge density $\rho$ correspond to the temporal components, and $A^i, J^i$ correspond to the spatial components of the respective four-vectors.

For a spacetime with spatial dimension $d$, given an electromagnetic potential field $A$ and a Lorentz current density field $J$, the variational gradient of the free current potential is the functional derivative of the integrated interaction potential with respect to $A$. It is defined as: $$\frac{\delta}{\delta A} \int \langle A(x), J(x) \rangle_m \, dx$$ where $\langle A(x), J(x) \rangle_m$ is the Minkowski inner product (the free current potential) at the spacetime point $x$. The result is a vector field that maps each spacetime point $x$ to a Lorentz vector.

For a spatial dimension $d$, let $A$ be an infinitely differentiable ($C^\infty$) electromagnetic potential field and $J$ be an infinitely differentiable Lorentz current density field. The variational gradient of the free current potential with respect to the potential $A$ is the vector field defined by: $$\frac{\delta}{\delta A} \int \langle A(x), J(x) \rangle_m \, dx = \sum_{\mu} (\eta_{\mu\mu} J^\mu) \mathbf{e}_\mu$$ where $\langle \cdot, \cdot \rangle_m$ denotes the Minkowski inner product, $\eta = \mathrm{diag}(1, -1, \dots, -1)$ is the Minkowski matrix, $J^\mu$ are the components of the current density field, and $\mathbf{e}_\mu$ are the standard basis vectors of the Lorentz vector space.

Let $d$ be the number of spatial dimensions and $\mathcal{F}$ be the physical parameters of free space, including the speed of light $c$. Let $A$ be an infinitely differentiable ($C^\infty$) electromagnetic potential field and $J$ be an infinitely differentiable Lorentz current density field. At any spacetime point $x$, the variational gradient of the free current potential with respect to $A$ (the functional derivative of the integrated interaction potential) is given by: $$\left( \frac{\delta}{\delta A} \int \langle A(x), J(x) \rangle_m \, dx \right)(x) = c \rho(x) \mathbf{e}_0 - \sum_{i=1}^d j_i(x) \mathbf{e}_i$$ where $\rho(x)$ is the charge density at $x$, $j_i(x)$ are the components of the 3-current density $\mathbf{j}(x)$ at $x$, $\mathbf{e}_0$ is the temporal basis vector, and $\mathbf{e}_i$ are the spatial basis vectors of the Lorentz vector space.

For a spatial dimension $d$, let $A$ be an infinitely differentiable electromagnetic potential field and $J$ be an infinitely differentiable Lorentz current density field. For any point in spacetime $x$ and any index $\nu \in \{0, 1, \dots, d\}$, the $\nu$-th component of the variational gradient of the free current potential with respect to $A$ is given by: $$\left( \frac{\delta}{\delta A} \int \langle A(y), J(y) \rangle_m \, dy \right)_\nu(x) = \eta_{\nu\nu} J^\nu(x)$$ where $\langle \cdot, \cdot \rangle_m$ is the Minkowski inner product and $\eta = \text{diag}(1, -1, \dots, -1)$ is the Minkowski matrix. In this identity, the $\nu$-th component of the current density $J^\nu(x)$ is evaluated by converting the vector $J(x)$ to its equivalent tensor representation.

For a spatial dimension $d$, let $\mathcal{F}$ denote the physical parameters of free space. Given an electromagnetic 4-potential field $A$, a Lorentz 4-current density field $J$, and a point in spacetime $x$, the electromagnetic Lagrangian density $\mathcal{L}(x)$ is defined as the difference between the kinetic term of the field and the interaction potential with the current: $$\mathcal{L}(x) = \mathcal{L}_{\text{kin}}(x) - \langle A(x), J(x) \rangle_m$$ where $\mathcal{L}_{\text{kin}}(x)$ is the kinetic energy density of the electromagnetic field (typically $-\frac{1}{4\mu_0} F_{\mu\nu} F^{\mu\nu}$) and $\langle A(x), J(x) \rangle_m$ is the free current potential (the Minkowski inner product $A_\mu J^\mu$).

