Fundamentals of Lagrangian mechanics

Abstract Motivated by filling the gap we felt after years of teaching analytical mechanics, a non-relativistic, classical introduction to Lagrangian mechanics has accordingly been provided here, which covers all possible forms of Euler–Lagrange equation, derived through dealing with different kinds of forces including conservative forces, forces of constraint, velocity-dependent forces, and non-conservative/dissipative forces. Hamiltonian mechanics has also been concluded as a reformulation of Lagrangian mechanics via applying Legendre transformation. Ignorable coordinates have finally been introduced, leading to Hamilton–Jacobi formalism, from which an equivalence between dynamics of a classical point particle and that of a plane wave has been inferred. We have showed that such an equivalence had long laid the required theoretical ground for the advent of wave mechanics; therefore, a number of landmark advancements in theoretical physics, including Hamiltonian mechanics, canonical transformations, and formulations of quantum mechanics have roots in Lagrangian mechanics.