Hamiltoniana ou energia total? A dinâmica da miçanga ao longo de uma haste.

Abstract

Newtonian mechanics is a milestone in the history of physics and modern science. However, limitations arise when constrained systems are analyzed, which is why many physicists and mathematicians sought an alternative solution developed by Newton. Thus, analytical mechanics arises, with the lagrangean and hamiltonian formalisms being the exponents of this new way of describing dynamical systems. In this work, we analyze the dynamics, under the light of analytical mechanics, of harmonic oscillators (simple, damped and forced) and, mainly, of the problems of the beads sliding along a rod (rod with uniform rotation, accelerated rotation and oscillating). Furthermore, we will study in detail the relationship between total and hamiltonian energy, describing all existing cases, including emphatic discussions about the physical quantities that are conserved or not.