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
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