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

Introduction of Generalized Coordinates

This chapter provides an introduction of Generalized Coordinates. Topics include introduction of generalized position, velocity and momentum; Lagrange Equations and Hamilton Equations in generalized coordinates; Legendre Transformation.

Generalized Coordinates and Generalized Velocity

Simple Pendulum Motion in Generalized Coordinates

Hamilton's Principle in Generalized Coordinates

Lagrange Equations in Generalized Coordinates

Lagrange Equations on Simple Pendulum

Takeaways:

- Generalized position q and generalized velocity q' can be transformed to Cartesian position r and Cartesian velocity r' by a transformation function r(t) = r(q(t)).
- Hamilton's Principle in Generalized Coordinates stays in the same form as d(S[q(t)]) / d(q(t)) = 0.
- Lagrange Equations in Generalized Coordinates stay in the same form as d(∂L/∂q')/dt = ∂L/∂q.
- Generalized momentum is defined as: p = ∂L/∂q'.
- Legendre Transformation H = p∙q' - L gives conversion between Lagrangian L and Hamiltonian H.
- Hamilton Equations in Generalized Coordinates stay in the same form as ∂H/∂q = -p', ∂H/∂p = q'.

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