Princípio de Hamilton no ensino medio: proposta baseada na análise qualitativa da trajetória da luz e da braquistócrona

Abstract

This research aims to make available to high school teachers a proposal for approaching Hamilton’s Principle (sometimes called the Principle of Least Action) based on the qualitative analysis of the phenomena of the trajectory of light and the brachistochrone. The research details the development of this proposal, which includes a didactic sequence and support material, using Chevallard’s Theory of Didactic Transposition as a methodological framework to adapt scholarly knowledge (the formalism of Analytical Mechanics) into accessible teaching knowledge. The methodology employs historical analysis and adapted problem solving, illustrating the unifying power and economy of the principle without resorting to variational calculus, but instead using analogies and geometric reasoning. As results, the structured didactic sequence (Appendix A), aligned with BNCC skills, two questionnaires (Appendix B) one diagnostic and one evaluative that assist teachers in mediating the content and the commented solution of the final problem proposed in the questionnaires (Appendix C), which exemplifies the practical application of the discussed concepts, are presented. It is concluded that this approach provides a viable path to explore fundamental principles of Physics in high school, contributing to teaching practice and to students’ scientific formation.