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On Reflection Symmetry and Its Application to the Euler-Lagrange Equations in Fractional Mechanics

[+] Author Affiliations
Malgorzata Klimek

Czestochowa University of Technology, Czestochowa, Poland

Paper No. DETC2011-47721, pp. 241-250; 10 pages
doi:10.1115/DETC2011-47721
From:
  • ASME 2011 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference
  • Volume 3: 2011 ASME/IEEE International Conference on Mechatronic and Embedded Systems and Applications, Parts A and B
  • Washington, DC, USA, August 28–31, 2011
  • Conference Sponsors: Design Engineering Division and Computers and Information in Engineering Division
  • ISBN: 978-0-7918-5480-8
  • Copyright © 2011 by ASME

abstract

We study the properties of fractional differentiation with respect to the reflection symmetry in a finite interval. The representation and integration formulas are derived for the symmetric and anti-symmetric fractional derivatives, both of the Riemann -Liouville and Caputo type. The action dependent on the left -sided Caputo derivatives of orders in range (1.2) is considered and we derive the Euler-Lagrange equations for the symmetric and anti-symmetric part of the trajectory. The procedure is illustrated with an example of the action dependent linearly on fractional velocities. For the obtained Euler-Lagrange system we discuss its localization resulting from the subsequent sym-metrization of the action.

Copyright © 2011 by ASME
Topics: Reflection , Equations

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