Classical Mechanics on Fractal Curves

Abstract

Fractal analogue of Newton, Lagrange, Hamilton, and Appell's mechanics are suggested. The fractal alpha-velocity and alpha-acceleration are defined in order to obtain the Langevin equation on fractal curves. Using the Legendre transformation, Hamilton's mechanics on fractal curves is derived for modeling a non-conservative system on fractal curves with fractional dimensions. Fractal differential equations have solutions that are non-differentiable in the sense of ordinary derivatives and explain space and time with fractional dimensions. The illustrated examples with graphs present the details.