Dynamics Induced by a Magnetic Field

Abstract

§1 shows that a particle sensible to a magnetic field obeys a new Lorentz-world-force law with respect to a special geometrical structure. Precisely, the energy of the magnetic field is incorporated into a Riemann-Jacobi metric and the Lorentz equation of charged particle motion results from this metric and from the rotor of the magnetic field (external electromagnetic field). Prom geometrical point of view the magnetic lines belong to a class of geodesics, and the magnetic surfaces appears like ruled surfaces generated by geodesics. §2 studies the stable manifold and the unstable manifold of an hyperbolic equilibrium point of a stationary magnetic dynamical system as ruled submanifolds (a surface, and a geodesic), and gives properties of a stationary magnetic field with heteroclinic structure. §3 analyses the morphology of an elementary magnetic field which admit two hyperbolic equilibrium points whose flow invariant manifolds intersect giving rise to a heteroclinic structure and a family of spatial polycyles.