Classical Field Theory: Lagrangian and Hamiltonian Formulation

This lecture extends the concept of field theory from mechanics to classical field theory, focusing on Lagrangian and Hamiltonian formulations. It explains the notion of fields in a space with dynamical variables, using examples like electromagnetic fields and fluid dynamics. The lecture delves into the Lagrangian formulation of fluids and introduces the action principle for fields, emphasizing the property of locality in spacetime. It discusses the Euler-Lagrange equations as the fundamental dynamics of field systems, highlighting the connection between Lagrangian formulation and modifications in the system. The lecture concludes by illustrating the application of the action principle in field theory through concrete examples and addressing questions related to boundary conditions and local variations.