Associated Curves of Timelike Magnetic Curves

A magnetic vector field in three-dimensional space is divergence-free. The trajectories it generates are called magnetic curves. This paper characterises the Mannheim, Bertrand and involute–evolute structures carried by timelike magnetic curves in Minkowski 3-space. For a timelike N -magnetic curve, a single lemma governs all three cases. A Mannheim pair occurs exactly when the magnetic field lies in the rectifying plane of the trajectory and the relation κ = λ ( τ 2 − κ 2 ) holds with λ constant. The same condition rules out Bertrand pairs. Involutes never occur. The Mannheim structure has codimension one. It is therefore unstable under perturbations of the magnetic field. The trajectory is a slant helix if and only if its Mannheim partner is a general helix, and the two invariants are reciprocal. A closed-form example is given. Timelike B -magnetic curves behave differently. There the governing orthogonality is automatic, and Bertrand pairs do exist. An explicit example is constructed.