The quadruple pendulum: an analytical solution

Abstract In this work, we obtained the normal frequencies for small oscillations of four particular cases of the quadruple pendulum using an analytical and didactic approach. We employed the Tartaglia-Cardano and Ferrari methods to solve the characteristic equations of the triple and quadruple pendulums, respectively, considering equal masses and lengths. We ascertained that the characteristic frequencies for these cases are given by ω i = λ i ( g / l ) 1/2 for i = 1, 2, …, n , where λ i corresponds to the square root of the i -th zero of the Laguerre polynomials L n ( x ), and n being the number of degrees of freedom of the system. In addition, we demonstrated an important property for the sum of the characteristic frequencies not established in the literature. This approach relates fundamental concepts of physics to advanced mathematical tools, and can be incorporated in intermediate-level university courses on Classical Mechanics and Mathematical Methods for Physicists.