Magnetic trajectories on the Heisenberg group of dimension three

Início: 25/08/2023 17:00

Término: 25/08/2023 18:00

Palestrante: Mauro Subils (UN Rosario)

E-mail do Palestrante: subils@fceia.unr.edu.ar

Resumo: A magnetic trajectory is a curve $g a m m a$ on a Riemannian manifold $(M, g)$ satisfying the equation: $n a b l a_{g a m m a^{'}} g a m m a^{'} = q F g a m m a^{'}$ where $n a b l a$ is the corresponding Levi-Civita connection and $F$ is a skew-symmetric $(1, 1)$ -tensor such that the corresponding 2-form $g (F c d o t, c d o t)$ is closed. In this talk we are going to describe all magnetic trajectories on the Heisenberg Lie group of dimension three $H_{3}$ for any invariant Lorentz force. We will write explicitly the magnetic equations and show that the solutions are described by Jacobi's elliptic functions. As a consequence, we will prove the existence and characterize the periodic magnetic trajectories. Then we will induce the Lorentz force to a compact quotient $H_{3} / G a m m a$ and study the periodic magnetic trajectories there, proving its existence for any energy level when $F$ is non-exact. This is a joint work with Gabriela Ovando (UNR).