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Czechoslovak Journal of Physics

, Volume 52, Issue 11, pp 1247–1253 | Cite as

Lagrangean and Hamiltonian fractional sequential mechanics

  • Małgorzata Klimek
Article

Abstract

The models described by fractional order derivatives of Riemann-Liouville type in sequential form are discussed in Lagrangean and Hamiltonian formalism. The Euler-Lagrange equations are derived using the minimum action principle. Then the methods of generalized mechanics are applied to obtain the Hamilton’s equations. As an example free motion in fractional picture is studied. The respective fractional differential equations are explicitly solved and it is shown that the limitα→1+ recovers classical model with linear trajectories and constant velocity.

PACS

03.20.+i 46.10.+z 02.30.-f 46.90.+s 

Key words

fractional derivative fractional integral fractional mechanics Euler-Lagrange equations Hamilton equation non-conservative systems 

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References

  1. [1]
    K.B. Oldham and J. Spanier:The Fractional Calculus. Academic Press, New York, 1974.zbMATHGoogle Scholar
  2. [2]
    K.S. Miller and B. Ross:An Introduction to the Fractional Calculus and Fractional Differential Equations. John Wiley & Sons, New York, 1993.zbMATHGoogle Scholar
  3. [3]
    S.G. Samko, A.A. Kilbas, and O.I. Marichev:Fractional Derivatives and Integrals. Theory and Applications. Gordon and Breach, Amsterdam, 1993.Google Scholar
  4. [4]
    E.R. Love and L.C. Young: Proc. London Math. Soc.44 (1938) 1.zbMATHCrossRefGoogle Scholar
  5. [5]
    F. Riewe: Phys. Rev. E53 (1996) 1890;55 (1997) 3581.CrossRefADSMathSciNetGoogle Scholar
  6. [6]
    H. Goldstein:Classical Mechanics. Addison-Wesley, Reading (MA), 1950;Google Scholar
  7. [6a]
    M. De Leon and P.R. Rodriguez:Generalized Classical Mechanics and Field Theory: A Geometrical Approach of Lagrangian and Hamiltonian Formalisms Involving Higher Order Derivatives. North-Holland, Amsterdam, 1985.zbMATHGoogle Scholar
  8. [7]
    M. Klimek: Czech. J. Phys.51 (2001) 1348.zbMATHCrossRefADSMathSciNetGoogle Scholar

Copyright information

© Springer 2002

Authors and Affiliations

  • Małgorzata Klimek
    • 1
  1. 1.Institute of Mathematics and Computer ScienceTechnical University of CzęstochowaCzęstochowaPoland

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