Canonical momentum ($p_i$)

The canonical momentum conjugate to a generalized coordinate $q_i$ is $p_i=\partial L/\partial {\dot{q}}_i$.

For an axisymmetric magnetic vector potential, the conserved angular canonical momentum is $p_{\theta }=\gamma m{\rho }^2\dot{\theta }+q\rho A_{\theta }$. A particle arriving parallel to a solenoid axis from a field-free region has $p_{\theta }=0$. In a uniform field, $A_{\theta }=B\rho /2$ and $\dot{\theta }=-qB/(2\gamma m)$ wherever $\rho \ne 0$. The mechanical angular momentum changes through the fringe while the canonical quantity remains constant. Static magnetic fields also conserve $\gamma m{c}^2$; the azimuthal kick therefore reduces axial kinetic motion rather than increasing total speed.

For the affine action $S=-\frac{m}{2}\int g_{\mu \nu }{\dot{x}}^{\mu }{\dot{x}}^{\nu }d\tau$ with signature $(-+++)$, differentiating with respect to velocity gives the displayed canonical momentum. Future-oriented proper time in a conformally flat metric yields $P_i^c=-a\gamma m{v}_i$, whereas the conventional future-directed mechanical momentum has $q_i=a\gamma m{v}_i$. Changing the action's overall sign interchanges these canonical labels but leaves its classical geodesics unchanged. Both labels are conserved by spatial translation invariance, and either gives the same physical momentum magnitude proportional to $a^{-1}$. The sign should be stated when comparing a worldline canonical covector with a positive-frequency field-mode label.

If the spatially integrated Lagrangian changes by $dB[\Phi ]/dt$, where $B$ depends on fields and their spatial derivatives but not velocities, the canonical momentum components change by ${\pi }_a\mapsto {\pi }_a+\delta B/\delta {\Phi }_a$. This follows by varying $dB/dt$ and using spatial integration by parts. The canonical one-form changes by the exact variation $\delta B$, so the symplectic form and canonical commutation relations are preserved. For the two Feynman gauge Maxwell densities, $B=\int A_0{\partial }_i{A}_i{d}^{3}x$ gives shifts $\Delta {\pi }^0={\partial }_i{A}_i$ and $\Delta {\pi }^i=-{\partial }_i{A}_0$. Boundary flux must vanish for this spatial functional calculation.