Hamiltonian on Simple Pendulum Motion

This section provides an example of calculating the Hamiltonian on a mechanical system of an single object in simple pendulum motion and applying the Law of Conservation of Energy.

What Is Simple Pendulum Motion? A Simple Pendulum Motion is an object of mass m hanging on a string from a pivot point so that it is constrained to move on a circle of a fixed radius.

If we map the space in Cartesian coordinates, the Simple Pendulum Motion is a 2-dimensional problem, both the position, r, and the speed, v, have 2 components:

r = (x, y)
v = (x', y')

If we introduce an extra variable θ as the angular position of the object from the vertical line, r and v can be expressed as:

r = (l*sin(θ), -l*cos(θ))
v = (l*cos(θ)*θ', l*sin(θ)*θ')

Now the object's kinetic energy, T, can be expressed as:

T = m*|v|**2

  T = 0.5*m*l*l*(cos(θ)**2)+sin(θ)**2)*θ'*θ'
    # Pythagorean Theorem applied

  T = 0.5*m*l*l*θ'*θ'
    # Since cos(θ)**2)+sin(θ)**2 = 1

The potential energy, V, can be expressed as:

V = m*g*y

  V = m*g*(-l*cos(θ))

  V = -m*g*l*cos(θ)

So the Hamiltonian, H, can be expressed as:

H = T + V                          (H.1)

  H = 0.5*m*l*l*θ'*θ' - m*g*l*cos(θ)
    # Replaced T and V with their expressions

Since this simple pendulum motion can be considered as an isolated conservative system, we can apply the Law of Conservation of Energy:

H = constant

  0.5*m*l*l*θ'*θ' - m*g*l*cos(θ) = constant

  d(0.5*m*l*l*θ'*θ' - m*g*l*cos(θ))/dt = 0
    # Since d(constant)/dt = 0

   m*l*l*θ'*θ" + m*g*l*sin(θ)*θ' = 0
   l*θ" + g*sin(θ) = 0
   θ" = - g*sin(θ)/l

Cool. We got the simplest form of the equations for the simple pendulum motion.

Object as a Simple Pendulum
Object as a Simple Pendulum

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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