**Physics Notes - Herong's Tutorial Notes** - v3.24, by Herong Yang

Introduction of Lagrangian

This chapter provides an introduction of Lagrangian Mechanics. Topics include introduction of Lagrangian; Action, Hamilton Principle, Stationary Action, Lagrange Equations, examples of free fall motion; simple harmonic motion, simple pendulum motion.

Action - Integral of Lagrangian

Action - Functional of Position Function x(t)

Hamilton's Principle - Stationary Action

Other Proofs of the Lagrange Equation

Lagrange Equation on Free Fall Motion

Lagrange Equation on Simple Harmonic Motion

Takeaways:

- Lagrangian Function is defined as the difference of kinetic and potential energy: H = T - V.
- Action is defined as the integral of the Lagrangian function L(t)
between two time instances, t
_{1}and t_{2}. - Hamilton's Principle states that the true position function x(t) of a system is a stationary point of the Action S[x(t)] functional.
- Lagrange Equations state that the time derivative of the partial derivative Lagrangian L against velocity equals to the partial derivative Lagrangian L against position: d(∂L/∂r')/dt = ∂L/∂r.

Table of Contents

Introduction of Frame of Reference

Introduction of Special Relativity

Time Dilation in Special Relativity

Length Contraction in Special Relativity

The Relativity of Simultaneity

Minkowski Spacetime and Diagrams

Introduction of Generalized Coordinates