October  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, 1, Springer, Berlin, 1988.  Google Scholar

[2]

T. M. Atanacković, S. Konjik and S. Pilipović, Variational problems with fractional derivatives: Euler-Lagrange equations, J. Phys. A, 41 (2008), 095201, 12 pp. doi: 10.1088/1751-8113/41/9/095201.  Google Scholar

[3]

T. M. Atanacković and B. Stanković, On a numerical scheme for solving differential equations of fractional order, Mech. Res. Comm., 35 (2008), 429-438. doi: 10.1016/j.mechrescom.2008.05.003.  Google Scholar

[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), 834-846. doi: 10.1016/j.jmaa.2007.01.013.  Google Scholar

[5]

N. J. Ford and M. L. Morgado, Fractional boundary value problems: Analysis and numerical methods, Fract. Calc. Appl. Anal., 14 (2011), 554-567. doi: 10.2478/s13540-011-0034-4.  Google Scholar

[6]

N. J. Ford and M. L. Morgado, Distributed order equations as boundary value problems, Comput. Math. Appl., 64 (2012), 2973-2981. doi: 10.1016/j.camwa.2012.01.053.  Google Scholar

[7]

B. Georgieva and R. Guenther, First Noether-type theorem for the generalized variational principle of Herglotz, Topol. Methods Nonlinear Anal., 20 (2002), 261-273.  Google Scholar

[8]

B. Georgieva and R. Guenther, Second Noether-type theorem for the generalized variational principle of Herglotz, Topol. Methods Nonlinear Anal., 26 (2005), 307-314.  Google Scholar

[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), 3911-3927. doi: 10.1063/1.1597419.  Google Scholar

[10]

H. Goldstein, Classical Mechanics, Addison-Wesley Press, Inc., Cambridge, MA, 1951.  Google Scholar

[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, 1, Juliusz Schauder Center for Nonlinear Studies, Nicholas Copernicus University, Toruń, 1996. Google Scholar

[12]

G. Herglotz, Berührungstransformationen, Lectures at the University of Göttingen, Göttingen, 1930. Google Scholar

[13]

A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, North-Holland Mathematics Studies, 204, Elsevier Science B. V., Amsterdam, 2006.  Google Scholar

[14]

M. Klimek, On Solutions of Linear Fractional Differential Equations of a Variational Type, The Publishing Office of Czenstochowa University of Technology, Czestochowa, 2009. Google Scholar

[15]

C. Lánczos, The Variational Principles of Mechanics, Fourth edition, Mathematical Expositions, No. 4, Univ. Toronto Press, Toronto, ON, 1970.  Google Scholar

[16]

A. B. Malinowska and D. F. M. Torres, Introduction to the Fractional Calculus of Variations, Imp. Coll. Press, London, 2012. doi: 10.1142/p871.  Google Scholar

[17]

A. B. Malinowska, A formulation of the fractional Noether-type theorem for multidimensional Lagrangians, Appl. Math. Lett., 25 (2012), 1941-1946. doi: 10.1016/j.aml.2012.03.006.  Google Scholar

[18]

V. J. Menon, N. Chanana and Y. Singh, A fresh look at the BCK frictional lagrangian, Prog. Theor. Phys., 98 (1997), 321-329. doi: 10.1143/PTP.98.321.  Google Scholar

[19]

I. Podlubny, Fractional Differential Equations, Mathematics in Science and Engineering, Mathematics in Science and Engineering, 198, Academic Press, Inc., San Diego, CA, 1999.  Google Scholar

[20]

S. Pooseh, R. Almeida and D. F. M. Torres, Approximation of fractional integrals by means of derivatives, Comput. Math. Appl., 64 (2012), 3090-3100. doi: 10.1016/j.camwa.2012.01.068.  Google Scholar

[21]

F. Riewe, Nonconservative Lagrangian and Hamiltonian mechanics, Phys. Rev. E, 53 (1996), 1890-1899. doi: 10.1103/PhysRevE.53.1890.  Google Scholar

[22]

S. G. Samko, A. A. Kilbas and O. I. Marichev, Fractional Integrals and Derivatives, translated from the 1987 Russian original, Gordon and Breach, Yverdon, 1993.  Google Scholar

[23]

S. Santos, N. Martins and D. F. M. Torres, Higher-order variational problems of Herglotz type, Vietnam Journal of Mathematics, (2013), 1-11. doi: 10.1007/s10013-013-0048-9.  Google Scholar