For a spatial dimension $d$ and free space parameters $\mathcal{F}$, let $A$ be an electromagnetic potential field, $J$ be a Lorentz current density field, $c$ be a constant Lorentz vector, and $x$ be a point in spacetime. The electromagnetic Lagrangian density $\mathcal{L}$ evaluated at $x$ for the potential shifted by $c$ (defined by the field $x \mapsto A(x) + c$) is equal to the Lagrangian density for $A$ at $x$ minus the Minkowski inner product of $c$ and $J(x)$: $$\mathcal{L}(\mathcal{F}, A + c, J, x) = \mathcal{L}(\mathcal{F}, A, J, x) - \langle c, J(x) \rangle_m$$ where $\langle \cdot, \cdot \rangle_m$ denotes the Minkowski inner product.

For a spatial dimension $d$, let $\mathcal{F}$ denote the physical parameters of free space, including the vacuum permittivity $\epsilon_0$ and the vacuum permeability $\mu_0$. Given a twice-differentiable electromagnetic 4-potential field $A$, a Lorentz 4-current density field $J$, and a point in spacetime $x$ with time $t$ and spatial position $\mathbf{x}$, the electromagnetic Lagrangian density $\mathcal{L}$ at $x$ is expressed as: $$\mathcal{L}(x) = \frac{1}{2} \left( \epsilon_0 \|\mathbf{E}(t, \mathbf{x})\|^2 - \frac{1}{2\mu_0} \sum_{i,j=1}^d |B_{ij}(t, \mathbf{x})|^2 \right) - \Phi(t, \mathbf{x}) \rho(t, \mathbf{x}) + \sum_{i=1}^d A^i(t, \mathbf{x}) j^i(t, \mathbf{x})$$ where: - $\mathbf{E}$ is the electric field vector. - $B_{ij}$ are the components of the magnetic field matrix. - $\Phi$ is the scalar potential and $\rho$ is the charge density. - $A^i$ are the components of the vector potential and $j^i$ are the components of the current density.

For a spatial dimension $d$, let $\mathcal{F}$ denote the physical parameters of free space. Let $A$ be an infinitely differentiable ($C^\infty$) electromagnetic potential field and $J$ be an infinitely differentiable Lorentz current density field. The functional that maps the potential $A$ to the electromagnetic Lagrangian density $\mathcal{L} = \mathcal{L}_{\text{kin}} - \langle A, J \rangle_m$ possesses a variational gradient at $A$. This gradient is equal to the difference between the variational gradient of the kinetic term and the variational gradient of the free current potential: $$\frac{\delta \mathcal{L}}{\delta A} = \frac{\delta \mathcal{L}_{\text{kin}}}{\delta A} - \frac{\delta}{\delta A} \int \langle A(x), J(x) \rangle_m \, dx$$ where $\mathcal{L}_{\text{kin}}$ is the kinetic energy density of the field and $\langle A, J \rangle_m$ is the interaction potential with the current density.

Given a spatial dimension $d$, physical parameters of free space $\mathcal{F}$, an electromagnetic 4-potential field $A$, and a Lorentz 4-current density field $J$, this definition represents the variational gradient (functional derivative) of the action functional $S[A] = \int \mathcal{L}(x) \, dx$ with respect to the potential $A$. The resulting function maps each point $x$ in spacetime to a Lorentz vector, corresponding to the Euler–Lagrange expression for the electromagnetic field.

For a spacetime with spatial dimension $d$, let $\mathcal{F}$ be the physical parameters of free space. Let $A$ be an infinitely differentiable ($C^\infty$) electromagnetic potential field and $J$ be an infinitely differentiable Lorentz current density field. Then the variational gradient of the electromagnetic Lagrangian $\mathcal{L}$ with respect to the potential $A$ is equal to the difference between the variational gradient of the kinetic term and the variational gradient of the free current potential: $$\frac{\delta \mathcal{L}}{\delta A} = \frac{\delta \mathcal{L}_{\text{kin}}}{\delta A} - \frac{\delta}{\delta A} \int \langle A, J \rangle_m \, dx$$ where $\mathcal{L}_{\text{kin}}$ is the kinetic energy density of the electromagnetic field and $\langle A, J \rangle_m$ is the Minkowski interaction term.

