Physics Notes - Herong's Tutorial Notes

∟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.

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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

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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

 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

 Phase Space and Phase Portrait

 Poisson Bracket Expression

 References

 Full Version in PDF/ePUB

Introduction of Generalized Coordinates - Updated in 2024, by Herong Yang