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, 1, Springer, Berlin, 1988.  Google Scholar

[2]

T. M. Atanacković, S. Konjik and S. Pilipović, Variational problems with fractional derivatives: Euler-Lagrange equations, J. Phys. A, 41 (2008), 095201, 12 pp. doi: 10.1088/1751-8113/41/9/095201.  Google Scholar

[3]

T. M. Atanacković and B. Stanković, On a numerical scheme for solving differential equations of fractional order, Mech. Res. Comm., 35 (2008), 429-438. doi: 10.1016/j.mechrescom.2008.05.003.  Google Scholar

[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), 834-846. doi: 10.1016/j.jmaa.2007.01.013.  Google Scholar

[5]

N. J. Ford and M. L. Morgado, Fractional boundary value problems: Analysis and numerical methods, Fract. Calc. Appl. Anal., 14 (2011), 554-567. doi: 10.2478/s13540-011-0034-4.  Google Scholar

[6]

N. J. Ford and M. L. Morgado, Distributed order equations as boundary value problems, Comput. Math. Appl., 64 (2012), 2973-2981. doi: 10.1016/j.camwa.2012.01.053.  Google Scholar

[7]

B. Georgieva and R. Guenther, First Noether-type theorem for the generalized variational principle of Herglotz, Topol. Methods Nonlinear Anal., 20 (2002), 261-273.  Google Scholar

[8]

B. Georgieva and R. Guenther, Second Noether-type theorem for the generalized variational principle of Herglotz, Topol. Methods Nonlinear Anal., 26 (2005), 307-314.  Google Scholar

[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), 3911-3927. doi: 10.1063/1.1597419.  Google Scholar

[10]

H. Goldstein, Classical Mechanics, Addison-Wesley Press, Inc., Cambridge, MA, 1951.  Google Scholar

[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, 1, Juliusz Schauder Center for Nonlinear Studies, Nicholas Copernicus University, Toruń, 1996. Google Scholar

[12]

G. Herglotz, Berührungstransformationen, Lectures at the University of Göttingen, Göttingen, 1930. Google Scholar

[13]

A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, North-Holland Mathematics Studies, 204, Elsevier Science B. V., Amsterdam, 2006.  Google Scholar

[14]

M. Klimek, On Solutions of Linear Fractional Differential Equations of a Variational Type, The Publishing Office of Czenstochowa University of Technology, Czestochowa, 2009. Google Scholar

[15]

C. Lánczos, The Variational Principles of Mechanics, Fourth edition, Mathematical Expositions, No. 4, Univ. Toronto Press, Toronto, ON, 1970.  Google Scholar

[16]

A. B. Malinowska and D. F. M. Torres, Introduction to the Fractional Calculus of Variations, Imp. Coll. Press, London, 2012. doi: 10.1142/p871.  Google Scholar

[17]

A. B. Malinowska, A formulation of the fractional Noether-type theorem for multidimensional Lagrangians, Appl. Math. Lett., 25 (2012), 1941-1946. doi: 10.1016/j.aml.2012.03.006.  Google Scholar

[18]

V. J. Menon, N. Chanana and Y. Singh, A fresh look at the BCK frictional lagrangian, Prog. Theor. Phys., 98 (1997), 321-329. doi: 10.1143/PTP.98.321.  Google Scholar

[19]

I. Podlubny, Fractional Differential Equations, Mathematics in Science and Engineering, Mathematics in Science and Engineering, 198, Academic Press, Inc., San Diego, CA, 1999.  Google Scholar

[20]

S. Pooseh, R. Almeida and D. F. M. Torres, Approximation of fractional integrals by means of derivatives, Comput. Math. Appl., 64 (2012), 3090-3100. doi: 10.1016/j.camwa.2012.01.068.  Google Scholar

[21]

F. Riewe, Nonconservative Lagrangian and Hamiltonian mechanics, Phys. Rev. E, 53 (1996), 1890-1899. doi: 10.1103/PhysRevE.53.1890.  Google Scholar

[22]

S. G. Samko, A. A. Kilbas and O. I. Marichev, Fractional Integrals and Derivatives, translated from the 1987 Russian original, Gordon and Breach, Yverdon, 1993.  Google Scholar

[23]

S. Santos, N. Martins and D. F. M. Torres, Higher-order variational problems of Herglotz type, Vietnam Journal of Mathematics, (2013), 1-11. doi: 10.1007/s10013-013-0048-9.  Google Scholar

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