Magnetic systems represent a classical topic which is still of great mathematical interest. They systematically appear in contemporary physic, from collider accelerators and plasma fusion, where the magnetic field confines the particle on a desired region of the space; to Earth’s magnetosphere and mass spectrometry, where the magnetic field acts as a mirror or a prism. From a mathematical point of view, a magnetic system is the pure toy model for the motion of a charged particle moving on a Riemannian Manifold endowed with a (static) magnetic force. In this course we put in evidence some geometrical aspect of such dynamics.
Part 1 – Magnetic systems and magnetic geodesics – Hamiltonian and variational setting – The existence of closed magnetic geodesics – The magnetic exponential map – Magnetic systems on surfaces.
Part 2 – Magnetic curvature operator – Sectional, Ricci and Scalar magnetic curvature – Jacobi equation and the second variation of the magnetic energy.
Part 3 – Magnetic flows positively curved – The Bonnet-Myers Theorem in the magnetic case – Application to the existence of closed magnetic geodesic – Some topological discussion for magnetic flows with positive magnetic curvatures.
Part 4 – Magnetic flows without conjugate points – Magnetic systems negatively curved – Magnetic CartanHadamard Theorem – The magnetic Hopf-Green Theorem – Magnetic flatness.
The students will select and present one of the above topics. The schedule of the lectures as well as references and basic information will appear soon.