Velocity-dependent potentials

Let L=T−U, where U may depend on position, velocity, and time. The generalized force represented by U is minus its position derivative plus the time derivative of its velocity derivative. Using only minus the position gradient would omit part of the force. This representation does not cover every dissipative force; ordinary linear drag is usually kept as a nonconservative generalized force or described with a Rayleigh dissipation function. A velocity-dependent term should be justified by the intended dynamics, not inserted merely because a force contains velocity.

For a particle with charge e, choose U=eΦ−e A·v in SI conventions. The scalar potential Φ and vector potential A define E=−grad Φ−∂tA and B=curl A. The kinetic energy is mv²/2 in the nonrelativistic regime. Differentiate the Lagrangian carefully: the velocity derivative gives m v+e A, while spatial derivatives also act on the components of A. Combining those terms produces the magnetic cross product rather than an extra scalar-potential force.

In Cartesian components, the acceleration equals e times the electric field plus e times velocity crossed with magnetic field, divided by mass. The antisymmetric combination of vector-potential derivatives generates the cross product. This gives a useful sign check: for positive charge moving along positive x in a positive z magnetic field, the magnetic force points along negative y. Reversing the charge reverses the deflection. The magnetic force is perpendicular to velocity and therefore does no instantaneous mechanical work.

The Lagrangian energy is the sum of velocity times canonical momentum minus L. Its linear vector-potential terms cancel, leaving kinetic energy plus eΦ in this nonrelativistic model. This quantity is conserved when the chosen Lagrangian has no explicit time dependence. With time-dependent potentials, differentiate it rather than assuming conservation from the word magnetic. Gauge-dependent potential terms and externally maintained fields require care when interpreting the energy of the particle alone; the field-producing apparatus is outside this reduced description.

Under A→A+grad χ and Φ→Φ−∂tχ, the Lagrangian changes by e times the total time derivative of χ. Fixed-endpoint path variations therefore give the same equations of motion. Canonical momentum shifts by e grad χ, while mechanical momentum and the electromagnetic fields remain unchanged. Consequently, comparing canonical momenta computed in two gauges without transforming them is meaningless. A coordinate that appears cyclic in one gauge can encode its physical symmetry through a boundary term in another.

The vector lab supports the right-angle geometry behind the no-work statement only indirectly. Set a vector angle, predict which perpendicular direction a magnetic force would have using the right-hand rule, then compare the displayed projections with your sketch. The lab itself contains just one planar vector, not a magnetic field, cross-product calculator, or charged-particle orbit. The worked example supplies the field and gauge analytically. Keeping those limits explicit avoids confusing a vector illustration with a Lorentz-force simulation.