Magnetic Dynamical Systems and Sabba Ştefănescu Conjectures

Abstract

Section 10.1.1 recalls known facts about the magnetic field H produced by the Bioh Savart-Laplace law for a massive conductor D̄. Section 10.1.2 proves that, generally, the part of a magnetic line that lies in ext D̄ is a trajectory of a potential dynamical system of order two (a geodesic of the Riemann-Jacobi structure), and the part that lies in int D̄ is a trajectory of a nonpotential dynamical system of order two (a geodesic of a Riemann-Jacobi-Lagrange structure). Consequently, we have discovered new variants of Lorentz world-force laws describing nonclassical magnetic dynamics. This section presents also some properties of magnetic traps, two significant examples, and formulates an open problem. Section 10.1.3 describes the magnetic dynamical systems that can be reduced to 2-dimensional Hamiltonian systems. Section 10.1.4 analyses the magnetic fields to determine which ones are symmetric or antisymmetric with respect to some symmetries.