Physics Notes - Herong's Tutorial Notes

∟Introduction of Hamiltonian

∟Hamilton Equations in Cartesian Coordinates

This section provides a quick introduction to the Hamilton Equations in Cartesian Coordinates, partial derivative equations on the Hamiltonian function against position r and momentum p for an isolated conservative system.

What Are Hamilton Equations? Hamilton Equations are partial derivative equations on the Hamiltonian function against position r and momentum p for an isolated conservative system.

First let's take the partial derivative against position r. 

∂H/∂r = ∂T/∂r + ∂V/∂r
  # This is a vector equation with 3 components

or:
  ∂H/∂r = ∂(0.5*|p|2/m)/∂r + ∂V(r)/∂r
    # H.12 applied

or:
  ∂H/∂p = 0 + ∂V(r)/∂r
    # Since 0.5*|p|2/m is independent of r
or:
  ∂H/∂r = -p'                     (H.15)
    # H.14 applied

Now take the partial derivative against momentum p. It's very easy to do. 

∂H/∂p = ∂T/∂p + ∂V/∂p
  # This is a vector equation with 3 components

or:
  ∂H/∂p = ∂(0.5*|p|2/m)/∂p + ∂V(r)/∂p
    # H.12 applied

or:
  ∂H/∂p = ∂(0.5*|p|2/m)/∂p + 0
    # Since V(r) is independent of p

or:
  ∂H/∂p = p/m
    # The chain rule for derivatives applied

or:
  ∂H/∂p = r'                      (H.16)
    # H.7 applied

Putting them together, we have the Hamilton Equations for an isolated conservative system: 

∂H/∂r = -p'                       (H.15)
∂H/∂p = r'                        (H.16)

Again Hamilton Equations are also called Hamilton's Law of Motion, which is equivalent to Newton's Second Law of Motion, which is equivalent to the Law of Conservation of Energy.

Now we can describe a moving object without using force and acceleration:

By the way, many text books derive Hamilton Equations from Lagrangian Equations. They are actually equivalent, as we will see later.

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

 References

 Full Version in PDF/ePUB