Physlib.Electromagnetism.Dynamics.Hamiltonian

For a spatial dimension $d$, let $\mathcal{F}$ denote the physical parameters of free space. Given an electromagnetic potential field $A$ and a Lorentz current density field $J$, the **canonical momentum** at a spacetime point $x$ is the Lorentz vector defined by the difference of two gradients: $$\Pi(x) = \left. \nabla_v \mathcal{L}(A + x^0 v, J, x) \right|_{v=0} - x^0 \left. \nabla_v \mathcal{L}(A + v, J, x) \right|_{v=0}$$ where $\mathcal{L}$ is the electromagnetic Lagrangian density, $x^0$ is the time component of the spacetime point $x$, and $v$ is a Lorentz vector. This construction isolates the conjugate momentum density of the field, which is mathematically equivalent to the partial derivative of the Lagrangian density with respect to the time derivative of the potential: $$\Pi^\mu = \frac{\partial \mathcal{L}}{\partial (\partial_0 A_\mu)}$$

For a spatial dimension $d$, let $\mathcal{F}$ be the physical parameters of free space, $A$ be a twice-differentiable electromagnetic potential field, and $J$ be a Lorentz current density. The canonical momentum $\Pi(x)$ at a spacetime point $x$ is equal to the gradient with respect to a Lorentz vector $v$, evaluated at $v = 0$, of the kinetic Lagrangian density $\mathcal{L}_{\text{kin}}$ for a modified potential field $x' \mapsto A(x') + x'^0 v$: $$\Pi(x) = \left. \nabla_v \mathcal{L}_{\text{kin}}(\mathcal{F}, A', x) \right|_{v=0}$$ where $A'(x') = A(x') + x'^0 v$, and $x'^0$ denotes the time component of the spacetime point $x'$.

For a spatial dimension $d$, let $\mathcal{F}$ denote the parameters of free space with vacuum permeability $\mu_0$. Given a twice-differentiable electromagnetic potential field $A$ and a Lorentz current density $J$, the $\mu$-th component of the canonical momentum $\Pi^\mu(x)$ at a spacetime point $x$ is given by: $$\Pi^\mu(x) = \frac{1}{\mu_0} \eta_{\mu\mu} F_{\mu 0}(x)$$ where $\eta$ is the Minkowski matrix and $F_{\mu 0}(x)$ is the component of the electromagnetic field strength tensor (Faraday tensor) corresponding to the index $\mu$ and the temporal index $0$.

For a spatial dimension $d$, let $\mathcal{F}$ denote the physical parameters of free space, characterized by the vacuum permeability $\mu_0$ and the speed of light $c$. Given a twice-differentiable electromagnetic potential field $A$ and a Lorentz current density $J$, the components of the canonical momentum field $\Pi^\mu(x)$ at a spacetime point $x$ are related to the electric field $E$ as follows: 1. The temporal component ($\mu = 0$) is zero: $$\Pi^0(x) = 0$$ 2. The spatial components ($\mu = i \in \{1, \dots, d\}$) are given by: $$\Pi^i(x) = -\frac{1}{\mu_0 c} E_i(x)$$ where $E_i(x)$ is the $i$-th component of the electric field at the spacetime point $x$.

For a given spatial dimension $d$ and the physical parameters of free space $\mathcal{F}$, let $A$ be an electromagnetic 4-potential field, $J$ be a Lorentz 4-current density field, and $x$ be a point in spacetime. The **Hamiltonian density** $\mathcal{H}$ at the point $x$ is defined as the sum over the spacetime indices $\mu$ of the product of the canonical momentum density components and the time derivatives of the potential components, minus the Lagrangian density: $$\mathcal{H}(x) = \sum_{\mu} \Pi^\mu(x) \frac{\partial A_\mu(x)}{\partial x^0} - \mathcal{L}(x)$$ where $\Pi^\mu(x)$ is the $\mu$-th component of the canonical momentum density $\Pi$ at $x$, $\frac{\partial A_\mu(x)}{\partial x^0}$ is the partial derivative of the $\mu$-th component of the potential $A$ with respect to the time coordinate $x^0$ (represented as `Sum.inl 0`), and $\mathcal{L}(x)$ is the electromagnetic Lagrangian density at $x$.

For a spatial dimension $d$, let $\mathcal{F}$ denote the physical parameters of free space with speed of light $c$ and vacuum permeability $\mu_0$. Given a twice-differentiable electromagnetic potential field $A$ and a Lorentz current density $J$, the Hamiltonian density $\mathcal{H}$ at a spacetime point $x$ is given by: $$\mathcal{H}(x) = -\frac{1}{c^2 \mu_0} \sum_{i=1}^d E_i(x) \frac{\partial \mathbf{A}_i(x)}{\partial t} - \mathcal{L}(x)$$ where $E_i(x)$ is the $i$-th component of the electric field, $\mathbf{A}_i(x)$ is the $i$-th component of the vector potential, $\frac{\partial}{\partial t}$ denotes the partial derivative with respect to time, and $\mathcal{L}(x)$ is the electromagnetic Lagrangian density at $x$.

For a spatial dimension $d$, let $\mathcal{F}$ denote the physical parameters of free space with speed of light $c$ and vacuum permeability $\mu_0$. Given a twice-differentiable electromagnetic potential field $A$, a Lorentz current density $J$, and a point in spacetime $x$, the Hamiltonian density $\mathcal{H}$ at $x$ is given by: $$\mathcal{H}(x) = \frac{1}{c^2 \mu_0} \left( \|\mathbf{E}(x)\|^2 + \langle \mathbf{E}(x), \nabla \phi(x) \rangle \right) - \mathcal{L}(x)$$ where $\mathbf{E}(x)$ is the electric field, $\phi(x)$ is the scalar potential, $\nabla$ is the spatial gradient operator, $\langle \cdot, \cdot \rangle$ is the Euclidean inner product, $\|\cdot\|$ is the Euclidean norm, and $\mathcal{L}(x)$ is the electromagnetic Lagrangian density at $x$.

For a spatial dimension $d$, let $\mathcal{F}$ denote the physical parameters of free space, including the speed of light $c$, vacuum permittivity $\epsilon_0$, and vacuum permeability $\mu_0$. Given a twice-differentiable electromagnetic potential field $A$, a Lorentz current density $J$, and a point in spacetime $x$ with time $t$ and spatial position $\mathbf{x}$, the Hamiltonian density $\mathcal{H}$ at $x$ is expressed in terms of the electric and magnetic fields as: $$\begin{aligned} \mathcal{H}(x) = & \frac{1}{2}\epsilon_0 \left( \|\mathbf{E}(t, \mathbf{x})\|^2 + \frac{c^2}{2} \sum_{i,j=1}^d |B_{ij}(t, \mathbf{x})|^2 \right) + \epsilon_0 \langle \mathbf{E}(t, \mathbf{x}), \nabla \phi(t, \mathbf{x}) \rangle \\ & + \phi(t, \mathbf{x})\rho(t, \mathbf{x}) - \sum_{i=1}^d A^i(t, \mathbf{x}) j^i(t, \mathbf{x}) \end{aligned}$$ where: - $\mathbf{E}$ is the electric field and $B_{ij}$ are the components of the magnetic field matrix. - $\phi$ is the scalar potential and $\nabla \phi$ is its spatial gradient. - $\rho$ is the charge density and $j^i$ are the components of the current density. - $A^i$ are the components of the vector potential.