As long we known, the non-relativistic Lagrangian definition, L = T − U with T and U as being, respectively, kinetic and potential energy, it is not valid at the relativistic level. This issue partially obstructs the college student’s comprehension of special theory of relativity from the Lagrangian formalism. Further, it is well known that the space-time world demands essentials principles of special theory of relativity: four-vectors and Lorentz invariance. In this work, we departure from these basics concepts of space-time world in order to get a relativistic Lagrangian definition parallel to the non-relativistic one and, as a consequence, this can help the college student to better understand special theory of relativity [1].

It is well known that hypothetical tunnels net within the Earth is usually considered as an illustration in a gravitation course for undergraduate students. In this context, an uniform mass distribution is usually considered and, consequently, a vehicle inside the tunnel oscillates in a simple harmonic motion (SHM). In this work, we investigate this problem and we argue that, instead to illustrate this problem using Earth, it is better to consider the Moon. In order to support this proposition, we present a twofold argumentation: first, we show that there is a quite discrepancy between the results obtained from Earth uniform mass distribution model with those obtained considering realistic Earth models (PREM model), whose mass is nonuniformly distributed; second, when the Moon is considered, we show that there is not a discrepancy between the results computed from uniform mass distribution model with those computed from nonuniform mass distribution. Indeed, we revealed that a vehicle inside the tunnel within Moon oscillates approximately in a SHM.

<p>Apresentamos uma abordagem essencialmente geométrica do problema da braquistócrona. Utilizamos conceitos de trigonometria básica, acessíveis a estudantes de Física Geral e Experimental I e II, de modo a permitir que o problema já seja apresentado num primeiro momento ainda no primeiro ano de um curso de física ou engenharia. Mostramos que a aceleração resultante sobre uma partícula descrevendo uma braquistócrona tem módulo <italic>g</italic> da aceleração gravitacional e aponta na direção do centro do círculo que gera a cicloide. Apresentamos um método geométrico que permite deduzir a equação diferencial da cicloide sem lançar mão do cálculo variacional. Mostramos que o tempo de descida ao longo desta trajetória é sempre o mesmo, independentemente da posição inicial.</p>

<p>We present an essentially geometric approach to the brachistochrone problem. We apply basic knowledge of trigonometry, accessible to the students in a one or two semester basic physics program, in such a way to allow the problem to be introduced at the beginning of an undergraduate course of Physics or Engineering. We show that the resulting acceleration on a particle describing a brachistochrone has gravitational acceleration of magnitude <italic>g</italic>and points towards the center of the circle that generates the cycloid. We present a geometrical method to deduce the differential equation for the cycloid, without using variational calculus. We show that the descending time along this trajectory is always the same, independently of the starting point.</p>