What Is Hamiltonian

This section provides a quick introduction to the Hamiltonian, which is a derived property of a mechanical system defined as the total energy, the sum of kinetic energy and potential energy of the system.

What Is Hamiltonian? Hamiltonian, H, is a derived property of a a mechanical system defined as the total energy, or the sum of kinetic energy and potential energy, of the system:

H = T + V                          (H.1)
  # H is the total energy
  # T is the kinetic energy
  # V is the potential energy

When a mechanical system changes over time, t, the value of Hamiltonian may change too. The kinetic energy, the potential energy and Hamiltonian of the system can be viewed as functions of t. So Hamiltonian can also be expressed as:

H(t) = T(t) + V(t)                 (H.1)

Based on the law of conservation of energy, the Hamiltonian should be a constant for an isolated conservative system. In other words, the derivative of Hamiltonian against time is zero.

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

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

Hamiltonian was introduced by Sir William Rowan Hamilton (1805-1865), an Irish mathematician.

Portrait of William Rowan Hamilton
Portrait of William Rowan Hamilton

Hamiltonian for a Single Object

For a single object, the kinetic energy, T, can be expressed as:

T = 0.5*m*v*v
  # m is the mass of the object
  # v is the velocity of the object

So applying the Law of Conservation of Energy on the Hamiltonian, we get:

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

  dT/dt = -dV/dt                   (H.3)

  d(0.5*m*v*v)/dt = -dV/dt
    # kinetic energy is applied.

  m*v*dv/dt = -dV/dt
  m*v*a = -dV/dt

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