Simple Pendulum Motion in Generalized Coordinates

This section provides an example of applying the Hamiltonian and the Law of Conservation of Energy to an object in simple pendulum motion using generalized coordinates.

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 radius L.

Let's introduce 2 generalized coordinates to describe this motion, θ and l:

q = (θ, l)

  # θ is the angular position from the vertical line
  # l is the length of the string

The transformation functions are:

r = (x, y)

r = (r1(), r2())

r = (l*sin(θ), -l*cos(θ))

Then the velocity is:

r' = dr/dt

  r' = (l*cos(θ)*θ', l*sin(θ)*θ')

Then the kinetic energy is:

T = m*v*v/2

  T = 0.5*m*r'*r'

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

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

Then the potential energy is:

V = m*g*y

  V = m*g*(-l*cos(θ))

  V = -m*g*l*cos(θ)

Now put them into the Hamiltonian:

H = T + V                          (H.1)

  H = 0.5*m*l*l*θ'*θ' - m*g*l*cos(θ)

Apply the Law of Conservation of Energy:

T + V = constant                   (H.2)

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

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

   m*l*l*θ'*θ" + m*g*l*sin(θ)*θ' = 0

   l*θ" + g*sin(θ) = 0
   # Cancel out common factors

As you can see, using generalized coordinates can given a simple form for the equation of motion of a given physical system.

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

 Introduction of Lagrangian

Introduction of Generalized Coordinates

 Generalized Coordinates and Generalized Velocity

Simple Pendulum Motion in Generalized Coordinates

 Hamilton's Principle in Generalized Coordinates

 Lagrange Equations in Generalized Coordinates

 Lagrange Equations on Simple Pendulum

 What Is Generalized Momentum

 What Is Legendre Transformation

 Hamilton Equations in Generalized Coordinates

 Phase Space and Phase Portrait


 Full Version in PDF/ePUB