A 4-vector formalism for classical mechanics

Abstract

We present a matrix formalism, inspired by the Minkowski four-vectors of special relativity, useful to solve classical physics problems related to both mechanics and thermodynamics. The formalism turns out to be convenient to deal with exercises involving non-conservative forces and production or destruction of mechanical energy. On the other hand, it provides a framework to treat straightforwardly changes of inertial reference frames, since it embodies the Principle of Relativity. We apply the formalism to a few cases to better show how it works.

Relativistic mechanical–thermodynamical formalism—description of inelastic collisions

“…In the next section we examine, through an example, the usefulness and the predictions that may be obtained from equation (8). But, before that, in the next subsections we shall discuss several aspects related to the construction of a relativistic thermodynamical theory.…”

“…Even if we were interested in processes occurring at low velocities, the special theory of relativity would offer the appropriate framework to develop a formalism that integrates dynamics and thermodynamics on equal footing [8], therefore avoiding conceptual difficulties associated with that unification, as noted by some authors [9]. In the relativistic framework, one has to regard the energy variations, the heat, and the work as components of four-vectors which obey a mechanical-thermodynamical fundamental equation [10].…”

“…This paper is a contribution to incorporate relativity in thermodynamics. We start with the special relativity formulation using four-vectors in the Minkowski space [13,14] and incorporate, in that known formalism, conservation laws such as the first law of thermodynamics [10,15]. We illustrate how the formalism works in practice, using the example of a deformable ball inelastically colliding against a wall.…”

Sobre el flujo de calor en el tensor energía momento de un fluido relativista

“…INTRODUCCI ÓN es posible construir una termodinámica relativista; así es para Ter Haar y Wergeland [7]. J. Güémez [8] discute la necesidad conceptual de construir una termodinámica relativista Lorentziana, como generalización, aplicando la teoría especial de la relatividad de Einstein, de una termodinámica Galileana, y discute sobre las posibles bases con las que se podrían construir 4-vectores de energía interna, trabajo y calor.…”

“…Puesto que el calor es una forma de que se transmita la energía y que en Termodinámica Clásica no lleva asociado momento, en Relatividad debe existir un marco de referencia privilegiado en el que la energía emitida en forma de calor tenga momento cero (teniendo momento distinto de cero en otros marcos de referencia). Para esto, se debe hacer la hipótesis de que debe existir un sistema de referencia privilegiado en el que el calor intercambiado por el sistema con su entorno lo sea de tal manera que el momento (relativista) sea nulo [8].…”