Action - Integral of Lagrangian

This section provides an introduction of Action, the integral of Lagrangian of a given system between two time instances.

What Is Action? Action, S, is the integral of the Lagrangian L(t) of a given system between two time instances, t1 and t2.

S =  L(t)dt between t1 and t2      (G.2)

To understand the meaning of Action, S, let's apply it to a single object in Free Fall Motion. Assuming the object moving on a vertical line represented by coordinate x pointing upward, the position of object at any given time, t, can be represented by a function, x(t).

Assuming that the object is moving under the gravity force, we know how to calculate the kinetic energy and the potential energy:

T(t) = m*x'(t)*x'(t)/2
  # t is time
  # m is the mass of the object
  # x'(t) is the velocity of the object

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

So Lagrangian L(t) can be expressed as:

L(t) = T(t) - V(t)                                     (G.1)

or:
  L(t) = m*x'(t)*x'(t)/2 - m*g*x(t)                    (G.3)

Using Newton's Second Law of Motion, we can get analytical solutions for the position x(t) and the velocity x'(t):

x(t) = -g*t*t/2 - x'1*t + x1
  # x1 is the position at t1
  # x'1 is the velocity at t1

x'(t) = -g*t - x'1

Applying above solutions to (G.3), we can get a analytical solution for Lagrangian L(t):

L(t) = 0.5*m*(-g*t - x'1)*(-g*t - x'1)
       - m*g*(-g*t*t/2 - x'1*t + x1)

or:
  L(t) = 0.5*m*g2*t2 + m*g*x'1*t + 0.5*m*x'12
       + 0.5*m*g2*t2 + m*g*x'1*t - m*g*x1

or:
  L(t) = m*g2*t2 + 2*m*g*x'1*t + 0.5*m*x'12 - m*g*x1    (G.4)

With the analytical solution of Lagrangian (G.4), the Action S becomes the area under a second order polynomial of time t, between t1 and t2:

S =  L(t)dt between t1 and t2                         (G.2)

or:
  S =  (m*g2*t2 + 2*m*g*x'1*t + 0.5*m*x'12 - m*g*x1)dt
    # Between t1 and t2

Cool, we can easily calculate the Action S of a single object in Free Fall Motion between between any two time instances!

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

 What Is Lagrangian

Action - Integral of Lagrangian

 Action - Functional of Position Function x(t)

 Hamilton's Principle - Stationary Action

 What Is Lagrange Equation

 Other Proofs of the Lagrange Equation

 Lagrange Equation on Free Fall Motion

 Lagrange Equation on Simple Harmonic Motion

 Lagrangian in Cartesian Coordinates

 Lagrange Equations in Cartesian Coordinates

 Introduction of Generalized Coordinates

 Phase Space and Phase Portrait

 References

 Full Version in PDF/ePUB