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The Hamilton–Jacobi Analysis of Powers of Singular Lagrangians: A Connection Between the Modified Schrödinger and the Navier–Stokes Equations

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Abstract

Non-standard Lagrangians have gained recently an increasing interest in the theory of nonlinear differential equations, classical and quantum nonlinear dynamical systems. In this work, we discuss a number of dynamical systems characterized by powers of singular Lagrangians identified to non-standard Lagrangians based on the resulting Hamilton–Jacobi equation. A number of dynamical problems were addressed and a number of statements which support the non-standard Lagrangians formalism were postulated. After connecting the action to a certain complex wave function and in particular for the case of linear potentials, a link is established between the resulting modified Schrödinger equation which describes specific classes of quantum mechanical systems and the Navier–Stokes equation which describes the motion of viscous fluid matters.

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References

  1. Alekseev, A.I., Arbuzov, B.A.: Classical Yang–Mills field theory with nonstandard Lagrangian. Theor. Math. Phys. 59, 372–378 (1984)

    Article  Google Scholar 

  2. Alekseev, A.I., Vshivtsev, A.S., Tatarintsev, A.V.: Classical non-abelian solutions for non-standard Lagrangians. Theor. Math. Phys. 77(2), 1189–1197 (1988)

    Article  Google Scholar 

  3. Allison, A., Pearce, C.E.M., Abbott, D.: A variational approach to the analysis of dissipative electromechanical systems. PloS One 2(9), 1–12 (2014)

    Google Scholar 

  4. Ambrosetti, A., Zelati, V.C.: Periodic Solutions of Singular Lagrangian Systems. Birkhauser, Springer, Berlin (1993)

    Book  MATH  Google Scholar 

  5. Balseiro, P., Marrero, J.C., Martin De Diego, D., Padron, E.: A unified framework for mechanics: Hamilton–Jacobi equation and applications. Nonlinearity 23, 1887–1918 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  6. Bateman, H.: On dissipative systems and related variational principles. Phys. Rev. 38(4), 815–819 (1931)

    Article  MATH  Google Scholar 

  7. Batlle, C., Gomis, J., Pons, J.M., Román-Roy, N.: Lagrangian and Hamiltonian constraints for second order singular Lagrangians. J. Phys. A Math. Gen. 21(12), 2693–2703 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  8. Bauer, P.S.: Dissipative dynamical systems. PNAS 5(17), 311–314 (1931)

    Article  MATH  Google Scholar 

  9. Bender, C.M., Holm, D., Hook, D.W.: Complex trajectories of a simple pendulum. J. Phys. A Math. Gen. 40, F81–F90 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  10. Bender, C.M., Holm, D., Hook, D.W.: Complexified dynamical systems. J. Phys. A Math. Gen. 40, F793–F804 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  11. Bender, C.M., Brody, D.C., Hook, D.W.: Quantum effects in classical systems having complex energy. J. Phys. A Math. Theor. 41, 352003–352018 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  12. Blatt, F.J.: Modern Physics. McGraw-Hill, New York (1992)

    Google Scholar 

  13. Bohm, D.: A suggested interpretation of the quantum theory in terms of ‘hidden variables’ I. Phys. Rev. 85, 166–179 (1952)

    Article  MathSciNet  MATH  Google Scholar 

  14. Carinena, J.F., Ranada, M.F., Santander, M.: Lagrangian formalism for nonlinear second-order Riccati systems: one-dimensional integrability and two-dimensional superintegrability. J. Math. Phys. 46, 062703–062721 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  15. Carinena, J.F., Nunez, J.F.: Geometric approach to dynamics obtained by deformation of Lagrangians. Nonlinear Dyn. 83(1), 457–461 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  16. Carinena, J.F., Nunez, J.F.: Geometric approach to dynamics obtained by deformation of time-dependent Lagrangians. Nonlinear Dyn. 86(2), 1285–1291 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  17. Carinera, J.F.: Theory of singular Lagrangians. Fortschr. Phys. 38(9), 641–679 (1990)

