# American Institute of Mathematical Sciences

2014, 19(8): 2367-2381. doi: 10.3934/dcdsb.2014.19.2367

## Fractional variational principle of Herglotz

 1 CIDMA — Center for Research and Development in Mathematics and Applications, Department of Mathematics, University of Aveiro, 3810-193 Aveiro 2 Faculty of Computer Science, Bialystok University of Technology, 15-351 Białystok, Poland

Received  April 2014 Revised  April 2014 Published  August 2014

The aim of this paper is to bring together two approaches to non-conservative systems -- the generalized variational principle of Herglotz and the fractional calculus of variations. Namely, we consider functionals whose extrema are sought, by differential equations that involve Caputo fractional derivatives. The Euler--Lagrange equations are obtained for the fractional variational problems of Herglotz-type and the transversality conditions are derived. The fractional Noether-type theorem for conservative and non-conservative physical systems is proved.
Citation: Ricardo Almeida, Agnieszka B. Malinowska. Fractional variational principle of Herglotz. Discrete & Continuous Dynamical Systems - B, 2014, 19 (8) : 2367-2381. doi: 10.3934/dcdsb.2014.19.2367
##### References:
 [1] D. V. Anosov, S. K. Aranson, V. I. Arnold, I. U. Bronshtein, V. Z. Grines and Y. S. Ilyashenko, Ordinary Differential Equations and Smooth Dynamical Systems,, Encyclopaedia of Mathematical Sciences, (1988). [2] T. M. Atanacković, S. Konjik and S. Pilipović, Variational problems with fractional derivatives: Euler-Lagrange equations,, J. Phys. A, 41 (2008). doi: 10.1088/1751-8113/41/9/095201. [3] T. M. Atanacković and B. Stanković, On a numerical scheme for solving differential equations of fractional order,, Mech. Res. Comm., 35 (2008), 429. doi: 10.1016/j.mechrescom.2008.05.003. [4] G. S. F. Frederico and D. F. M. Torres, A formulation of Noethers theorem for fractional problems of the calculus of variations,, J. Math. Anal. Appl., 334 (2007), 834. doi: 10.1016/j.jmaa.2007.01.013. [5] N. J. Ford and M. L. Morgado, Fractional boundary value problems: Analysis and numerical methods,, Fract. Calc. Appl. Anal., 14 (2011), 554. doi: 10.2478/s13540-011-0034-4. [6] N. J. Ford and M. L. Morgado, Distributed order equations as boundary value problems,, Comput. Math. Appl., 64 (2012), 2973. doi: 10.1016/j.camwa.2012.01.053. [7] B. Georgieva and R. Guenther, First Noether-type theorem for the generalized variational principle of Herglotz,, Topol. Methods Nonlinear Anal., 20 (2002), 261. [8] B. Georgieva and R. Guenther, Second Noether-type theorem for the generalized variational principle of Herglotz,, Topol. Methods Nonlinear Anal., 26 (2005), 307. [9] B. Georgieva, R. Guenther and T. Bodurov, Generalized variational principle of Herglotz for several independent variables. First Noether-type theorem,, J. Math. Phys., 44 (2003), 3911. doi: 10.1063/1.1597419. [10] H. Goldstein, Classical Mechanics,, Addison-Wesley Press, (1951). [11] R. B. Guenther, C. M. Guenther and J. A. Gottsch, The Herglotz Lectures on Contact Trans- formations and Hamiltonian Systems,, Lecture Notes in Nonlinear Analysis, (1996). [12] G. Herglotz, Berührungstransformationen,, Lectures at the University of Göttingen, (1930). [13] A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations,, North-Holland Mathematics Studies, (2006). [14] M. Klimek, On Solutions of Linear Fractional Differential Equations of a Variational Type,, The Publishing Office of Czenstochowa University of Technology, (2009). [15] C. Lánczos, The Variational Principles of Mechanics,, Fourth edition, (1970). [16] A. B. Malinowska and D. F. M. Torres, Introduction to the Fractional Calculus of Variations,, Imp. Coll. Press, (2012). doi: 10.1142/p871. [17] A. B. Malinowska, A formulation of the fractional Noether-type theorem for multidimensional Lagrangians,, Appl. Math. Lett., 25 (2012), 1941. doi: 10.1016/j.aml.2012.03.006. [18] V. J. Menon, N. Chanana and Y. Singh, A fresh look at the BCK frictional lagrangian,, Prog. Theor. Phys., 98 (1997), 321. doi: 10.1143/PTP.98.321. [19] I. Podlubny, Fractional Differential Equations, Mathematics in Science and Engineering,, Mathematics in Science and Engineering, (1999). [20] S. Pooseh, R. Almeida and D. F. M. Torres, Approximation of fractional integrals by means of derivatives,, Comput. Math. Appl., 64 (2012), 3090. doi: 10.1016/j.camwa.2012.01.068. [21] F. Riewe, Nonconservative Lagrangian and Hamiltonian mechanics,, Phys. Rev. E, 53 (1996), 1890. doi: 10.1103/PhysRevE.53.1890. [22] S. G. Samko, A. A. Kilbas and O. I. Marichev, Fractional Integrals and Derivatives,, translated from the 1987 Russian original, (1987). [23] S. Santos, N. Martins and D. F. M. Torres, Higher-order variational problems of Herglotz type,, Vietnam Journal of Mathematics, (2013), 1. doi: 10.1007/s10013-013-0048-9.