Let $d$ be the spatial dimension and $\mathcal{F}$ denote the physical parameters of free space. Let $A$ be an infinitely differentiable ($C^\infty$) electromagnetic potential field and $J$ be an infinitely differentiable Lorentz current density field. The functional that maps the potential $A$ to the electromagnetic Lagrangian density $\mathcal{L}(A, J) = \mathcal{L}_{\text{kin}} - \langle A, J \rangle_m$ possesses a variational gradient at $A$. This variational gradient is given by the function $\frac{\delta S}{\delta A}$, which maps each point in spacetime to a Lorentz vector representing the Euler–Lagrange expression for the electromagnetic field.

Let $d$ be the spatial dimension and $\mathcal{F}$ be the parameters of free space with magnetic permeability $\mu_0$. For an infinitely differentiable ($C^\infty$) electromagnetic potential field $A$ and an infinitely differentiable Lorentz current density field $J$, the variational gradient of the Lagrangian density $\mathcal{L}$ with respect to $A$ at a spacetime point $x$ is given by: $$\frac{\delta \mathcal{L}}{\delta A}(x) = \sum_{\nu} \eta_{\nu\nu} \left( \frac{1}{\mu_0} \sum_{\mu} \partial_\mu F^{\mu\nu}(x) - J^\nu(x) \right) \mathbf{e}_\nu$$ where $F^{\mu\nu}$ are the components of the electromagnetic field strength matrix (Faraday tensor), $\eta = \text{diag}(1, -1, \dots, -1)$ is the Minkowski metric, $J^\nu$ is the $\nu$-th component of the current density, and $\mathbf{e}_\nu$ are the standard basis vectors for the Lorentz vector space.

Let $d$ be the number of spatial dimensions and $\mathcal{F}$ be the physical parameters of free space, including the speed of light $c$, the vacuum permittivity $\varepsilon_0$, and the vacuum permeability $\mu_0$. Let $A$ be an infinitely differentiable ($C^\infty$) electromagnetic potential field and $J$ be an infinitely differentiable Lorentz current density field. At any spacetime point $x$ with time coordinate $t = x^0/c$ and spatial coordinates $\mathbf{x}$, the variational gradient of the electromagnetic Lagrangian $\mathcal{L}$ with respect to $A$ (the Euler–Lagrange expression) is given by: $$\left( \frac{\delta \mathcal{L}}{\delta A} \right)(x) = \left( \frac{1}{\mu_0 c} \nabla \cdot \mathbf{E}(t, \mathbf{x}) - c \rho(t, \mathbf{x}) \right) \mathbf{e}_0 + \sum_{i=1}^d \left( \frac{1}{\mu_0} \left( \varepsilon_0 \mu_0 \frac{\partial E_i}{\partial t} - \sum_{j=1}^d \frac{\partial M_{ji}}{\partial x_j} \right) + j_i(t, \mathbf{x}) \right) \mathbf{e}_i$$ where $\mathbf{E}$ is the electric field, $M_{ji}$ is the magnetic field matrix (related to the components of $\mathbf{B}$), $\rho$ is the charge density, $j_i$ are the components of the current density $\mathbf{j}$, $\mathbf{e}_0$ is the temporal basis vector, and $\mathbf{e}_i$ are the spatial basis vectors. In the case where $\mu_0 = 1$ and $c = 1$, this recovers the expressions for Gauss's Law and Ampère's Law.

For a spatial dimension $d$, let $\mathcal{F}$ be the physical parameters of free space with permeability $\mu_0$. Let $A$ be an infinitely differentiable electromagnetic potential field and $J$ be an infinitely differentiable Lorentz current density field. For any point in spacetime $x$ and any index $\nu \in \{0, 1, \dots, d\}$, the $\nu$-th component of the variational gradient of the electromagnetic Lagrangian with respect to $A$ is given by: $$\left( \frac{\delta \mathcal{L}}{\delta A} \right)_\nu(x) = \eta_{\nu\nu} \left( \frac{1}{\mu_0} \partial_\mu F^{\mu\nu}(x) - J^\nu(x) \right)$$ where $\eta = \text{diag}(1, -1, \dots, -1)$ is the Minkowski matrix and $F^{\mu\nu}$ is the electromagnetic field strength tensor. In this identity, the term $\frac{1}{\mu_0} \partial_\mu F^{\mu\nu} - J^\nu$ is evaluated by mapping the resulting tensor expression back to its Lorentz vector representation.

