Euler-Lagrange equation: derivation and application

Classical Mechanics and Relativity: Lecture 3 0:00 Introduction 0:51 Principle of Least Action and the Lagrangian 6:01 Generalized Coordinates 11:38 Derivation of the Euler-Lagrange equation in generalized coordinates 25:29 Generalized momentum and generalized force 27:55 Polar coordinates 38:33 Example: pendulum 46:18 Newtonian vs Lagrangian mechanics 49:47 Global vs Local approach 57:47 The Hamiltonian 1:03:18 Legendre transformation 1:07:26 Hamilton's Equations 1:08:44 The Hamiltonian and Energy 1:12:27 Conservation of Energy Theoretical physicist Dr Andrew Mitchell presents an undergraduate lecture course on Classical Mechanics and Relativity at University College Dublin. This is a complete and self-contained course in which everything is derived from scratch. In this lecture I use the Principle of Least Action to derive the Euler-Lagrange Equation of Motion in generalized coordinates and perform the Legendre transformation to obtain Hamilton's equations. We will explore the connection between the classical Hamiltonian and the energy, and show that it is conserved. The concepts are illustrated with simple examples. Full lecture course playlist: https://www.youtube.com/playlist?list=PLotxEOxVaaoKDo2GFPZsg04p3OnlwYBkR Course textbooks: "Classical Mechanics" by Goldstein, Safko, and Poole "Classical Mechanics" by Morin "Relativity" by Rindler