On Noether's Theorem and Applications in Classical Mechanics and Quantum Field Theory

The role symmetries play in the laws of physics is explored in these papers.It covers systems from Newtonian mechanics to modern physics, such as the Standard Model.The math involved focuses on Lagrangian, Hamiltonian, and calculus of variations, which are foundational to understanding Noether's theorem.This paper uses the theorem to show how continuous symmetries lead to conservation laws, including conservation of momentum, angular momentum, energy, and charge.The paper then advances to topics such as gauge symmetry, complex scalar fields, and scalar quantum electrodynamics (QED).The paper emphasizes how symmetry provides a unifying framework for physics and its applications across classical and modern physics.