    Article  MathSciNet  Google Scholar 

  18. Chandrasekar, V.K., Pandey, S.N., Senthilvelan, M., Lakshmanan, M.: Simple and unified approach to identify integrable nonlinear oscillators and systems. J. Math. Phys. 47, 023508–023545 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  19. Chandrasekar, V.K., Senthilvelan, M., Lakshmanan, M.: On the Lagrangian and Hamiltonian description of the damped linear harmonic oscillator. Phys. Rev. E 72, 066203–066211 (2005)

    Article  Google Scholar 

  20. Cieslinski, J.L., Nikiciuk, T.: A direct approach to the construction of standard and non-standard Lagrangians for dissipative. J. Phys. A Math. Gen. 43, 175205–175222 (2010)

    Article  MATH  Google Scholar 

  21. Cisneros-Parra, J.U.: On singular Lagrangians and Dirac’s method. Rev. Mex. Fis. 58, 61–68 (2012)

    MathSciNet  MATH  Google Scholar 

  22. Cortes, V., Haupt, A.S.: Mathematical methods of classical physics, arXiv: 1612.03100v1

  23. Curtright, T., Mezincescu, L.: Biorthogonal quantum systems. J. Math. Phys. 48, 092106 (2007). (34 pages)

    Article  MathSciNet  MATH  Google Scholar 

  24. Dector, A., Morales-Tecotl, H.A., Urrutia, L.F., Vergara, J.D.: An alternative canonical approach to the ghost problem in a complexified extension of the Pais–Uhlenbeck oscillator, SIGMA5, 053 (22 pages) (2009)

  25. Deng, Y., Jin, L., Peng, S.: Solutions of Schrödinger equations with inverse square potential and critical nonlinearity. J. Differ. Equ. 253, 1376–1398 (2012)

    Article  MATH  Google Scholar 

  26. Denman, H.H., Buch, L.H.: Solution of the Hamilton–Jacobi equation for certain dissipative classical mechanical systems. J. Math. Phys. 14, 326–329 (1973)

    Article  MATH  Google Scholar 

  27. Dimitrijevic, D.D., Milosevic, M.: About non-standard Lagrangians in cosmology. AIP Conf. Proc. Phys. Conf. 1472, 41–46 (2012)

    Article  Google Scholar 

  28. Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, Yeshiva University, New York (1964)

    Google Scholar 

  29. El-Nabulsi, R.A.: Non-linear dynamics with non-standard Lagrangians. Qual. Theory Dyn. Syst. 12(2), 273–291 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  30. El-Nabulsi, R.A.: Non-standard non-local-in-time Lagrangians in classical mechanics. Qual. Theory. Dyn. Syst. 13(1), 149–160 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  31. El-Nabulsi, R.A.: Non-standard power-law Lagrangians in classical and quantum dynamics. Appl. Math. Lett. 43, 120–127 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  32. El-Nabulsi, R.A.: Lagrangian and Hamiltonian dynamics with imaginary time. J. Appl. Anal. 18(2), 283–296 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  33. El-Nabulsi, R.A.: Fractional quantum Euler–Cauchy equation in the Schrödinger picture, complexified oscillators and emergence of complexified Lagrangian and Hamiltonian dynamics. Mod. Phys. Lett. A 23, 3369–3386 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  34. El-Nabulsi, R.A.: Fractional variational approach for dissipative mechanical systems. Anal. Theor. Appl. 30(3), 1–10 (2014)

    MathSciNet  MATH  Google Scholar 

  35. Fanelli, L., Felli, V., Fontelos, M.A., Primo, A.: Time decay of scaling critical electromagnetic Schrödinger flows. Commun. Math. Phys. 324, 1033–1067 (2013)

    Article  MATH  Google Scholar 

  36. Fernandez de Cordoba, P., Isidro, J.M., Vazquez Molina, J.: Schrödinger vs. Navier Stokes. Entropy 18, 34 (2016). (12 pages)

    Article  Google Scholar 

  37. Frank, W.M., Land, D.J., Spector, R.M.: Singular potentials. Rev. Mod. Phys. 43, 36–98 (1971)

    Article  MathSciNet  Google Scholar 

  38. Grillo, G., Hovarik, H.: Weighted dispersive estimates for two-dimensional Schrödinger operators with Aharonov–Bohm magnetic field. J. Differ. Equ. 256(12), 3889–3911 (2014)

