20 nov 2020 -- 16:50
Politecnico di Torino - PRIN seminar (online)
Abstract.
In 1882, Sophus Lie formulated the task to describe two-dimensional metrics admitting non-trivial symmetries that preserve geodesics up to reparametrisation. Such symmetries are called projective. Lie's Problem has been resolved in recent years in terms of a classification up to diffeomorphisms (Bryant-Manno-Matveev 2008, Matveev 2012 and Manno-V 2020).
The talk will focus on a distinct subclass of these metrics, namely those that are superintegrable with quadratic integrals of motion. Generally speaking a metric is superintegrable if it admits a maximal amount of independent constants of motion. Matveev's geometries are a particular example, in which case the projective symmetry is unique. It turns out that all of Matveev's geometries share the same geodesics up to reparametrisation (in other words, they are projectively equivalent). The associated superintegrable systems are of non-degenerate type meaning that they admit a four-parameter potential.