Symmetry in geometry and mechanics

Abstract

[en] In this work we have tried to give a theoretical description, as well as a historical review, of the differential geometry behind different areas of physics (Classical Mechanics, Electromagnetism, Yang-Mills...) trying to capture the deep geometric nature of modern theoretical physics. Firstly we start by introducing classical differential-geometric notions from the modern viewpoint (emphasizing the role played by Bundles and Sections). Then we will devote to the study of Hamiltonian Systems both in the Symplectic and in the Pois- son formalisms, in here we will see two results that we believe are specially important, geodesic flow from the dynamics point of view (where we will analyze the notion of geodesics through the lenses of physical theories, and a proof of Clairaut’s formula in classical differential geometry by the use of a fundamental concept in modern geometric mechanics, the moment map.