Lagrange Equation on Free Fall Motion

∟Lagrange Equation on Free Fall Motion

This section provides an example of applying the Lagrange Equation to an object in free fall motion.

Can we use Lagrange Equation to solve the problem of Free Fall Motion? The answer is of course.

Consider an object with mass m in free fall motion (source: owlcation.com):

The object's kinetic energy, T, and potential energy, V, can be expressed as below, assuming axis x is pointing upward.

T = m*x'*x'/2
  # m is the mass of the object
  # x' is the velocity of the object

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

The Lagrangian function L becomes:

L = T - V                          (G.1)

or:
  L = m*x'*x'/2 - m*g*x           (G.17)

The Lagrangian Equation becomes:

d(∂L/∂x')/dt = ∂L/∂x               (G.5)

or:
  d(m*x')/dt = -m*g

or:
  m*x" = -m*g
    # Since d(m*x')/dt = m*x"

or:
  x" = -g                         (G.18)