Relation of Momentum and Hamiltonian

This section describes the Relation of Momentum and Hamiltonian.

Hamiltonian, H, is related to momentum, p, through the kinetic energy, T.

Let's use a single object as an example:

H = T + V                          (H.1)

  H = 0.5*m*v*v + V
    # Since T = 0.5*m*v*v

  H = 0.5*p*v + V
    # p = m*v applied

  H = 0.5*p*p/m + V
    # p = m*v applied again

If we apply the law of conservation of energy, we can have:

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

  dH/dt = 0
  dT/dt + dV/dt = 0
  dT/dt = -dV/dt                   (H.3)

  d(0.5*p*p/m)/dt = - dV/dt
  p*p'/m = -dV/dt

  v*p' = -dV/dt
    # p = m*v applied

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


 Full Version in PDF/ePUB