Principi variazionali, leggi di conservazione e stabilità nelle teorie di campo classiche

The convenient setting in which is possible to discuss classical field theories and problems concerning conservation laws, field equations and Noether's theorem is the so called geometric variational calculus over jet bundles. After reviewing some geometrical concepts already known in the literature (such as Principal Bundles, Jet bundles and their Contact Structure) we present the Lagrangian formulation of Classical Field Theories with a particular emphasis on conserved quantities and Noether's theorem then we discuss some application in physics (General Relativity,Yang-Mills, Klein-Gordon, Spinor theories...). The aim of the thesis is to adress, in the aforementioned setting, what means for a solution of field equations for a given field theory to be stable. We give a general definition of stability for a classical field theory studying the sign of the second variation of the so called Energy Functional (which is defined through Noether's theorem using the ADM splitting procedure). To verify the correctness of our definition, we obtain, in this framework, a well known and established result in physical litterature concerning the stability of a Klein-Gordon field over a fixed AdS background which is the famous Breitenlohner-Freedman as well as discussing stability in classical mechanics. We also present some results concerning variational principles over the Composite Bundles.