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Fractional Analogous Models in Mechanics and Gravity Theories

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Fractional Dynamics and Control

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Notes

  1. 1.

    We recommend readers to consult in advance the above cited papers on details, notation conventions, and bibliography.

  2. 2.

    The symbol T is underlined to emphasize that we shall associate the approach with a fractional Caputo derivative.

  3. 3.

    We shall put a left label L to certain geometric objects if it is necessary to emphasize that they are induced by Lagrange generating function. Nevertheless, such labels will be omitted (to simplify the notations) if that will not result in ambiguities.

  4. 4.

    For integer dimensions, we contract “horizontal” and “vertical” indices following the rule: i = 1 is \(a = n + 1;\) i = 2 is \(a = n + 2;\) ... i = n is \(a = n + n.\)

  5. 5.

    A nonholonomic manifold is a manifold endowed with a non-integrable (equivalently, nonholonomic, or anholonomic) distribution. There are three useful (for our considerations) examples when (1) V is a (pseudo) Riemannian manifold; (2) V = E(M), or (3) V = TM, for a vector, or tangent, bundle on a base manifold M. We also emphasize that in this chapter we follow the conventions from [7,  1,  2] when left indices are used as labels and right indices may be abstract ones or running certain values.

References

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Acknowledgements

This chapter summarizes the results presented in our corresponding talk at Conference ”New Trends in Nanotechnology and Nonlinear Dynamical Systems”, 25–27 July, 2010, Çankaya University, Ankara, Turkey.

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Authors and Affiliations

  1. Department of Mathematics and Computer Sciences, Cankaya University, 06530, Ankara, Turkey

    Dumitru Baleanu

  2. Institute of Space Sciences, MG-23, R 76900, Magurele–Bucharest, Romania

    Dumitru Baleanu

  3. Science Department, University “Al. I. Cuza” Iaşi, 54, Lascar Catargi street, Iaşi, Romania, 700107

    Sergiu I. Vacaru

Authors
  1. Dumitru Baleanu
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  2. Sergiu I. Vacaru
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Corresponding author

Correspondence to Dumitru Baleanu .

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Editors and Affiliations

  1. Faculty of Art and Sciences, Mathematics and Computer Sciences, Cankaya University, Ankara, Turkey

    Dumitru Baleanu

  2. Institute of Engineering, Department of Electrical Engineering, Polytechnic Institute of Porto, Rua Dr António Bernadino Almeida 431, Porto, 4200-072, Portugal

    José António Tenreiro Machado

  3. Edwardsville, School of Engineering, Southern Illinois University, Room: EB2064, Edwardsville, 62026-1805, USA

    Albert C. J. Luo

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Baleanu, D., Vacaru, S.I. (2012). Fractional Analogous Models in Mechanics and Gravity Theories. In: Baleanu, D., Machado, J., Luo, A. (eds) Fractional Dynamics and Control. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-0457-6_16

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  • DOI: https://doi.org/10.1007/978-1-4614-0457-6_16

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  • Publisher Name: Springer, New York, NY

  • Print ISBN: 978-1-4614-0456-9

  • Online ISBN: 978-1-4614-0457-6

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