    Article  MATH  Google Scholar 

  39. Ghosh, S., Modak, S.J.: Classical oscillator with position-dependent mass in a complex domain. Phys. Lett. A 373, 1212–1217 (2009)

    Article  MATH  Google Scholar 

  40. Graefe, E.M., Honing, M., Korsch, H.J.: Classical limit of non-Hermitian quantum dynamics-a generalized canonical structure. J. Phys. A Math. Theor. 43, 075306 (2010). (19 pages)

    Article  MathSciNet  MATH  Google Scholar 

  41. Goldberg, M., Vega, L., Visciglia, N.: Counter examples of Strichartz inequalities for Schrödinger equations with repulsive potentials. Int. Math. Res. Not. 2006, article ID13927 (2006)

  42. Goldstein, H.: Classical Mechanics, 2nd edn. Addison-Wesley, Reading (1980)

    MATH  Google Scholar 

  43. Gutzwiller, M.C.: Chaos in Classical and Quantum Mechanics. Springer, New York (1990)

    Book  MATH  Google Scholar 

  44. Hassan, E.H.: Hamilton–Jacobi formalism for singular Lagrangians with linear accelerations. Mod. Appl. Sci. 8(3), 31–36 (2014)

    Google Scholar 

  45. Heifetz, E., Cohen, E.: Toward a thermo-hydrodynamic like description of Schrödinger equation via the Madelung formulation and Fisher information. Found. Phys. 45, 1514–1525 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  46. Hojman, S., Urrutia, L.F.: On the inverse problem of the calculus of variations. J. Math. Phys. 22, 1896–1903 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  47. Jarab’ah, O.A., Nawafleh, K.I., Ghassib, H.B.: A Hamilton–Jacobi treatment of dissipative systems. Eur. Sci. J. 9(30), 70–81 (2013)

    Google Scholar 

  48. Jungel, A., Milisic, J.-P.: Full compressible Navier–Stokes equations for quantum fluids: derivation and numerical solutions. Kinetic Relat. Mod. 4, 785–807 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  49. Kovarik, H., Truc, F.: Schrödinger operators on a half-line with inverse square potentials, IF PREPUB. hal-00959561 (2014)

  50. Kulish, V.V., Lage, J.L.: On the relationship between fluid velocity and de Broglie’s wave function and the implications to the Navier–Stokes equation. Int. J. Fluid Mech. Res. 29(1), 40–52 (2002)

    MathSciNet  Google Scholar 

  51. Lipovka, A.: Nature of the quantum potential. J. Appl. Math. Phys. 4, 897–902 (2016)

    Article  Google Scholar 

  52. Liu, S., Guan, F., Wang, Y., Liu, C., Guo, Y.: The nonlinear dynamics based on the nonstandard Hamiltonians. Nonlinear Dyn. 88, 1229–1236 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  53. Ma, X., Soukoulis, C.M.: Schrödinger equation with imaginary potential. Phys. B296, 107–111 (2001)

    Google Scholar 

  54. Madelung, E.: Quantentheorie in hydrodynamischer form. Z. Phys. A40, 322–326 (1927)

    Article  MATH  Google Scholar 

  55. Markus, F., Gambar, K.: Generalized Hamilton–Jacobi equation for simple dissipative processes. Phys. Rev. E 70, 016123–6 (2004)

    Article  Google Scholar 

  56. Musielak, Z.E.: Standard and non-standard Lagrangians for dissipative dynamical systems with variable coefficients. J. Phys. A Math. Theor. 41, 055205–055222 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  57. Musielak, Z.E.: General conditions for the existence of non-standard Lagrangians for dissipative dynamical systems. Chaos Solitons Fractals 42(15), 2645–2652 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  58. Parthasarathy, R.: Relativistic Quantum Mechanics. Alpha Science International, Oxford (2010)