For a given spatial dimension $d$, this linear map represents the variational gradient of the free current potential with respect to the electromagnetic potential. It maps a distributional Lorentz current density $J$ to a vector-valued distribution. For a test function $\varepsilon$ in the Schwartz space $\mathcal{S}(\text{SpaceTime } d, \mathbb{R})$, the resulting Lorentz vector is given by \[ \sum_{\mu} \eta_{\mu\mu} \langle J^\mu, \varepsilon \rangle \mathbf{e}_\mu \] where $\eta_{\mu\mu}$ are the diagonal entries of the Minkowski metric $\text{diag}(1, -1, \dots, -1)$, $J^\mu$ is the $\mu$-th component of the distributional current density, and $\mathbf{e}_\mu$ is the standard basis for Lorentz vectors.

For a distributional Lorentz current density $J$ and a test function $\varepsilon \in \mathcal{S}(\text{SpaceTime } d, \mathbb{R})$, the variational gradient of the free current potential evaluated at $\varepsilon$ is given by the weighted sum: \[ (\text{gradFreeCurrentPotential } J) \varepsilon = \sum_{\mu} \eta_{\mu\mu} \langle J^\mu, \varepsilon \rangle \mathbf{e}_\mu \] where $\eta_{\mu\mu}$ are the diagonal entries of the Minkowski metric $\text{diag}(1, -1, \dots, -1)$, $\langle J^\mu, \varepsilon \rangle$ denotes the $\mu$-th component of the distribution $J$ acting on the test function $\varepsilon$, and $\mathbf{e}_\mu$ is the $\mu$-th standard basis vector for Lorentz vectors.

For a given spatial dimension $d$, let $c$ be the speed of light in a free space environment. For any distributional Lorentz current density $J$ and any test function $\varepsilon$ in the Schwartz space $\mathcal{S}(\text{SpaceTime } d, \mathbb{R})$, the temporal component (the component at index $0$) of the variational gradient of the free current potential, when evaluated at $\varepsilon$, is equal to the speed of light $c$ multiplied by the value of the distributional charge density $\rho$ applied to $\varepsilon$: \[ [(\text{gradFreeCurrentPotential } J) \varepsilon]_0 = c \langle \rho, \varepsilon \rangle \] where the index $0$ refers to the temporal component in the Lorentz vector basis.

For a given spatial dimension $d$ and a free space environment $\mathcal{F}$, let $J$ be a distributional Lorentz current density and $\varepsilon \in \mathcal{S}(\text{SpaceTime } d, \mathbb{R})$ be a test function in the Schwartz space. The $i$-th spatial component (indexed by $i \in \{0, \dots, d-1\}$) of the variational gradient of the free current potential with respect to the electromagnetic potential, when applied to $\varepsilon$, is equal to the negative of the $i$-th component of the spatial current density distribution $\mathbf{j}$ evaluated at $\varepsilon$. Mathematically, this is expressed as: \[ (\text{gradFreeCurrentPotential } J) \varepsilon (\text{inr } i) = - \mathbf{j}_i(\varepsilon) \] where $\text{inr } i$ denotes the spatial indices in the $d+1$ dimensional space-time, and $\mathbf{j}$ represents the spatial part of the Lorentz current density $J$.

For any spatial dimension $d$, let $J$ be a distributional Lorentz current density and $\varepsilon \in \mathcal{S}(\text{SpaceTime } d, \mathbb{R})$ be a test function in the Schwartz space. For a spacetime index $\nu \in \{0, 1, \dots, d\}$, the $\nu$-th component of the variational gradient of the free current potential evaluated at $\varepsilon$ is equal to the product of the $\nu$-th diagonal element of the Minkowski metric $\eta$ and the $\nu$-th component of the distribution $J$ acting on $\varepsilon$: $$[(\text{gradFreeCurrentPotential } J) \varepsilon]_\nu = \eta_{\nu\nu} J^\nu(\varepsilon)$$ where $\eta = \text{diag}(1, -1, \dots, -1)$. The right-hand side is expressed using the tensor representation of the Lorentz vector components.

