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

PHYSICS · Multi-part · Fermi University Physics · Mechanics & Dynamical Systems · Lagrangian mechanics · 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 a scalar potential

\phi(\mathbf r,t)

and a vector potential

\mathbf A(\mathbf r,t)

, with

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

and

\mathbf B=\nabla\times\mathbf A

. Its motion follows from the generalised potential

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

through

\mathcal Q_k = \dfrac{\rm d}{{\rm d}t}\left(\dfrac{\partial U}{\partial \dot q_k}\right)-\dfrac{\partial U}{\partial q_k}

, with

q_k=x,y,z

and

\dot q_k

the corresponding velocity component. ((a)) Starting from this definition, derive the

y

-component

\mathcal Q_y

of the generalised force and show that it equals

Q_{\rm e}\left(E_y+(\mathbf v\times\mathbf B)_y\right)

. ((b)) A proton (charge

+\mathrm e

, mass

m_{\rm p}

) moves with

\mathbf E=0

in a uniform field

\mathbf B=B\hat{\mathbf z}

, described in the gauge

\mathbf A_1=(-By,0,0)

. Write the Lagrangian

L_1=T-U_1

and its momentum

p_y=\partial L_1/\partial\dot y

. State, with a reason drawn from the form of

L_1

, whether

p_y

is conserved in this gauge. ((c)) The gauge function

\Lambda=Bxy

gives a second vector potential

\mathbf A_2=\mathbf A_1+\nabla\Lambda=(0,Bx,0)

for the same field. Write

L_2=T-U_2

, find

p_y

in this gauge, and show it is conserved. Verify that

L_2-L_1=\mathrm e\,{\rm d}\Lambda/{\rm d}t

, and explain in one or two sentences why the equations of motion, and hence the proton's orbit, are the same in both gauges even though

p_y

is not.