Physics Notes - Herong's Tutorial Notes - v3.25, by Herong Yang
Lagrange Equation on Simple Harmonic Motion
This section provides a quick introduction to the Hamiltonian function, the total energy of the system, which is the sum of kinetic energy and potential energy of the system.
Can we use Lagrange Equation to solve the problem of Simple Harmonic Motion? The answer is of course.
Consider an object with mass m in simple harmonic motion (source: slideserver.com):
The object's kinetic energy, T, and potential energy, V, can be expressed as below, assuming x is object's displacement from the equilibrium position.
T = m*x'*x'/2 # m is the mass of the object # x' is the velocity of the object V = k*x*x/2 # g is the standard gravity (9.80665) # x is the displacement of the object
The Lagrangian function L becomes:
L = T - V (G.1) or: L = m*x'*x'/2 - k*x*x/2 (G.19)
The Lagrangian Equation becomes:
d(∂L/∂x')/dt = ∂L/∂x (G.5) or: d(m*x')/dt = -k*x or: m*x" = -k*x # Since d(m*x')/dt = m*x" or: m*x" = -k*x (G.20)
Cool. Equation G.20 matches perfectly with Newton's second law of motion.
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