PL EN

Preferencje
Język
Widoczny [Schowaj] Abstrakt
Liczba wyników
Czasopismo

## Annales Universitatis Mariae Curie-Skłodowska, sectio A – Mathematica

2011 | 65 | 1 |
Tytuł artykułu

### Extended fractional calculus of variations, complexified geodesics and Wong’s fractional equations on complex plane and on Lie algebroids

Autorzy
Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
In this work, we communicate the topic of complex Lie algebroids based on the extended fractional calculus of variations in the complex plane. The complexified Euler-Lagrange geodesics and Wong’s fractional equations are derived. Many interesting consequences are explored.
Słowa kluczowe
EN
Rocznik
Tom
Numer
Opis fizyczny
Daty
wydano
2011
online
2016-07-25
Twórcy
Bibliografia
• Agrawal, O. P., Fractional variational calculus and the transversality conditions, J. Phys. A 39 (2006), 10375-10384.
• Agrawal, O. P., Fractional variational calculus in terms of Riesz fractional derivatives, J. Phys. A 40 (2007), 6287-6303.
• Agrawal, O. P., Tenreiro Machado, J. A. and Sabatier, J. (Editors), Fractional
• Derivatives and their Applications, Nonlinear Dynamics 38, no. 1-4 (2004).
• Almeida, R., Malinowska, A. B. and Torres, D. F. M., A fractional calculus of variations for multiple integrals with application to vibrating string, J. Math. Phys. 51, no. 3, 033503 (2010), 12 pp.
• Atanackovic, T. M., Konjik, S. and Pilipovic, S., Variational problems with fractional derivatives: Euler-Lagrange equations, J. Phys. A 41, no. 9, 095201 (2008), 12 pp.
• Baleanu, D., Agrawal, O. P., Fractional Hamilton formalism within Caputo’s derivative, Czechoslovak J. Phys. 56, no. 10-11 (2006), 1087-1092.
• Cannas da Silva, A., Weinstein, A., Geometric models for noncommutative algebras, Berkeley Mathematics Lecture Notes, 10. American Mathematical Society, Providence, 1999.
• Connes, A., A survey of foliations and operator algebras. Operator algebras and applications, Part I, pp. 521-628, Proc. Sympos. Pure Math. 38, Amer. Math. Soc., Providence, R. I., (1982).
• El-Nabulsi, R. A., A fractional approach to nonconservative Lagrangian dynamical systems, Fizika A 14, no. 4, (2005) 289-298.
• El-Nabulsi, R. A., A fractional action-like variational approach of some classical, quantum and geometrical dynamics, Int. J. Appl. Math. 17 (2005), 299-317.
• El-Nabulsi, R. A., Fractional field theories from multi-dimensional fractional variational problems, Int. J. Geom. Methods Mod. Phys. 5 (2008), 863-892.
• El-Nabulsi, R. A., Complexified dynamical systems from real fractional actionlike with time-dependent fractional dimension on multifractal sets, Invited contribution to The 3rd International Conference on Complex Systems and Applications, University of Le Havre, Le Havre, Normandy, France, June 29-July 02, (2009).
• El-Nabulsi, R. A., Fractional variational problems from extended exponentially fractional integral, Appl. Math. Comp. 217, no. 22 (2011), 9492-9496.
• El-Nabulsi, R. A., Fractional action-like variational problems in holonomic, nonholonomic and semi-holonomic constrained and dissipative dynamical systems, Chaos Solitons Fractals 42 (2009), 52-61.
• El-Nabulsi, R. A., Fractional action-like variational problems in holonomic, nonholonomic and semi-holonomic constrained and dissipative dynamical systems, Chaos Solitons Fractals 42 (2009), 52-61.
• El-Nabulsi, R. A., The fractional calculus of variations from extended Erdelyi-Kober operator, Int. J. Mod. Phys. B 23 (2009), 3349-3361.
• El-Nabulsi, R. A., Fractional quantum Euler-Cauchy equation in the Schrodinger picture, complexified harmonic oscillators and emergence of complexified Lagrangian and Hamiltonian dynamics, Mod. Phys. Lett. B 23 (2009), 3369-3386.
