Resolução dos pêndulos simples, duplo e triplo por meio das equações de Euler-Lagrange e diagonalização de matrizes

Abstract

We present, in this work, a mathematical treatment for the analytical resolution of the simple, double and triple pendulum, using the Lagrangian formalism and matrix diagonalization with a very detailed and didactic approach. In the first step, we use Lagrangian mechanics to obtain the equations of motion of the three pendulum systems, obtaining second-order nonlinear ODEs. In the case of double and triple pendulums, we transform the system of equations into a matrix ODE. In the second step, we assume small oscillations, obtaining linear ODEs, and solve the ODEs. We use matrix diagonalization to solve matrix ODEs. In the third step, we obtained the constants as a function of the initial conditions. In the fourth step, we obtained the graphs of θ(t) assuming some values for the masses, pendulum lengths and initial conditions. In some calculations we use Wolfram Mathematica Software. We verified the sensitivity of the initial conditions of these systems. We also analyze the interferences between the masses of the coupled systems