Mandelbrot Scaling and Parametrization Invariant Theories

Abstract

Fractional variational principles have gained considerable importance during the last decade due to their applications in several areas of sciences and engineering. In this paper we will adapt this variational principle to obtain the Euler-Lagrange equation of motion, by considering two different cases. In the first case we used the scaling concepts of Mandelbrot of fractals in variational problems of mechanical systems in order to re-write the action function as an integration over a scaling measure. After that we parameterize the time in the action integral to obtain the equations of motion. It is shown that the genuine Euler-Lagrange equations of motion are those which are obtained using the Mandelbrot scaling of space/and or time.