Lagrangian and Hamiltonian mechanics on fractals subset of real-line. (English) Zbl 1282.70033

Summary: A discontinuous media can be described by fractal dimensions. Fractal objects has special geometric properties, which are discrete and discontinuous structure. A fractal-time diffusion equation is a model for subdiffusive. In this work, we have generalized the Hamiltonian and Lagrangian dynamics on fractal using the fractional local derivative, so one can use as a new mathematical model for the motion in the fractal media. More, Poisson bracket on fractal subset of real line is suggested.


70H03 Lagrange’s equations
70H05 Hamilton’s equations
28A80 Fractals
26A33 Fractional derivatives and integrals
Full Text: DOI


[1] Mandelbrot, B.B.: The Fractal Geometry of Nature. Freeman, New York (1977) · Zbl 0504.28001
[2] Bunde, A., Havlin, S. (eds.): Fractal in Science. Springer, Berlin (1995) · Zbl 0796.00013
[3] Barlow, M.T., Perkins, E.A.: Brownian motion on the Sierpinsski gasket. Probab. Theory Relat. Fields 79, 543-623 (1988) · Zbl 0635.60090
[4] Ben-Avraham, D., Havlin, S.: Diffusion and Reactions in Fractals and Disordered Systems. Cambridge University Press, Cambridge (2000) · Zbl 1075.82001
[5] Kigami, J.: Analysis on Fractals. Cambridge Tracts in Mathematics, vol. 143. Cambridge University Press, Cambridge (2001) · Zbl 0998.28004
[6] Falconer, K.: The Geometry of Fractal Sets. Cambridge University Press, Cambridge (1985) · Zbl 0587.28004
[7] Falconer, K.: Fractal Geometry: Mathematical Foundations and Applications. Wiley, New York (1990) · Zbl 0689.28003
[8] Falconer, K.: Techniques in Fractal Geometry. Wiley, New York (1997) · Zbl 0869.28003
[9] Edgar, G.A.: Integral, Probability and Fractal Measures. Springer, New York (1998) · Zbl 0893.28001
[10] Yang, X.J.: Advanced Local Fractional Calculus and Its Applications. World Science Publisher, New York (2012)
[11] Yang, X.J., Baleanu, D.: Fractal heat conduction problem solved by local fractional variation iteration method. Therm. Sci. 17(2), 625 (2013)
[12] Ming-Sheng, H., Agarwal Ravi, P., Yang, X.J.: Local fractional Fourier series with application to wave equation in fractal vibrating string. Abstr. Appl. Anal. (2012). doi:10.1155/2012/567401 · Zbl 1257.35193
[13] Yang, X.J., Srivastava, H.M., He, J.H., Baleanu, D.: Cantor-type cylindrical-coordinate method for differential equations with local fractional derivatives. Phys. Lett. A 377, 1696 (2013) · Zbl 1298.35243
[14] Su, W.H., Yang, X.J., Jafari, H., Baleanu, D.: Fractional complex transform method for wave equations on Cantor sets within local fractional differential operator. Adv. Differ. Equ. 2013(1), 97 (2013) · Zbl 1380.35163
[15] Satin, S., Gangal, A.D.: 2011, Random walk and broad distributions on fractal curves. arXiv preprint. arXiv:1103.5249 · Zbl 1448.60101
[16] Kolwankar, K.M., Gangal, A.D.: Fractional differentiability of nowhere differentiable functions and dimensions. Chaos 6, 505 (1996) · Zbl 1055.26504
[17] Kolwankar, K.M., Gangal, A.D.: Local fractional Fokker-Planck equation. Phys. Rev. Lett. 80, 214 (1998) · Zbl 0945.82005
[18] Adda, F.B., Cresson, J.: About non-differentiable functions. J. Math. Anal. Appl. 263, 721-737 (2001) · Zbl 0995.26006
[19] Baleanu, D., Golmankhaneh, K.A., Golmankhaneh, K.A., Baleanu, M.C.: Fractional electromagnetic equations using fractional forms. Int. J. Theor. Phys. 48, 3114-3123 (2009) · Zbl 1184.83024
[20] Golmankhaneh, K.A., Yengejeh, M.A., Baleanu, D.: On the fractional Hamilton and Lagrange mechanics. Int. J. Theor. Phys. 51, 2909-2916 (2012) · Zbl 1256.35192
[21] Golmankhaneh, K.A., Fazlollahi, V., Baleanu, D.: Newtonian mechanics on fractals subset of real-line. Rom. Rep. Phys. 65, 84-93 (2013)
[22] Golmankhaneh, K.A., Golmankhaneh, K.A., Baleanu, D.: About Maxwell’s equations on fractal subsets of R3. Cent. Eur. J. Phys. (2013). doi:10.2478/s11534-013-0192-6 · Zbl 1274.74062
[23] Rocco, A., West, B.J.: Fractional calculus and evolution of fractal phenomena. Physica A 265, 535 (1999)
[24] Barlow, M.T.: Diffusion on Fractals. Lecture Notes Math., vol. 1690. Springer, Berlin (1998) · Zbl 0916.60069
[25] Metzler, R., Klafter, J.: The random walks guide to anomalous diffusion a fractional dynamics approach. Phys. Rep. 339, 1-77 (2000) · Zbl 0984.82032
[26] Zaslavsky, G.M.: Fractional kinetic equation for Hamiltonian chaos. Physica D 76, 110 (1994) · Zbl 1194.37163
[27] Hilfer, R.: Application of Fractional Calculus in Physics. World Scientific, Singapore (2000) · Zbl 0998.26002
[28] Samko, S.G., Kilbas, A.A., Marichev, O.I.: Fractional Integrals and Derivative—Theory and Applications. Gordon and Breach, New York (1993) · Zbl 0818.26003
[29] Metzler, R., Barkai, E., Klafter, J.: Anomalus diffusion and relaxation close thermal equilibrium: a fractional Fokker-Planck equation approach. Phys. Rev. Lett. 83, 3563 (1999)
[30] Metzler, R., Glöckle, W.G., Nonnenmacher, T.F.: Fractional model equation for anomalous diffusion. Physica A 13, 211 (1994)
[31] Freiberg, U., Zähle, M.: Harmonic calculus on fractals—a measure geometric approach II. Trans. Am. Math. Soc. 357.9, 3407-3423 (2005) · Zbl 1071.28008
[32] Dalrymple, K., Strichartz, R.S., Vinson, J.P.: Fractal differential equations on the Sierpinski gasket. J. Fourier Anal. Appl. 5, 205 (1999) · Zbl 0937.31010
[33] Strichartz, R.S.: Differential Equations of Fractals. Princeton University Press, New Jersey (2006) · Zbl 1190.35001
[34] Parvate, A., Gangal, A.D.: Calculus on fractal subsets of real line-I: formulation. Fractals 17, 53-81 (2009) · Zbl 1173.28005
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.