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

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1. Agrawal, O. P., Fractional variational calculus and the transversality conditions, J. Phys. A 39 (2006), 10375-10384. Zbl1097.49021
2. Agrawal, O. P., Fractional variational calculus in terms of Riesz fractional derivatives, J. Phys. A 40 (2007), 6287-6303. Zbl1125.26007
3. Agrawal, O. P., Tenreiro Machado, J. A. and Sabatier, J. (Editors), Fractional Derivatives and their Applications, Nonlinear Dynamics 38, no. 1-4 (2004).[Crossref] 
4. 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. Zbl1309.49003
5. Atanacković, T. M., Konjik, S. and Pilipović, S., Variational problems with fractional derivatives: Euler-Lagrange equations, J. Phys. A 41, no. 9, 095201 (2008), 12 pp. Zbl1175.49020
6. Baleanu, D., Agrawal, O. P., Fractional Hamilton formalism within Caputo's derivative, Czechoslovak J. Phys. 56, no. 10-11 (2006), 1087-1092. Zbl1111.37304
7. Cannas da Silva, A., Weinstein, A., Geometric models for noncommutative algebras, Berkeley Mathematics Lecture Notes, 10. American Mathematical Society, Providence, 1999. Zbl1135.58300
8. 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).[Crossref] 
9. El-Nabulsi, R. A., A fractional approach to nonconservative Lagrangian dynamical systems, Fizika A 14, no. 4, (2005) 289-298. 
10. 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. Zbl1137.70349
11. El-Nabulsi, R. A., Fractional field theories from multi-dimensional fractional variational problems, Int. J. Geom. Methods Mod. Phys. 5 (2008), 863-892. Zbl1172.26305
12. 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). 
13. El-Nabulsi, R. A., Fractional variational problems from extended exponentially fractional integral, Appl. Math. Comp. 217, no. 22 (2011), 9492-9496. Zbl1220.26004
14. El-Nabulsi, R. A., Fractional action-like variational problems in holonomic, non-holonomic and semi-holonomic constrained and dissipative dynamical systems, Chaos Solitons Fractals 42 (2009), 52-61. Zbl1198.70010
15. El-Nabulsi, R. A., Fractional action-like variational problems in holonomic, non-holonomic and semi-holonomic constrained and dissipative dynamical systems, Chaos Solitons Fractals 42 (2009), 52-61. Zbl1198.70010
16. El-Nabulsi, R. A., The fractional calculus of variations from extended Erdélyi-Kober operator, Int. J. Mod. Phys. B 23 (2009), 3349-3361. 
17. El-Nabulsi, R. A., Fractional quantum Euler-Cauchy equation in the Schrödinger picture, complexified harmonic oscillators and emergence of complexified Lagrangian and Hamiltonian dynamics, Mod. Phys. Lett. B 23 (2009), 3369-3386. Zbl1185.81089
18. 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. Zbl1179.58002
19. 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. Zbl1195.35296
20. El-Nabulsi, R. A., Modifications at large distances from fractional and fractal arguments, Fractals 18 (2010), 185-190. Zbl1196.28014
21. El-Nabulsi, R. A., Oscillating flat FRW dark energy dominated cosmology from periodic functional approach, Comm. Theor. Phys. 54, no. 01 (2010), 16-20. Zbl1213.83147
22. 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. Zbl1325.37036
23. 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. 
24. 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 (α, β), Math. Methods Appl. Sci. 30 (2007), 1931-1939. Zbl1177.49036
25. El-Nabulsi, R. A, Torres, D. F. M., Fractional actionlike variational problems, J. Math. Phys. 49, no. 5, 053521 (2008), 7 pp. Zbl1152.81422
26. 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. Zbl1145.30303
27. 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. Zbl1103.49012
28. 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. 
