Physics Notes - Herong's Tutorial Notes

∟Introduction of Generalized Coordinates

∟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/∂q1
  d(∂L/∂q'2)/dt = ∂L/∂q2
  d(∂L/∂q'3)/dt = ∂L/∂q3

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/∂q1             (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/∂q1)/dt
  # Since ∂r'/∂q' = ∂r/∂q from G.29

d(∂L/∂q'1)/dt = d(∂L/∂r')/dt∙∂r/∂q1 + ∂L/∂r'∙d(∂r/∂q1)/dt
  # Chain rule applied again

d(∂L/∂q'1)/dt = d(∂L/∂r')/dt∙∂r/∂q1 + ∂L/∂r'∙∂r'/∂q1
  # Since d(∂r/∂q1)/dt = ∂r'/∂q1 by derivative switching

d(∂L/∂q'1)/dt = ∂L/∂r∙∂r/∂q1 + ∂L/∂r'∙∂r'/∂q1
  # Since d(∂L/∂r')/dt = ∂L/∂r by Lagrange Equations in Cartesian coordinates

d(∂L/∂q'1)/dt = ∂L/∂q1             (C.4)
  # Reverse chain rule applied

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

Table of Contents

 About This Book

 Introduction of Space

 Introduction of Frame of Reference

 Introduction of Time

 Introduction of Speed

 Newton's Laws of Motion

 Introduction of Special Relativity

 Time Dilation in Special Relativity

 Length Contraction in Special Relativity

 The Relativity of Simultaneity

 Introduction of Spacetime

 Minkowski Spacetime and Diagrams

 Introduction of Hamiltonian

 Introduction of Lagrangian

►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 Generalized Momentum

 What Is Legendre Transformation

 Hamilton Equations in Generalized Coordinates

 Phase Space and Phase Portrait

 References

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