T-23 · Magnetism

Fields that turn.

A magnetic field never speeds a charge up; it only turns it. Boris's integrator does exactly that, by a rotation whose cosine and sine come from a Pythagorean triple, so its speed never drifts. Currents make fields that circle them: a loop's, built from straight sides, against its closed form; and Ampère's law, as a circulation.

Boris turns by a triple

Boris's step splits the electric push in two halves and puts between them a pure rotation of the velocity, by the angle whose tangent of half is t = Bh/2. Its cosine and sine are (1 − t²)/(1 + t²) and 2t/(1 + t²): Euclid's formula for a Pythagorean triple. In exact fractions the speed comes back identical, step after step.

cos, sin of each turn	3/5, 4/5
cos² + sin² = 1	9/25 + 16/25 exactly
\|v\|² over 26 exact steps	13/4 every step, exactly

Three integrators

The same field, start and step for each. Euler multiplies the speed by √(1 + (Bh)²) every step and spirals out. Runge–Kutta 4 is very accurate per step but its speed slowly sags. Boris keeps the speed exactly, and errs only in phase: it turns by 2 arctan(Bh/2) instead of Bh.

integrator

speed change

miss at the end

Euler

…

Runge–Kutta 4

…

Boris

…

The orbit

Without an electric field the charge circles at radius |v|/B, once every 2π/B. With E across B it drifts at E × B/B², whatever its charge or mass.

radius \|v\|/B	1.802776
speed kept, this integrator	residual …
drift by Boris, 4000 steps	… residual …