The mechanics of Lagrange and Hamilton constitute a powerful tool in the study of various physical systems, however, they were developed for ideal systems, without taking into account forces of a dissipative nature. As a result, this work sought to build Lagrangian’s functions and Hamiltonian’s that allow to obtain the equations of motion of the vertical launch of a sphere subjected to the resistive force of air proportional to linear and quadratic speed, respectively. For this, they were used as mathematical tools the Rayleigh dissipation function, the principles of equivalent lagrangian and fractional calculus. As a result of this work, we were able to satisfactorily find the equations of motion and, moreover, a Lagrangian function was formulated for the first time with fractional derivative for a dissipative force proportional to the square of the velocity.
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