We examined the fractional second-order singular Lagrangian systems. We wrote the action principal function and equations of motion as fractional total differential equations. Also, we constructed the set of Hamilton-Jacobi partial differential equations (HJPDEs) within fractional calculus. We formulated the fractional path integral quantization for these systems. A mathematical example is examined with first- and second-class constraints.
Dirac, P.A.M. (1964) Lectures on Quantum Mechanics. Lectures, Belfer Graduate School of Science, Yeshiva University.
[3]
Rabei, E.M. and Güler, Y. (1992) Hamilton-Jacobi Treatment of Second-Class Constraints. Physical Review A, 46, 3513-3515. https://doi.org/10.1103/physreva.46.3513
[4]
Muslih, S.I. (2001) Path Integral Formulation of Constrained Systems with Singular Higher-Order Lagrangians. Hadronic Journal, 24, 713-721. https://doi.org/10.48550/arXiv.math-ph/0009015
[5]
Pimentel, B.M. and Teixeira, R.G. (1996) Hamilton-Jacobi Formulation for Singular Systems with Second-Order Lagrangians. Il Nuovo Cimento B, 111, 841-854. https://doi.org/10.1007/bf02749015
[6]
Rabei, E.M., Hasan, E.H. and Ghassib, H.B. (2004) Hamilton-Jacobi Treatment of Constrained Systems with Second-Order Lagrangians. International Journal of Theoretical Physics, 43, 1073-1096. https://doi.org/10.1023/b:ijtp.0000048601.92005.fe
[7]
Rabei, E.M., Hassan, E.H., Ghassib, H.B. and Muslih, S. (2005) Quantization of Second-Order Constrained Lagrangian Systems Using the WKB Approximation. International Journal of Geometric Methods in Modern Physics, 2, 485-504. https://doi.org/10.1142/s0219887805000661
[8]
Hasan, E.H., Rabei, E.M. and Ghassib, H.B. (2004) Quantization of Higher-Order Constrained Lagrangian Systems Using the WKB Approximation. International Journal of Theoretical Physics, 43, 2285-2298. https://doi.org/10.1023/b:ijtp.0000049027.45011.37
[9]
Samko, S.G., Kilbas, A.A. and Marichev, O.I. (1993) Fractional Integrals and Derivatives: Theory and Applications. Gordon and Breach Science Publishers.
[10]
Riewe, F. (1996) Nonconservative Lagrangian and Hamiltonian Mechanics. Physical Review E, 53, 1890-1899. https://doi.org/10.1103/physreve.53.1890
[11]
Rabei, E.M., Nawafleh, K.I., Hijjawi, R.S., Muslih, S.I. and Baleanu, D. (2007) The Hamilton Formalism with Fractional Derivatives. Journal of Mathematical Analysis and Applications, 327, 891-897. https://doi.org/10.1016/j.jmaa.2006.04.076
[12]
Agrawal, O.P. (2002) Formulation of Euler-Lagrange Equations for Fractional Variational Problems. Journal of Mathematical Analysis and Applications, 272, 368-379. https://doi.org/10.1016/s0022-247x(02)00180-4
[13]
Hasan, E.H. (2016) Fractional Variational Problems of Euler-Lagrange Equations with Holonomic Constrained Systems. Applied Physics Research, 8, 60-65. https://doi.org/10.5539/apr.v8n3p60
[14]
Hasan, E.H. (2016) Fractional Quantization of Holonomic Constrained Systems Using Fractional WKB Approximation. Advanced Studies in Theoretical Physics, 10, 223-234. https://doi.org/10.12988/astp.2016.6313
[15]
Hasan, E.H. and Asad, J.H. (2017) Remarks on Fractional Hamilton-Jacobi Formalism with Second-Order Discrete Lagrangian Systems. Journal of Advanced Physics, 6, 430-433. https://doi.org/10.1166/jap.2017.1335
[16]
Hasan, E.H. (2018) On Fractional Solutions of Euler-Lagrange Equations with Second-Order Linear Lagrangians. Journal of Advanced Physics, 7, 110-113. https://doi.org/10.1166/jap.2018.1388
[17]
Rabei, E.M. and Al Horani, M. (2018) Quantization of Fractional Singular Lagrangian Systems Using WKB Approximation. International Journal of Modern Physics A, 33, Article ID: 1850222. https://doi.org/10.1142/s0217751x18502226
[18]
Hasan, E.H. (2020) Path Integral Quantization of Singular Lagrangians Using Fractional Derivatives. International Journal of Theoretical Physics, 59, 1157-1164. https://doi.org/10.1007/s10773-020-04395-3
[19]
Hasan, E.H. (2023) Fractional Quantization of Singular Lagrangian Systems with Second-Order Derivatives Using WKB Approximation. https://doi.org/10.48550/arXiv.2301.08133
[20]
Ostrogradski, M. (1850) Mémoires sur les équations différentielles, relatives au problème des isopérimètres. Mémoires de l’Académie Impériale des Sciences de St-Pétersbourg VI, 4, 385-517.
