Lagrange Equation on Simple Harmonic Motion

∟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)