Physics Notes - Herong's Tutorial Notes - v3.24, by Herong Yang
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
Introduction of Frame of Reference
Introduction of Special Relativity
Time Dilation in Special Relativity
Length Contraction in Special Relativity
The Relativity of Simultaneity
Minkowski Spacetime and Diagrams
►Action - Integral of Lagrangian
Action - Functional of Position Function x(t)
Hamilton's Principle - Stationary Action
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