Given a spatial dimension $d$, a free space environment $\mathcal{F}$ (providing physical constants such as $\mu_0$ and $c$), a distributional electromagnetic potential $A$, and a distributional Lorentz current density $J$, the variational gradient of the Lagrangian density $\mathcal{L}$ with respect to $A$ is a Lorentz vector-valued distribution. It is defined as the difference between the variational gradient of the kinetic term (associated with the field strength tensor $F_{\mu\nu}$) and the variational gradient of the free current potential (associated with the interaction $A_\mu J^\mu$): \[ \frac{\delta \mathcal{L}}{\delta A} = \frac{\delta \mathcal{L}_{\text{kinetic}}}{\delta A} - \frac{\delta \mathcal{L}_{\text{interaction}}}{\delta A} \] The resulting distribution maps test functions on space-time $\text{SpaceTime } d$ to Lorentz vectors.

For any spatial dimension $d$, let $\mathcal{F}$ be a free space environment with magnetic permeability $\mu_0$ and speed of light $c$. Let $A$ be a distributional electromagnetic potential and $J$ be a distributional Lorentz current density. For any test function $\varepsilon$ in the Schwartz space $\mathcal{S}(\text{SpaceTime } d, \mathbb{R})$, the temporal component (index $0$) of the variational gradient of the Lagrangian density $\mathcal{L}$ evaluated at $\varepsilon$ is given by: \[ \left[ \left( \frac{\delta \mathcal{L}}{\delta A} \right) \varepsilon \right]_0 = \frac{1}{\mu_0 c} \langle \nabla \cdot \mathbf{E}, \varepsilon \rangle - c \langle \rho, \varepsilon \rangle \] where $\mathbf{E}$ is the distributional electric field associated with $A$, $\rho$ is the distributional charge density associated with $J$, and $\langle \cdot, \varepsilon \rangle$ denotes the application of the distribution to the test function.

Given a spatial dimension $d$ and a free space environment $\mathcal{F}$ (with permeability $\mu_0$ and speed of light $c$), let $A$ be a distributional electromagnetic potential and $J$ be a distributional Lorentz current density. For any test function $\varepsilon$ in the Schwartz space $\mathcal{S}(\text{SpaceTime } d, \mathbb{R})$ and any spatial index $i \in \{1, \dots, d\}$, the $i$-th spatial component of the variational gradient of the Lagrangian density $\mathcal{L}$ with respect to $A$, evaluated at $\varepsilon$, is given by: \[ \left( \frac{\delta \mathcal{L}}{\delta A} \right)_i(\varepsilon) = \frac{1}{\mu_0} \left( \frac{1}{c^2} \frac{\partial E_i}{\partial t}(\varepsilon) - \sum_{j=1}^d \frac{\partial M_{ji}}{\partial x_j}(\varepsilon) \right) + j_i(\varepsilon) \] where $E_i$ is the $i$-th component of the distributional electric field, $M_{ji}$ is the $(j, i)$-th component of the magnetic field matrix (representing the spatial components of the field strength tensor $F_{ji}$), and $j_i$ is the $i$-th component of the spatial current density distribution.

For any spatial dimension $d$, let $\mathcal{F}$ be a free space environment with magnetic permeability $\mu_0$. Given a distributional electromagnetic potential $A$, a distributional Lorentz current density $J$, and a test function $\varepsilon$ in the Schwartz space $\mathcal{S}(\text{SpaceTime } d, \mathbb{R})$, the $\nu$-th component of the variational gradient of the Lagrangian density $\mathcal{L}$ evaluated at $\varepsilon$ is given by: \[ \left[ \left( \frac{\delta \mathcal{L}}{\delta A} \right) \varepsilon \right]_\nu = \eta_{\nu\nu} \left( \frac{1}{\mu_0} (\partial_\kappa F^{\kappa\nu})(\varepsilon) - J^\nu(\varepsilon) \right) \] where $\eta = \text{diag}(1, -1, \dots, -1)$ is the Minkowski metric, $F^{\kappa\nu}$ is the electromagnetic field strength tensor, $J^\nu$ is the $\nu$-th component of the current density distribution, and $\partial_\kappa$ denotes the distributional derivative with respect to the $\kappa$-th coordinate (with implicit summation over $\kappa$).