Static electromagnetic coupling and typed canonical momentum

Let $\mathrm{\Omega }\subset {\mathbb{R}}^3$ be nonempty, bounded and open with compact closure, and let $U$ be an open neighborhood of $\bar{\mathrm{\Omega }}$. Let the static scalar potential be $\phi :U\to \mathbb{R}$ and the static vector potential be $A:U\to {\mathbb{R}}^3$, both of class $C^2$. Let $m>0$ kg, and let $q_{\mathrm{p}}\in \mathbb{R}$ C be the particle's signed charge; this model parameter is distinct from the exact elementary-charge constant $e$ in Framework-local SI units, the vacuum constants and the exact/measured status of their values. Let velocity $v\in {\mathbb{R}}^3$ have units m s${}^{-1}$. The adopted single-particle Lagrangian is the function $L:\mathrm{\Omega }\times {\mathbb{R}}^3\to \mathbb{R},L(x,v)=\frac{1}{2}m\mathrm{\mid }v{\mathrm{\mid }}^2+q_{\mathrm{p}}A(x)\cdot v-q_{\mathrm{p}}\phi (x),$ with $A$ in T m and $\phi$ in V, as in Framework-local SI units, the vacuum constants and the exact/measured status of their values. Its Legendre momentum is $p:={\mathrm{\partial }}_{v}L=mv+q_{\mathrm{p}}A(x),$ so for every $(x,p)\in T^{\ast }\mathrm{\Omega }\cong \mathrm{\Omega }\times {\mathbb{R}}^3$ the inverse is $v=(p-q_{\mathrm{p}}A(x))\mathrm{/}m$. The mechanical momentum is $\pi :=p-q_{\mathrm{p}}A(x)=mv$; the canonical momentum is the cotangent coordinate $p$ of The momentum of a potential presentation: the fibre derivative of the mechanical Lagrangian, the Cartesian momentum Mv and generalized momenta. The Legendre transform $H=p\cdot v-L$, in the convention of The Hamiltonian of a hyperregular potential presentation: the Legendre transform of the mechanical Lagrangian, its domain and its dimensions, is therefore $H(x,p)=\frac{\mathrm{\mid }p-q_{\mathrm{p}}A(x){\mathrm{\mid }}^2}{2m}+q_{\mathrm{p}}\phi (x)\mathrm{.}$ At fixed $x$, the velocity derivative has linear part $m{I}_3$, so the displayed affine momentum map has the explicit inverse $v=(p-q_{\mathrm{p}}A(x))\mathrm{/}m$. Substitution into $p\cdot v-L$ cancels the two $q_{\mathrm{p}}A\cdot v$ terms and gives $\frac{1}{2}m\mathrm{\mid }v{\mathrm{\mid }}^2+q_{\mathrm{p}}\phi$; using the inverse gives the displayed Hamiltonian. The supplier The Hamiltonian of a hyperregular potential presentation: the Legendre transform of the mechanical Lagrangian, its domain and its dimensions provides the $p\cdot v-L$ convention for its hyperregular $K-U$ mechanical presentation; the inverse branch for this velocity-linear electromagnetic coupling is the local calculation above, not a consequence of that supplier's $K-U$ scope. The units are $[q_{\mathrm{p}}A]=$ kg m s${}^{-1}$, $[q_{\mathrm{p}}\phi ]=$ J, and $[H]=$ J.

For a time-independent gauge function $\chi :U\to \mathbb{R}$ of class $C^3$ (units V s), adopt the static convention of Potential representations, gauge transformations and the Lorenz and Coulomb conditions: $A_{\chi }=A+\mathrm{\nabla }\chi$ and ${\phi }_{\chi }=\phi$. The corresponding canonical-coordinate change is $T_{\chi }(x,p)=(x,p+q_{\mathrm{p}}\mathrm{\nabla }\chi (x))$; the mechanical momentum is unchanged under this paired transformation. This definition specifies an external-potential model only and asserts no existence of a global potential representation on arbitrary regions.