• El-Nabulsi, R. A., Complexified fractional heat kernel and physics beyond the spectral triplet action in noncommutative geometry, Int. J. Geom. Methods Mod. Phys. 6 (2009), 941-963.
• El-Nabulsi, R. A., Fractional Dirac operators and left-right fractional Chamseddine-Connes spectral bosonic action principle in noncommutative geometry, Int. J. Geom. Methods Mod. Phys. 7 (2010), 95-134.
• El-Nabulsi, R. A., Modifications at large distances from fractional and fractal arguments, Fractals 18 (2010), 185-190.
• El-Nabulsi, R. A., Oscillating flat FRW dark energy dominated cosmology from periodic functional approach, Comm. Theor. Phys. 54, no. 01 (2010), 16-20.
• El-Nabulsi, R. A., A periodic functional approach to the calculus of variations and the problem of time-dependent damped harmonic oscillators, Appl. Math. Lett. 24, no. 10 (2011), 1647-1653.
• El-Nabulsi, R. A., Universal fractional Euler-Lagrange equation from a generalized fractional derivate operator, Central Europan J. Phys. 9, no. 1 (2011), 260-256.
• El-Nabulsi, R. A., Torres, D. F. M., Necessary optimality conditions for fractional action-like integrals of variational calculus with Riemann-Liouville derivatives of order $$(\alpha,\beta)$$, Math. Methods Appl. Sci. 30 (2007), 1931-1939.
• El-Nabulsi, R. A, Torres, D. F. M., Fractional actionlike variational problems, J. Math. Phys. 49, no. 5, 053521 (2008), 7 pp.
• Frasin, B. A., Some applications of fractional calculus operators to the analytic part of harmonic univalent functions, Hacet. J. Math. Stat. 34 (2005), 1-7.
• Frederico, G. S. F., Torres, D. F. M., Constants of motion for fractional action-like variational problems, Int. J. Appl. Math. 19 (2006), 97-104.
• Frederico, G. S. F., Torres, D. F. M., Non-conservative Noether’s theorem for fractional action-like variational problems with intrinsic and observer times, Int. J. Ecol. Econ. Stat. 9, no. F07 (2007), 74-82.
• Frederico, G. S. F., Torres, D. F. M., Conservation laws for invariant functionals containing compositions, Appl. Anal. 86 (2007), 1117-1126.
• Frederico, G. S. F., Torres, D. F. M., A formulation of Noether’s theorem for fractional problems of the calculus of variations, J. Math. Anal. Appl. 334 (2007), 834-846.
• Frederico, G. S. F., Torres, D. F. M., Fractional optimal control in the sense of Caputo and the fractional Noether’s theorem, Int. Math. Forum 3, no. 9-12 (2008), 479-493.
• Frederico, G. S. F., Torres, D. F. M., Fractional optimal control in the sense of Caputo and the fractional Noether’s theorem, Int. Math. Forum 3, no. 9-12 (2008), 479–493.
• Glockner, H., Lie group structures on quotient groups and universal complexifications for infinite-dimensional Lie groups, J. Funct. Anal. 194 (2002), 347-409.
• Glockner, H., Neeb, K. H., Banach-Lie quotients, enlargibility, and universal complexifications, J. Reine Angew. Math. 560 (2003), 1-28.
• Goldfain, E., Fractional dynamics, Cantorian space-time and the gauge hierarchy problem, Chaos Solitons Fractals 22 (2004), 513-520.
• Goldfain, E., Complexity in quantum field theory and physics beyond the standard model, Chaos Solitons Fractals 28 (2006), 913-922.
• Goldfain, E., Fractional dynamics and the standard model for particle physics, Commun. Nonlinear Sci. Numer. Simul. 13 (2008), 1397-1404.
• Gualtieri, M., Generalized complex geometry, arXiv:math/0703298v2.
• Gukov, S., Witten, E., Gauge theory, ramification, and the geometric Langlands program, Current developments in mathematics, 2006, 35-180, Int. Press, Somerville, MA, 2008.
• Herrmann, R., Fractional dynamic symmetries and the ground state properties of nuclei, arXiv:0806.2300v2.
