From point particles to gauge field theoriesa differential-geometrical approach to the structures of the space of solutions

Résumé

We study the geometry of the space of solutions of the equations of motion of (a large class of) classical field theories. In particular, we exhibit the existence of a canonical pre-symplectic structure on it and we argue whether and how it is possible to use it to define a Poisson structure. Within gauge theories we show how a construction related to the so-called coisotropic embedding theorem can be suitably used to this scope. Several concrete physical systems are considered, such as the free particle as an example of mechanical system and Klein-Gordon theory, free Electrodynamics, Yang-Mills theories and Palatini's Gravity as examples of field theories.