show all references

##### References:
 [1] D. V. Anosov, S. K. Aranson, V. I. Arnold, I. U. Bronshtein, V. Z. Grines and Y. S. Ilyashenko, Ordinary Differential Equations and Smooth Dynamical Systems,, Encyclopaedia of Mathematical Sciences, (1988). [2] T. M. Atanacković, S. Konjik and S. Pilipović, Variational problems with fractional derivatives: Euler-Lagrange equations,, J. Phys. A, 41 (2008). doi: 10.1088/1751-8113/41/9/095201. [3] T. M. Atanacković and B. Stanković, On a numerical scheme for solving differential equations of fractional order,, Mech. Res. Comm., 35 (2008), 429. doi: 10.1016/j.mechrescom.2008.05.003. [4] G. S. F. Frederico and D. F. M. Torres, A formulation of Noethers theorem for fractional problems of the calculus of variations,, J. Math. Anal. Appl., 334 (2007), 834. doi: 10.1016/j.jmaa.2007.01.013. [5] N. J. Ford and M. L. Morgado, Fractional boundary value problems: Analysis and numerical methods,, Fract. Calc. Appl. Anal., 14 (2011), 554. doi: 10.2478/s13540-011-0034-4. [6] N. J. Ford and M. L. Morgado, Distributed order equations as boundary value problems,, Comput. Math. Appl., 64 (2012), 2973. doi: 10.1016/j.camwa.2012.01.053. [7] B. Georgieva and R. Guenther, First Noether-type theorem for the generalized variational principle of Herglotz,, Topol. Methods Nonlinear Anal., 20 (2002), 261. [8] B. Georgieva and R. Guenther, Second Noether-type theorem for the generalized variational principle of Herglotz,, Topol. Methods Nonlinear Anal., 26 (2005), 307. [9] B. Georgieva, R. Guenther and T. Bodurov, Generalized variational principle of Herglotz for several independent variables. First Noether-type theorem,, J. Math. Phys., 44 (2003), 3911. doi: 10.1063/1.1597419. [10] H. Goldstein, Classical Mechanics,, Addison-Wesley Press, (1951). [11] R. B. Guenther, C. M. Guenther and J. A. Gottsch, The Herglotz Lectures on Contact Trans- formations and Hamiltonian Systems,, Lecture Notes in Nonlinear Analysis, (1996). [12] G. Herglotz, Berührungstransformationen,, Lectures at the University of Göttingen, (1930). [13] A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations,, North-Holland Mathematics Studies, (2006). [14] M. Klimek, On Solutions of Linear Fractional Differential Equations of a Variational Type,, The Publishing Office of Czenstochowa University of Technology, (2009). [15] C. Lánczos, The Variational Principles of Mechanics,, Fourth edition, (1970). [16] A. B. Malinowska and D. F. M. Torres, Introduction to the Fractional Calculus of Variations,, Imp. Coll. Press, (2012). doi: 10.1142/p871. [17] A. B. Malinowska, A formulation of the fractional Noether-type theorem for multidimensional Lagrangians,, Appl. Math. Lett., 25 (2012), 1941. doi: 10.1016/j.aml.2012.03.006. [18] V. J. Menon, N. Chanana and Y. Singh, A fresh look at the BCK frictional lagrangian,, Prog. Theor. Phys., 98 (1997), 321. doi: 10.1143/PTP.98.321. [19] I. Podlubny, Fractional Differential Equations, Mathematics in Science and Engineering,, Mathematics in Science and Engineering, (1999). [20] S. Pooseh, R. Almeida and D. F. M. Torres, Approximation of fractional integrals by means of derivatives,, Comput. Math. Appl., 64 (2012), 3090. doi: 10.1016/j.camwa.2012.01.068. [21] F. Riewe, Nonconservative Lagrangian and Hamiltonian mechanics,, Phys. Rev. E, 53 (1996), 1890. doi: 10.1103/PhysRevE.53.1890. [22] S. G. Samko, A. A. Kilbas and O. I. Marichev, Fractional Integrals and Derivatives,, translated from the 1987 Russian original, (1987). [23] S. Santos, N. Martins and D. F. M. Torres, Higher-order variational problems of Herglotz type,, Vietnam Journal of Mathematics, (2013), 1. doi: 10.1007/s10013-013-0048-9.
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