29. Frederico, G. S. F., Torres, D. F. M., Conservation laws for invariant functionals containing compositions, Appl. Anal. 86 (2007), 1117-1126.[Crossref] Zbl1190.49027
30. 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. Zbl1119.49035
31. 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. Zbl1154.49016
32. 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. Zbl1154.49016
33. Glöckner, H., Lie group structures on quotient groups and universal complexifications for infinite-dimensional Lie groups, J. Funct. Anal. 194 (2002), 347-409. Zbl1022.22021
34. Glöckner, H., Neeb, K. H., Banach-Lie quotients, enlargibility, and universal complexifications, J. Reine Angew. Math. 560 (2003), 1-28. Zbl1029.22029
35. Goldfain, E., Fractional dynamics, Cantorian space-time and the gauge hierarchy problem, Chaos Solitons Fractals 22 (2004), 513-520.[Crossref] Zbl1068.81633
36. Goldfain, E., Complexity in quantum field theory and physics beyond the standard model, Chaos Solitons Fractals 28 (2006), 913-922.[Crossref] Zbl1097.81928
37. Goldfain, E., Fractional dynamics and the standard model for particle physics, Commun. Nonlinear Sci. Numer. Simul. 13 (2008), 1397-1404.[Crossref] Zbl1221.81175
38. Gualtieri, M., Generalized complex geometry, arXiv:math/0703298v2. Zbl1235.32020
39. Gukov, S., Witten, E., Gauge theory, ramification, and the geometric Langlands program, Current developments in mathematics, 2006, 35-180, Int. Press, Somerville, MA, 2008. Zbl1237.14024
40. Herrmann, R., Fractional dynamic symmetries and the ground state properties of nuclei, arXiv:0806.2300v2. 
41. Herrmann, R., Fractional spin - a property of particles described with a fractional Schrödinger equation, arXiv:0805.3434v1. 
42. Hilfer, R. (Editor), Applications of Fractional Calculus in Physics, Word Scientific Publishing Co., River Edge, NJ, 2000. Zbl0998.26002
43. 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. Zbl1221.70022
44. Kapustin, A., Witten, E., Electric-magnetic duality and the geometric Langlands program, Commun. Number Theory Phys. 1 (2007), 1-236.[Crossref] Zbl1128.22013
45. 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. 
46. Landsman, L. P., Lie groupoids and Lie algebroids in physics and noncommutative geometry, J. Geom. Phys. 56 (2006), 24-54.[Crossref] Zbl1088.58009
47. Lübke, M., Okonek, C., Moduli spaces of simple bundles and Hermitian-Einstein connections, Math. Ann. 276 (1987), 663-674. Zbl0609.53009
48. Lübke, M., Teleman, A., The Kobayashi-Hitchin Correspondence, World Scientific Publishing Co., Inc., River Edge, NJ, 1995. Zbl0849.32020
49. 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. Zbl1078.58011
50. Martinez, E., Lagrangian mechanics on Lie algebroids, Acta Appl. Math. 67 (2001), 295-320.[Crossref] Zbl1002.70013
51. 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. 
52. Martinez, E., Lie algebroids in classical mechanics and optimal control, SIGMA Symmetry Integrability Geom. Methods Appl. 3 (2007), Paper 050, 17 pp. 
53. Miller, K. S., Ross, B., An Introduction to the Fractional Calculus and Fractional Differential Equations, John Wiley & Sons Inc., New York, 1993. Zbl0789.26002
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. Zbl0292.26011
55. Ortigueira, M. D., Tenreiro Machado, J. A. (Editors), Fractional Signal Processing and Applications, Signal Processing 83, no. 11 (2003). 
56. Ortigueira, M. D., Tenreiro Machado, J. A. (Editors), Fractional Calculus Applications in Signals and Systems, Signal Processing 86, no. 10 (2006).[Crossref] Zbl1172.94301
57. 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. Zbl0924.34008
58. Riewe, F., Nonconservative Lagrangian and Hamiltonian mechanics, Phys. Rev. E (3) 53 (1996), 1890-1899. 
59. Samko, S., Kilbas A. and Marichev, O., Fractional Integrals and Derivatives. Theory and Applications, Gordon and Breach Science Publishers, Yverdon, 1993. Zbl0818.26003
60. Tenreiro Machado, J. A. (Editor), Fractional Order Calculus and its Applications, Nonlinear Dynamics 29, no. 1-4 (2002).[Crossref] 
61. Tenreiro Machado, J. A., Barbosa, R. S. (Editors), Fractional Differentiation and its Applications, J. Vib. Control 14, no. 9-10 (2008). 
62. Tenreiro Machado, J. A., Luo, A. (Editors), Discontinuous and Fractional Dynamical Systems, ASME Journal of Computational and Nonlinear Dynamics 3, no. 2 (2008). 
63. Weinstein, A., Lagrangian mechanics and groupoids, Mechanics day (Waterloo, ON, 1992), 207-231, Fields Inst. Commun., 7, Amer. Math. Soc., Providence, RI, 1996. Zbl0844.22007
64. Weinstein, A., Poisson geometry. Symplectic geometry, Differential Geom. Appl. 9, no. 1-2 (1998), 213-238. 
65. Zavada, P., Operator of fractional derivative in the complex plane, Comm. Math. Phys. 192 (1998), 261-285. Zbl0981.26007