A particle of charge moves in a given electromagnetic field described by the scalar…

PHYSICS · Multi-part · Fermi University Physics · Mechanics & Dynamical Systems · Electricity, Magnetism & Electromagnetism · Lagrangian mechanics

Action and Symmetry, Ch Multipliers, non-holonomic constraints and forces outside the potential

Problem

A particle of charge

Q_{\rm e}

moves in a given electromagnetic field described by the scalar potential

\phi(\mathbf r,t)

and vector potential

\mathbf A(\mathbf r,t)

, with

\mathbf B=\nabla\times\mathbf A

and

\mathbf E=-\nabla\phi-\partial\mathbf A/\partial t

. Its generalised potential is

U=Q_{\rm e}\phi-Q_{\rm e}\mathbf v\cdot\mathbf A

, with

\mathbf v=(\dot x,\dot y,\dot z)

, and

L=T-U

. ((a)) Starting from

\mathcal Q_y=\mathrm d(\partial U/\partial\dot y)/\mathrm dt-\partial U/\partial y

, derive the

y

component of the Lorentz force,

Q_{\rm e}[E_y+(\mathbf v\times\mathbf B)_y]

, keeping every term in the total time derivative. ((b)) Under a gauge change

\mathbf A\to\mathbf A+\nabla\Lambda

\phi\to\phi-\partial\Lambda/\partial t

for some function

\Lambda(\mathbf r,t)

, show that

L\to L+Q_{\rm e}\,\mathrm d\Lambda/\mathrm dt

, and explain in one sentence why the particle's motion is unaffected even though the generalised momentum changes.