PHY.7:5.2 - Recover an interaction that energy alone would miss

A nonrelativistic particle with mass m and charge e moves in the xy plane in a prescribed uniform magnetic field B perpendicular to it. Electric fields, radiation reaction and the particle’s alteration of the source field are neglected. The electromagnetic coupling is the physical premise; the field does no mechanical work but changes the direction of motion.

Choose a vector potential A=(-B*y/2,B*x/2,0), whose curl is the specified magnetic field. With zero electric scalar potential, the physical Lagrangian is

L=m*(x_dot^2+y_dot^2)/2 + e*B*(x*y_dot-y*x_dot)/2.

Its derivatives give

d/dt(L_xdot)=m*x_ddot-e*B*y_dot/2; L_x=e*B*y_dot/2,

and the corresponding y expressions. The Euler-Lagrange equations are therefore

m*x_ddot=e*B*y_dot; m*y_ddot=-e*B*x_dot.

For e*B/m=2 per second, initial x_dot=3 m/s and y_dot=0, the initial acceleration is (0,-6) m/s^2. Substitution into the kinetic-energy derivative gives zero. Conservation of kinetic energy alone would also allow straight uniform motion; it does not determine the magnetic turning. Using only T with zero scalar potential would miss the interaction.

Changed representation. Let chi=B*x*y/2 and use A_new=A+grad(chi)=(0,B*x,0). The new Lagrangian is L_new=m*(x_dot^2+y_dot^2)/2+e*B*x*y_dot. Its difference from L is e*d(chi)/dt, so the fixed-endpoint equations are unchanged. The canonical momenta L_xdot and L_ydot do change; the physical velocity and magnetic field do not. Comparing those canonical expressions as though they were two observed mechanical momenta would invent a physical discrepancy.

The action’s stationary paths depend on the physical interaction; equivalent gauge descriptions preserve them under the stated endpoint rule.