Relation of Momentum and Potential Energy

This section describes the Relation of Momentum and Potential Energy.

With Hamiltonian expressed in terms of position r and momentum p, we can derive the relation of Momentum and Potential Energy by applying the Law of Conservation of Energy:

H = constant
T + V = constant                   (H.2)

  dH/dt = 0                        (H.3)

  d(0.5*|p|2/m + V(r))/dt = 0
    # H.12 applied

  p/m ∙ dp/dt + ∂V(r)/∂r ∙ dr/dt = 0
    # The chain rule for derivatives applied
    #  is the vector dot product

  p/m ∙ p' + ∂V(r)/∂r ∙ r' = 0
    # p' = dp/dt and r' = dr/dt applied

  r' ∙ p' + ∂V(r)/∂r ∙ r' = 0
    # H.13, p = m*r' applied

  p' + ∂V(r)/∂r = 0
    # Cancel out r'.
    # It become 3 equations, one in each dimension.

  ∂V/∂r = -p'                     (H.14)

Very nice. the law of conservation of energy says that the potential energy change in space equals to the negative change of momentum!

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

 What Is Hamiltonian

 Hamiltonian on Free Fall Motion

 Hamiltonian on Simple Harmonic Motion

 Hamiltonian on Simple Pendulum Motion

 What Is Momentum

 Relation of Momentum and Hamiltonian

 Hamiltonian in Cartesian Coordinates

Relation of Momentum and Potential Energy

 Hamilton Equations in Cartesian Coordinates

 Introduction of Lagrangian

 Introduction of Generalized Coordinates

 Phase Space and Phase Portrait


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