Question

a) Show that x-component of the Lorentz force, when written in terms of vector potentials as we did in class, can be expanded and then recast as

$F_{x}=-\frac{\partial U}{\partial x}+\frac{d}{dt}\frac{\partial U}{\partial \dot{x}}$

where

$U=qV-q\mathbf{\dot{r}}\cdot \boldsymbol{\mathbf{A}}$

Thus show that Lagrange’s equations still hold so long as U is taken to have this form, and that the Lagrangian for a non-relativistic particle in an electric and magnetic field can be written as equation (7.104) in the text:

$\textit{L}(\boldsymbol{r,\dot{r},\boldsymbol{\textit{t}}})=\frac{1}{2}m(\dot{x}^{2}+\dot{y}^{2}+\dot{z}^{2})-q(V-\dot{x}A_{x}-\dot{y}A_{y}-\dot{z}A{_{z}})$

b) Show from this Lagrangian and the analysis in Section 7.8 that the Hamiltonian for a charged particle in an electromagnetic field is

$H=\frac{1}{2m}\mathbf{p}-q\mathbf{A}^{2}+qV$

Answer 1

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6) 2仁 2m 2 m