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Complex backward-forward derivative operator in non-local-in-time Lagrangians mechanics. (English) Zbl 1499.70021

Summary: In this paper we introduce non-local-in-time complexified Lagrangians characterized by an expanded complex backward-forward derivative operator which generalize the classical complex derivative operator. We developed the Euler-Lagrange equations and solved them for some special case. We discuss their implications in Newtonian mechanics where a number of applications were illustrated.

MSC:

70S05 Lagrangian formalism and Hamiltonian formalism in mechanics of particles and systems
26A33 Fractional derivatives and integrals
49S05 Variational principles of physics
70H30 Other variational principles in mechanics
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[1] Alber, S., Marsden, J.E.: Semiclassical monodromy and the spherical pendulum as a complex Hamiltonian system. Fields Inst. Commun. 8, 1-18 (1996) · Zbl 0853.58045
[2] Ben Adda, F., Cresson, J.: Quantum derivatives and the Schrödinger equation. Chaos Solitons Fractals 19, 1323-1334 (2004) · Zbl 1053.81027 · doi:10.1016/S0960-0779(03)00339-4
[3] Bender, C.M., Holm, D.D., Hook, D.W.: Complexified dynamical systems. J. Phys. A 40, F793-F804 (2007) · Zbl 1120.37050 · doi:10.1088/1751-8113/40/32/F02
[4] Cresson, J.: Fractional embedding of differential operators and Lagrangian system. J. Math. Phys. 48(3), 033504-044534 (2007) · Zbl 1137.37322 · doi:10.1063/1.2483292
[5] Dryl, M., Torres, D.F.M.: The delta-nabla calculus of variations for composition functionals on time scales. Int. J. Differ. Equ. 8, 27-47 (2013)
[6] El-Nabulsi, R.A.: Non-standard non-local-in-time Lagrangians in classical mechanics. Qual. Theor. Dyn. Syst. 13, 149-160 (2014) · Zbl 1305.70043 · doi:10.1007/s12346-014-0110-3
[7] El-Nabulsi, R.A., Torres, D.F.M.: Fractional actionlike variational problems. J. Math. Phys. 49(5), 053521-053527 (2008) · Zbl 1152.81422 · doi:10.1063/1.2929662
[8] 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 \[(\alpha\] α, \[ \beta )\] β). Math. Methods Appl. Sci. 30(15), 1931-1939 (2007) · Zbl 1177.49036 · doi:10.1002/mma.879
[9] El-Nabulsi, R.A.: Lagrangian and Hamiltonian dynamics with imaginary time. J. Appl. Anal. 18, 283-295 (2012) · Zbl 1276.70013 · doi:10.1515/jaa-2012-0010
[10] Feynman, R.P.: Space-time approach to relativistic quantum mechanics. Rev. Mod. Phys. 20, 367-387 (1948) · Zbl 1371.81126 · doi:10.1103/RevModPhys.20.367
[11] Feynman, R.P., Hibbs, A.: Quantum Mechanics and Path Integrals. MacGraw-Hill, New York (1965) · Zbl 0176.54902
[12] Kaushal, R.S.: Classical and quantum mechanics of complex Hamiltonian systems: an extended complex phase space approach. PRAMANA J. Phys. 73(2), 287-297 (2009) · doi:10.1007/s12043-009-0120-x
[13] Li, Z.-Y., Fu, J.-L., Chen, L.-Q.: Euler-Lagrange equation from nonlocal-in-time kinetic energy of nonconservative system. Phys. Lett. A 374, 106-109 (2009) · Zbl 1235.70035 · doi:10.1016/j.physleta.2009.10.080
[14] Malinowska, A.B., Torres, D.F.M.: Springer Briefs in Electrical and Computer Engineering: Control, Automation and Robotics. Quantum variational calculus. Springer, New York (2014)
[15] Martins, N., Torres, D.F.M.: Higher-order infinite horizon variational problems in discrete quantum calculus. Comput. Math. Appl. 64, 2166-2175 (2012) · Zbl 1268.49031 · doi:10.1016/j.camwa.2011.12.006
[16] Martins, N., Torres, D.F.M.: Calculus of variations on time scales with nabla derivatives. Nonlinear Anal. 71, e763-e773 (2009) · Zbl 1238.49037 · doi:10.1016/j.na.2008.11.035
[17] Mohanasubha, R., Sheeba, J.H., Chandrasekar, V.K., Senthilvelan, M., Lakshmanan, M.: A nonlocal connection between certain linear and nonlinear ordinary differential equations—Part II: Complex nonlinear oscillators. Appl. Math. Comput. 224, 593-602 (2013) · Zbl 1334.34085
[18] Nelson, E.: Derivation of the Schrödinger equation from Newtonian mechanics. Phys. Rev. 150, 1079-1085 (1966) · doi:10.1103/PhysRev.150.1079
[19] Nottale, L.: Fractal Space-Time and Microphysics: Towards a Theory of Scale Relativity. World Scientific, New York (1993) · Zbl 0789.58003 · doi:10.1142/1579
[20] Sbitnev, V.I.: Bohmian trajectories and the path integral paradigm. Complexified Lagrangian mechanics. Int. J. Bifurn. Chaos 19, 2335-2346 (2009) · Zbl 1176.81003 · doi:10.1142/S0218127409024104
[21] Suykens, J.A.K.: Extending Newton’s law from nonlocal-in-time kinetic energy. Phys. Lett. A 373, 1201-1211 (2009) · Zbl 1228.70004 · doi:10.1016/j.physleta.2009.01.065
[22] Tritton, D.J.: Physical Fluid Dynamics, 2nd edn. Clarendon Press, Oxford (1988) · Zbl 0383.76001
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