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

Lagrange Equations in Generalized Coordinates

This section provides a quick introduction to Lagrange Equations in Generalized Coordinates.

It is amazing to see that Lagrange Equations stay in the same form as in Cartesian Coordinates:

d(∂L/∂q')/dt = ∂L/∂q (C.3) or: d(∂L/∂q'_{1})/dt = ∂L/∂q_{1}d(∂L/∂q'_{2})/dt = ∂L/∂q_{2}d(∂L/∂q'_{3})/dt = ∂L/∂q_{3}

Here is the proof using the first component of the vector only as provided in David Morin's book.

`d(∂L/∂q'`_{1})/dt = ∂L/∂q_{1} (C.4)
# To be approved

Start from left side term:

d(∂L/∂q'_{1})/dt = d(∂L/∂r'∙∂r'/∂q'_{1})/dt # Chain rule applied d(∂L/∂q'_{1})/dt = d(∂L/∂r'∙∂r/∂q_{1})/dt # Since ∂r'/∂q' = ∂r/∂q from G.29 d(∂L/∂q'_{1})/dt = d(∂L/∂r')/dt∙∂r/∂q_{1}+ ∂L/∂r'∙d(∂r/∂q_{1})/dt # Chain rule applied again d(∂L/∂q'_{1})/dt = d(∂L/∂r')/dt∙∂r/∂q_{1}+ ∂L/∂r'∙∂r'/∂q_{1}# Since d(∂r/∂q_{1})/dt = ∂r'/∂q_{1}by derivative switching d(∂L/∂q'_{1})/dt = ∂L/∂r∙∂r/∂q_{1}+ ∂L/∂r'∙∂r'/∂q_{1}# Since d(∂L/∂r')/dt = ∂L/∂r by Lagrange Equations in Cartesian coordinates d(∂L/∂q'_{1})/dt = ∂L/∂q_{1}(C.4) # Reverse chain rule applied

Cool. We have proved Lagrange Equations in Generalized Coordinates based Lagrange Equations in Cartesian Coordinates.

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

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

What Is Legendre Transformation

Hamilton Equations in Generalized Coordinates