Analytical Solution to the Inverted Elastic Pendulum via Lagrange Approach

The pendulum itself is one of the mechanical systems that have been studied the most in the field of mechanics and has innumerable applications in very different scopes. The inverted spring pendulum is analyzed. We derive the equations of motion using Lagrange mechanics approach. Several ways to solve it are provided. The pendulum is linearized near the equlibrium position to obtain a system that may be solved using Mathieu functions. We solve it by means of a simple ansatz method and the Krýlov-Bogoliúbov-Mitropólsky method. The results are illustrated through several examples.