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Non-differentiable mechanical model and its implications. (English) Zbl 1197.83085

Summary: 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.

MSC:

83D05 Relativistic gravitational theories other than Einstein’s, including asymmetric field theories
83C10 Equations of motion in general relativity and gravitational theory
37F35 Conformal densities and Hausdorff dimension for holomorphic dynamical systems
76E20 Stability and instability of geophysical and astrophysical flows
81S10 Geometry and quantization, symplectic methods
83A05 Special relativity
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[1] Madelbrot, B.: The Fractal Geometry of Nature. Freeman, San Francisco (1982)
[2] Gouyet, J.F.: Physique et Structures Fractals. Masson, Paris (1992) · Zbl 0773.58015
[3] Nottale, L.: Fractal Space-Time and Microphysics: Towards a Theory of Scale Relativity. World Scientific, Singapore (1993) · Zbl 0789.58003
[4] El Naschie, M.S., Rösler, O.E., Prigogine, I. (eds.): Quantum Mechanics, Diffusion and Chaotic Fractals. Elsevier, Oxford (1995) · Zbl 0830.58001
[5] Weibel, P., Ord, G., Rössler, G. (eds.): Space-time Physics and Fractality. Springer, New York (2005) · Zbl 1103.82005
[6] 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)
[7] 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)
[8] EL-Nabulsi, A.R.: Chaos Solitons Fractals 42(5), 2929–2933 (2009)
[9] Cresson, J., Ben Adda, F.: Chaos Solitons Fractals 19, 1323 (2004) · Zbl 1053.81027 · doi:10.1016/S0960-0779(03)00339-4
[10] Cresson, J.: J. Math. Anal. Appl. 307, 48 (2005) · Zbl 1077.49033 · doi:10.1016/j.jmaa.2004.10.006
[11] Nottale, L., Célérier, M.N., Lehner, T.: J. Math. Phys. 47, 032303 (2006) · Zbl 1111.81119 · doi:10.1063/1.2176915
[12] Célérier, M.N., Nottale, L.: J. Phys. A. Math. Gen. 37, 931 (2004) · Zbl 1098.81730 · doi:10.1088/0305-4470/37/3/026
[13] El Naschie, M.S.: Chaos Solitons Fractals 19(1), 209–236 (2004) · Zbl 1071.81501 · doi:10.1016/S0960-0779(03)00278-9
[14] El Naschie, M.S.: Chaos Solitons Fractals 25(5), 955–964 (2005) · Zbl 1071.81503 · doi:10.1016/j.chaos.2004.12.033
[15] El Naschie, M.S.: Chaos Solitons Fractals 38(5), 1318–1322 (2008) · doi:10.1016/j.chaos.2008.06.025
[16] Marek-Crnjac, L.: Chaos Solitons Fractals 41(5), 2697–2705 (2009) · doi:10.1016/j.chaos.2008.10.007
[17] Marek-Crnjac, L.: Chaos Solitons Fractals 41(5), 2471–2473 (2009) · doi:10.1016/j.chaos.2008.09.014
[18] Gottlieb, I., Agop, M., Ciobanu, G., Stroe, A.: Chaos Solitons Fractals 30, 380 (2006) · doi:10.1016/j.chaos.2005.11.018
[19] Agop, M., Ioannou, P.D., Nica, P.: J. Math. Phys. 46, 062110 (2005) · Zbl 1110.82315 · doi:10.1063/1.1904163
[20] Agop, M., Nica, P.E., Ioannou, P.D., Antici, A., Paun, V.P.: Eur. Phys. J. D 49, 239–248 (2008) · doi:10.1140/epjd/e2008-00161-8
[21] Agop, M., Nica, P., Girtu, M.: Gen. Relativ. Gravit. 40, 35 (2008) · Zbl 1136.83320 · doi:10.1007/s10714-007-0519-y
[22] Agop, M., Nica, P., Ioannou, P.D., Malandraki, O., Gavanas-Pahomi, I.: Chaos Solitons Fractals 34, 1704 (2007) · doi:10.1016/j.chaos.2006.05.014
[23] Nottale, L.: L’univers et la Lumiére. Cosmologie Classique et Mirages Gravitationnels. Flammarion, Paris (1993)
[24] He, J.H.: Chaos Solitons Fractals 36(3), 542–545 (2008) · doi:10.1016/j.chaos.2007.07.093
[25] He, J.H., Wu, G.C., Austin, F.: Nonlinear Sci. Lett. A 1, 1–30 (2010)
[26] Yang, C.D.: Nonlinear Sci. Lett. A 1, 31–37 (2010)
[27] Buzea, C.G., Rusu, I., Bulancea, V., Badarau, G., Paun, V.P., Agop, M.: Nonlinear Sci. Lett. A 1, 109–142 (2010)
[28] Chiroiu, V., Stiuca, P., Munteanu, L., Danescu, S.: Introduction in Nanomechanics. Romanian Academy Publishing House, Bucharest (2005)
[29] Ferry, D.K., Goodnick, S.M.: Transport in Nanostructures. Cambridge University Press, Cambridge (1997)
[30] Halbwachs, F.: Theorie Relativiste des Fluid a Spin. Gauthier-Villars, Paris (1960) · Zbl 0089.21204
[31] Wilhem, H.E.: Phys. Rev. D 1, 2278 (1970) · Zbl 0195.28201 · doi:10.1103/PhysRevD.1.2278
[32] Landau, L., Lifshitz, E.: Fluid Mechanics. Butterworth-Heinemann, Oxford (1987) · Zbl 0146.22405
[33] Spitzer, L.: Physics of Fully Ionized Gases. Wiley, New York (1962) · Zbl 0074.45001
[34] Turcu, I.C.E., Dance, J.B.: X-rays from Laser Plasmas. Wiley, Chichester (1998)
[35] Zienkievicz, O.C., Taylor, R.L.: The Finite Element Method. McGraw-Hill, New York (1991)
[36] Gurlui, S., Agop, M., Nica, P., Ziskind, M., Focsa, C.: Phys. Rev. E 78, 062706 (2008) · doi:10.1103/PhysRevE.78.026405
[37] Harilal, S.S., Bindhu, C.V., Tillack, M.S., Najmabadi, F., Gaeris, A.C.: J. Appl. Phys. 93, 2380 (2003) · doi:10.1063/1.1544070
[38] Bulgakov, A.V., Bulgakova, N.M.: J. Phys. D 31, 693 (1998) · doi:10.1088/0022-3727/31/6/017
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