    Google Scholar 

  59. Pimentel, B.M., Teixeira, R.G.: Hamilton–Jacobi formulation for singular Lagrangians with second order Lagrangians, Instituto de Física Teórica, Universidade Estadual Paulista, IFT-P.001/96, (1996)

  60. Raidal, M., Veermae, H.: On the quantization of complex higher derivative theories and avoiding the Ostrogradsky ghost. Nucl. Phys. B 916, 607–626 (2017)

    Article  MATH  Google Scholar 

  61. Riccia, C.D.: A Hamilton–Jacobi treatment of dissipative systems with one degree of freedom. In: Dynamical Systems and Microphysics, vol. 261 of the Series International Center For Mechanical Systems, pp. 291–300, Springer, Berlin (1980)

  62. Sanz, A.S.: Bohm’s approach to quantum mechanics: alternative theory or practical picture? arXiv: 1707.00609

  63. Saha, A., Talukdar, B.: Inverse variational problem for non-standard Lagrangians. Rep. Math. Phys. 3(3), 299–309 (2014)

    Article  MATH  Google Scholar 

  64. Samengo, I., Barrachina, R.O.: Rainbow and glory scattering in Coulomb trajectories starting from a point in space. Eur. J. Phys. 15, 300–308 (1994)

    Article  Google Scholar 

  65. Schrödinger, E.: Quantizierung als Eigenwertproblem (Erste Mitteilung). Ann. der Phys 79, 361–376 (1926)

    Article  MATH  Google Scholar 

  66. Schrödinger, E.: Uber das Verhaltnis der Heisenberg Born Jordanischen Quantenmechanik zu der meinen. Ann. der Phys. 79, 734–756 (1926)

    Article  MATH  Google Scholar 

  67. Scholze, M.: Classical Mechanics and Dynamical Systems with Applications in Mathematica. Department of Applied Mathematics, Faculty of Transportation Sciences Czech Technical University in Prague, (2012)

  68. Sonego, S.: Interpretation of the hydrodynamical formalism of quantum mechanics. Found. Phys. 21, 1135–1181 (1991)

    Article  MathSciNet  Google Scholar 

  69. Tsekov, R.: Thermo-quantum diffusion. Int. J. Theor. Phys. 48, 630–636 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  70. Vadasz, P.: Rendering the Navier-Stokes equations for a compressible fluid into the Schrödinger equation for quantum mechanics. Fluids 1, 18 (2016). (11 pages)

    Article  Google Scholar 

  71. White, F.M.: Fluid Mechanics. McGraw-Hill, New York (2003)

    Google Scholar 

  72. Yang, J., Ju, Q.: Existence of global weak solutions for Navier–Stokes–Poisson equations with quantum effect and convergence to incompressible Navier–Stokes equations. Math. Meth. Appl. Sci. 38(17), 3629–3641 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  73. Yang, J., Li, Y.: Global existence of weak solution for quantum Navier–Stokes–Poisson equations. J. Math. Phys. 58, 071507–071512 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  74. Yang, J., Ju, Q.: Convergence of the quantum Navier-Stokes-Poisson equations to the incompressible Euler equations for general initial data. Nonlinear Anal. Real Word Appl. 23, 148–159 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  75. Yariv, A.: Optical Electronics in Modern Communications. Oxford University Press, New York (1997)

    Google Scholar 

  76. Zhang, Y., Zhou, X.S.: Noether theorem and its inverse for nonlinear dynamical systems with nonstandard Lagrangians. Nonlinear Dyn. 84(2), 1867–1876 (2016)

    Article  MathSciNet  MATH  Google Scholar 

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My sincere appreciation goes to the anonymous referees for reading and criticizing my work.

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  1. Mathematics and Physics Divisions, Athens Institute for Education and Research, 8 Valaoritou Street, Kolonaki, 10671, Athens, Greece

    Rami Ahmad El-Nabulsi

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El-Nabulsi, R.A. The Hamilton–Jacobi Analysis of Powers of Singular Lagrangians: A Connection Between the Modified Schrödinger and the Navier–Stokes Equations. Qual. Theory Dyn. Syst. 17, 583–608 (2018). https://doi.org/10.1007/s12346-017-0257-9

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