Princípios variacionais na física

Abstract

This work aims to establish a relationship between geometric optics and classical mechanics. We saw this through the so-called short wavelength approximation that allows us to ignore microscopic effects such as diffraction and interference in the context of wave propagation in material media. With this, we immediately obtain Fermat’s principle, which was stated before the emergence of electromagnetism and is an immediate corollary of the study of electromagnetic waves. After this, we enter the mechanical principle enunciated in the 18th century, the Maupertuis principle, which in its first form implicitly hides its analogy with Fermat’s principle. Jacobi was responsible for enunciating a modified form that shows the similarity between these two principles. After that, the main results of the Hamilton-Jacobi theory were seen, which leads to the emergence of a supposed Eikonal equation for the context of particles in configuration space. It is concluded, therefore, that classical mechanics is a supposed approximation of a more general wave theory and, with Hamilton’s fundamental function, it is possible to construct the phase of this supposed wave that is propagating in phase space. We saw that energy is discretized and it is possible to obtain the De Broglie relation from this point of view, furthermore, we discovered that the Hamilton-Jacobi equation is the classical Schrödinger equation. Finally, we obtained the Euler-Lagrange equation for Fermat’s principle and saw that we were able to obtain the equivalent of Newton’s second law for the study of light beams in the context of geometric optics. Furthermore, we covered the study of Feynman path integrals that serve as a reinterpretation of Hyugens’ principle. Keywords: Electromagnetism, Hamilton-Jacobi theory, Fermat principle.