• Herrmann, R., Fractional spin – a property of particles described with a fractional Schrodinger equation, arXiv:0805.3434v1.
• Hilfer, R. (Editor), Applications of Fractional Calculus in Physics, Word Scientific Publishing Co., River Edge, NJ, 2000.
• Ivan, G., Ivan, M. and Opris, D., Fractional Euler–Lagrange and fractional Wong equations for Lie algebroids, Proceedings of the 4th International Colloquium Mathematics and Numerical Physics, October 6-8, 2006, Bucharest, Romania, 73-80.
• Kapustin, A., Witten, E., Electric-magnetic duality and the geometric Langlands program, Commun. Number Theory Phys. 1 (2007), 1-236.
• Kosyakov, B. P., The field of an arbitrarily moving colored charge, Teoret. Mat. Fiz. 87 (1991), 422-425 (Russian); translation in Theoret. and Math. Phys. 87 (1991), 632-635.
• Landsman, L. P., Lie groupoids and Lie algebroids in physics and noncommutative geometry, J. Geom. Phys. 56 (2006), 24-54.
• Lubke, M., Okonek, C., Moduli spaces of simple bundles and Hermitian-Einstein connections, Math. Ann. 276 (1987), 663-674.
• Lubke, M., Teleman, A., The Kobayashi-Hitchin Correspondence, World Scientific Publishing Co., Inc., River Edge, NJ, 1995.
• Mackenzie, K. C. H., General theory of Lie groupoids and Lie algebroids, London Mathematical Society Lecture Note Series, Vol. 213, Cambridge University Press, Cambridge, 2005.
• Martinez, E., Lagrangian mechanics on Lie algebroids, Acta Appl. Math. 67 (2001), 295-320.
• Martinez, E., Geometric formulation of mechanics on Lie algebroids, Proceedings of the VIII Fall Workshop on Geometry and Physics, Medina del Campo, 1999, Publicaciones de la RSME 2, (2001) 209-222.
• Martinez, E., Lie algebroids in classical mechanics and optimal control, SIGMA Symmetry Integrability Geom. Methods Appl. 3 (2007), Paper 050, 17 pp.
• Miller, K. S., Ross, B., An Introduction to the Fractional Calculus and Fractional Differential Equations, John Wiley & Sons Inc., New York, 1993.
• [54] Oldham, K. B., Spanier, J., The Fractional Calculus. Theory and Applications of Differentiation and Integration to Arbitrary Order, Acad. Press, New York-London, 1974.
• Ortigueira, M. D., Tenreiro Machado, J. A. (Editors), Fractional Signal Processing and Applications, Signal Processing 83, no. 11 (2003).
• Ortigueira, M. D., Tenreiro Machado, J. A. (Editors), Fractional Calculus Applications in Signals and Systems, Signal Processing 86, no. 10 (2006).
• Podlubny, I., Fractional Differential Equations. An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of their Solution and Some of their Applications, Academic Press Inc., San Diego, CA, 1999.
• Riewe, F., Nonconservative Lagrangian and Hamiltonian mechanics, Phys. Rev. E (3) 53 (1996), 1890-1899.
• Samko, S., Kilbas A. and Marichev, O., Fractional Integrals and Derivatives. Theory and Applications, Gordon and Breach Science Publishers, Yverdon, 1993.
• Tenreiro Machado, J. A. (Editor), Fractional Order Calculus and its Applications, Nonlinear Dynamics 29, no. 1-4 (2002).
• Tenreiro Machado, J. A., Barbosa, R. S. (Editors), Fractional Differentiation and its Applications, J. Vib. Control 14, no. 9-10 (2008).
• Tenreiro Machado, J. A., Luo, A. (Editors), Discontinuous and Fractional Dynamical Systems, ASME Journal of Computational and Nonlinear Dynamics 3, no. 2 (2008).
• Weinstein, A., Lagrangian mechanics and groupoids, Mechanics day (Waterloo, ON, 1992), 207-231, Fields Inst. Commun., 7, Amer. Math. Soc., Providence, RI, 1996.
• Weinstein, A., Poisson geometry. Symplectic geometry, Differential Geom. Appl. 9, no. 1-2 (1998), 213-238.
• Zavada, P., Operator of fractional derivative in the complex plane, Comm. Math. Phys. 192 (1998), 261-285.
Typ dokumentu
Bibliografia
Identyfikatory