Energy Invariants and Geometric Properties of Charged-Particle Motion Under the Lorentz Force: Workless Forces, Turning Bounds and Field-Integral Theorems

Why does a magnetic field change the direction of a charged particle's motion but never its kinetic energy? The textbook answer is that the magnetic force is perpendicular to the velocity, so . In this paper we take that answer as a starting point and ask what can be said in general about how much a magnetic field can turn a particle. First, we show that the forces that do no work on any motion are exactly those of the form , and that among forces linear in the velocity these are precisely the magnetic-type forces. Second, the curvature of every trajectory in an arbitrary static field is . As a consequence, the angle between any two velocities along the path is bounded by the field integral . Combined with the theorems of Fenchel and Fáry–Milnor, this gives minimum field integrals for closed and knotted orbits. Third, in any static unidirectional field, whatever its spatial profile, the velocity stays on a cone around the field direction, so the direction can change by at most twice the initial pitch angle, independent of the field strength. Fourth, for a magnetic barrier , transmission, exit angle and specular reflection depend only on the running flux per unit length , not on the profile. All results hold relativistically with and are verified by symbolic computation and numerical integration in random fields.