Abstract
Considering that the motions of the particles take place on fractals, a non-differentiable mechanical model is built. Only if the spatial coordinates are fractal functions, the Galilean version of our model is obtained: the geodesics satisfy a Navier-Stokes-type of equation with an imaginary viscosity coefficient for a complex speed field or respectively, a Schrödinger-type of equation or hydrodynamic equations, in the case of irrotational movements. Moreover, in this approach, the analysis of the fractal fluid dynamics generates conductive properties in the case of movements synchronization both on differentiable and fractal scales, and convective properties in the absence of synchronization (e.g. laser ablation plasma is analyzed). On the other hand, if both the spatial and temporal coordinates are fractal functions, it results that, the geodesics satisfy a Klein-Gordon-type of equation on a Minkowskian manifold.
Similar content being viewed by others
References
Madelbrot, B.: The Fractal Geometry of Nature. Freeman, San Francisco (1982)
Gouyet, J.F.: Physique et Structures Fractals. Masson, Paris (1992)
Nottale, L.: Fractal Space-Time and Microphysics: Towards a Theory of Scale Relativity. World Scientific, Singapore (1993)
El Naschie, M.S., Rösler, O.E., Prigogine, I. (eds.): Quantum Mechanics, Diffusion and Chaotic Fractals. Elsevier, Oxford (1995)
Weibel, P., Ord, G., Rössler, G. (eds.): Space-time Physics and Fractality. Springer, New York (2005)
Finkelstein, D.R., Saller, H., Tang, Z.: Quantum space-time. In: Pronin, P., Sardanashvily, G. (eds.) Gravity and Space-Time, pp. 145–171. World Scientific, Singapore (1996)
Finkelstein, D., Rodriguez, E.: Quantum time-space and gravity. In: Penrose, R., Isham, C.J. (eds.) Quantum Concepts in Space and Time, pp. 247–254. Oxford (1986)
EL-Nabulsi, A.R.: Chaos Solitons Fractals 42(5), 2929–2933 (2009)
Cresson, J., Ben Adda, F.: Chaos Solitons Fractals 19, 1323 (2004)
Cresson, J.: J. Math. Anal. Appl. 307, 48 (2005)
Nottale, L., Célérier, M.N., Lehner, T.: J. Math. Phys. 47, 032303 (2006)
Célérier, M.N., Nottale, L.: J. Phys. A. Math. Gen. 37, 931 (2004)
El Naschie, M.S.: Chaos Solitons Fractals 19(1), 209–236 (2004)
El Naschie, M.S.: Chaos Solitons Fractals 25(5), 955–964 (2005)
El Naschie, M.S.: Chaos Solitons Fractals 38(5), 1318–1322 (2008)
Marek-Crnjac, L.: Chaos Solitons Fractals 41(5), 2697–2705 (2009)
Marek-Crnjac, L.: Chaos Solitons Fractals 41(5), 2471–2473 (2009)
Gottlieb, I., Agop, M., Ciobanu, G., Stroe, A.: Chaos Solitons Fractals 30, 380 (2006)
Agop, M., Ioannou, P.D., Nica, P.: J. Math. Phys. 46, 062110 (2005)
Agop, M., Nica, P.E., Ioannou, P.D., Antici, A., Paun, V.P.: Eur. Phys. J. D 49, 239–248 (2008)
Agop, M., Nica, P., Girtu, M.: Gen. Relativ. Gravit. 40, 35 (2008)
Agop, M., Nica, P., Ioannou, P.D., Malandraki, O., Gavanas-Pahomi, I.: Chaos Solitons Fractals 34, 1704 (2007)
Nottale, L.: L’univers et la Lumiére. Cosmologie Classique et Mirages Gravitationnels. Flammarion, Paris (1993)
He, J.H.: Chaos Solitons Fractals 36(3), 542–545 (2008)
He, J.H., Wu, G.C., Austin, F.: Nonlinear Sci. Lett. A 1, 1–30 (2010)
Yang, C.D.: Nonlinear Sci. Lett. A 1, 31–37 (2010)
Buzea, C.G., Rusu, I., Bulancea, V., Badarau, G., Paun, V.P., Agop, M.: Nonlinear Sci. Lett. A 1, 109–142 (2010)
Chiroiu, V., Stiuca, P., Munteanu, L., Danescu, S.: Introduction in Nanomechanics. Romanian Academy Publishing House, Bucharest (2005)
Ferry, D.K., Goodnick, S.M.: Transport in Nanostructures. Cambridge University Press, Cambridge (1997)
Halbwachs, F.: Theorie Relativiste des Fluid a Spin. Gauthier-Villars, Paris (1960)
Wilhem, H.E.: Phys. Rev. D 1, 2278 (1970)
Landau, L., Lifshitz, E.: Fluid Mechanics. Butterworth-Heinemann, Oxford (1987)
Spitzer, L.: Physics of Fully Ionized Gases. Wiley, New York (1962)
Turcu, I.C.E., Dance, J.B.: X-rays from Laser Plasmas. Wiley, Chichester (1998)
Zienkievicz, O.C., Taylor, R.L.: The Finite Element Method. McGraw-Hill, New York (1991)
Gurlui, S., Agop, M., Nica, P., Ziskind, M., Focsa, C.: Phys. Rev. E 78, 062706 (2008)
Harilal, S.S., Bindhu, C.V., Tillack, M.S., Najmabadi, F., Gaeris, A.C.: J. Appl. Phys. 93, 2380 (2003)
Bulgakov, A.V., Bulgakova, N.M.: J. Phys. D 31, 693 (1998)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Agop, M., Niculescu, O., Timofte, A. et al. Non-Differentiable Mechanical Model and Its Implications. Int J Theor Phys 49, 1489–1506 (2010). https://doi.org/10.1007/s10773-010-0330-5
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10773-010-0330-5