Hamiltonian on Free Fall Motion

∟Hamiltonian on Free Fall Motion

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

What Is Free Fall Motion? Free Fall Motion is simple mechanical system of a single object with mass, m, falling freely to the ground under the force of gravity.

To calculate the Hamiltonian of the free fall motion system, we can express the object's kinetic energy, T, and potential energy, V, as:

T = m*v*v/2
  # m is the mass of the object
  # v is the velocity of the object

V = m*g*x
  # g is the standard gravity (9.80665)
  # x is the height of the object

So the Hamiltonian, H, of the free fall motion system can be expressed as:

H = T + V

or:
  H = m*v*v/2 + m*g*x

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

H = constant

or:
  m*v*v/2 + m*g*x = constant

or:
  d(m*v*v/2)/dt + d(m*g*x)/dt = 0
    # Since d(constant)/dt = 0

The last equation can be simplified as:

m*v*dv/dt + m*g*dx/dt = 0
  # The chain rule for derivatives applied

m*v*a - m*g*v = 0
  # a = dv/dt, is the acceleration of the object
  # v = -dx/dt, is the velocity of the object

a - g = 0
a = g                              (H.4)
  # Cancel out m*v from the equation

Cool. Equation H.4 matches perfectly with Newton's second law of motion:

F = m*a                            (H.5)
  # Newton's second law of motion

m*g = m*a
  # Gravity force, F = m*g, applied.

a = g                              (H.4)
  # Cancel out m from the equation

With equation H.4, we can easily figure out the position h, the velocity v, and the acceleration a as functions of time t:

a(t) = g

v(t) = g*t + v₀
  # v₀ is the initial velocity

h(t) = -g*t*t/2 - v₀*t + h₀
  # h₀ is the initial position

The following picture illustrates an object in free fall motion (source: owlcation.com):

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

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