DefinePK

Abstract

Studies in pendulums (oscillatory motion) are significant in physics, engineering, and other fields of knowledge and continue to be a topic of active research. Considering the Lagrangian formalism and using the technique of variation of canonical variables, the physical symmetrical pendulum was studied. Using the conservation laws, we introduce the effective one-dimensional potential, which depends on the dimensionless variable ? (related implicit with nutation angle) and parametrically on reduced energy e and two reduced angular momenta,p1 and p2. Only three modes (plane, elliptical, and conical) were studied using the effective potential. An advantage to using this approach was to obtain approximate analytical expressions to study the phase diagrams that show the pendulum's behavior; therefore, in the plane mode, the motion is on a plane, in the elliptical mode, the pendulum moves quasi elliptical, and in the conical mode, the pendulum vibrates at a single point - the three modes depending on the initial condition. An advantage of this approach was to obtain approximate analytical expressions to study the phase diagrams for these three modes. This approach simplifies and visibility a complex problem as the oscillations of the physical pendulum.