Classical Field Theory for Mathematicians (Graduate Studies in Mathematics)

The aim of this book is to give a comprehensive treatment of the majority of important classical field theory from the mathematics perspective.The opening Part 1 gives the exposition of classical mechanics and special relativity that are based on the Hamiltonian approach and emphasizes the Hamiltonian action of the relevant Lie groups. Part 2 bridges classical mechanics and classical field theory.The authors develop all necessary tools: Lagrangian formulation of classical field theory, conservation laws, the Noether theorem, and Hamiltonian formulation.They present all necessary facts about jet bundles, multivariable calculus of variations, etc. Part 3 discusses gauge field theory: Maxwell's theory with the abelian structure group $U(1)$, and Yang-Mills theory with the structure group being semisimple compact Lie groups.For the convenience of the reader, the authors collect all necessary facts about connections and curvature in vector and principal bundles. In Part 4 the authors briefl