This chapter explores classical mechanics of discrete systems, then field theories, reviewing the Lagrangian and Hamiltonian formalisms. The discussion extends to electrodynamics and the coupling of scalar matter fields with electromagnetic fields. It explores the minimal coupling scheme and its application to both Schrödinger and Klein-Gordon equations in the presence of electromagnetic potentials. Insights from prominent physicists are discussed, highlighting the interplay between theoretical elegance and experimental rigor in advancing our understanding of physical laws.