Physics Notes - Herong's Tutorial Notes

∟Introduction of Generalized Coordinates

∟What Is Legendre Transformation

This section provides a quick introduction to Legendre Transformation, which is an equation to transform Lagrangian L to Hamiltonian H.

What Is Legendre Transformation? Legendre Transformation is an equation to transform Lagrangian L to Hamiltonian H as shown below: 

H = p∙q' - L
  # H is the Hamiltonian
  # p is the generalized momentum
  # q' is the generalized velocity
  # ∙ is the dot product
  # L is the Lagrangian

From Legendre Transformation, we can get a nice definition of the kinetic energy: T = 0.5*p∙q'. 

H = p∙q' - L

or:
  T + V = p∙q' - (T - V)
    # Since H = T + V, L = T - V
    # T is the kinetic energy
    # V is the potential energy

or:
  T = p∙q' - T
    # Cancel out V

or:
  T = 0.5*p∙q'                    (C.5)

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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