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Lagrangian mechanics of fractional order, Hamilton–Jacobi fractional PDE and Taylor’s series of nondifferentiable functions

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  • Jumarie, Guy

Abstract

The paper proposes an extension of the Lagrange analytical mechanics to deal with dynamics of fractal nature. First of all, by using fractional difference, one introduces a slight modification of the Riemann–Liouville derivative definition, which is more consistent with self-similarity by removing the effect of the initial value, and then for the convenience of the reader, one gives a brief background on the Taylor’s series of fractional order f(x+h)=Eα(hαDxα)f(x) of nondifferentiable function, where Eα is the Mittag–Leffler function. The Lagrange characteristics method is extended for solving a class of nonlinear fractional partial differential equations. All this material is necessary to solve the problem of fractional optimal control and mainly to find the characteristics of its fractional Hamilton–Jacobi equation, therefore the canonical equations of optimality. Then fractional Lagrangian mechanics is considered as an application of fractional optimal control. In this framework, the use of complex-valued variables, as Nottale did it, appears as a direct consequence of the irreversibility of time.

Suggested Citation

  • Jumarie, Guy, 2007. "Lagrangian mechanics of fractional order, Hamilton–Jacobi fractional PDE and Taylor’s series of nondifferentiable functions," Chaos, Solitons & Fractals, Elsevier, vol. 32(3), pages 969-987.
  • Handle: RePEc:eee:chsofr:v:32:y:2007:i:3:p:969-987
    DOI: 10.1016/j.chaos.2006.07.053
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    References listed on IDEAS

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    1. Cioczek-Georges, R. & Mandelbrot, B. B., 1995. "A class of micropulses and antipersistent fractional Brownian motion," Stochastic Processes and their Applications, Elsevier, vol. 60(1), pages 1-18, November.
    2. Cioczek-Georges, R. & Mandelbrot, B. B., 1996. "Alternative micropulses and fractional Brownian motion," Stochastic Processes and their Applications, Elsevier, vol. 64(2), pages 143-152, November.
    3. Anh, V. V. & Leonenko, N. N., 2000. "Scaling laws for fractional diffusion-wave equations with singular data," Statistics & Probability Letters, Elsevier, vol. 48(3), pages 239-252, July.
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    1. Jumarie, Guy, 2009. "Probability calculus of fractional order and fractional Taylor’s series application to Fokker–Planck equation and information of non-random functions," Chaos, Solitons & Fractals, Elsevier, vol. 40(3), pages 1428-1448.
    2. Jumarie, Guy, 2009. "From Lagrangian mechanics fractal in space to space fractal Schrödinger’s equation via fractional Taylor’s series," Chaos, Solitons & Fractals, Elsevier, vol. 41(4), pages 1590-1604.
    3. Su, Xin & Yu, Keshu & Yu, Miao, 2019. "Research on early warning algorithm for economic management based on Lagrangian fractional calculus," Chaos, Solitons & Fractals, Elsevier, vol. 128(C), pages 44-50.

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