A Reappraisal of Lagrangians with Non-Quadratic Velocity Dependence and Branched Hamiltonians
Abstract
:1. Introduction
2. Some Illustrative Examples
3. Liénard Systems
3.1. Chiellini Condition and Nonstandard Lagrangians
3.2. Hamiltonian Aspects
3.3. A Concrete Example
3.3.1. Case with
3.3.2. Case with
4. A Generalized Class of Lagrangians Yielding Branched Hamiltonians
4.1. The Model
4.2. Velocity-Independent Potentials
4.3. Velocity-Dependent Potentials
A Special Case
5. Three More Forms of Hamiltonians
5.1. Higher Power Lagrangians
Special Case
5.2. Rational Function Lagrangians
5.3. Relativistic Free Particle
6. Concluding Remarks
Author Contributions
Funding
Data Availability Statement
Acknowledgments
Conflicts of Interest
Appendix A. Jacobi Last Multiplier
References
- Shapere, A.; Wilczek, F. Branched Quantization. Phys. Rev. Lett. 2012, 109, 200402. [Google Scholar] [CrossRef]
- Shapere, A.; Wilczek, F. Classical Time Crystals. Phys. Rev. Lett. 2012, 109, 160402. [Google Scholar] [CrossRef]
- Wilczek, F. Quantum Time Crystals. Phys. Rev. Lett. 2012, 109, 160401. [Google Scholar] [CrossRef]
- Henneaux, M.; Teitelboim, C.; Zanelli, J. Quantum mechanics for multivalued Hamiltonians. Phys. Rev. A 1987, 36, 4417. [Google Scholar] [CrossRef]
- Bagchi, B.; Modak, S.; Panigrahi, P.K.; Ruzicka, F.; Znojil, M. Exploring branched Hamiltonians for a class of nonlinear systems. Mod. Phys. Lett. A 2015, 30, 1550213. [Google Scholar] [CrossRef]
- Mitsopoulos, A.; Tsamparlis, M. Cubic first integrals of autonomous dynamical systems in E2 by an algorithmic approach. J. Math. Phys. 2023, 64, 012701. [Google Scholar] [CrossRef]
- Bender, C.M.; Dorey, P.E.; Dunning, C.; Fring, A.; Hook, D.W.; Jones, H.F.; Kuzhel, S.; Lévai, G.; Tateo, R. PT Symmetry: In Quantum and Classical Physics; World Scientific: Hackensack, NJ, USA, 2019. [Google Scholar]
- Mandal, B.P.; Mourya, B.K.; Ali, K.; Ghatak, A. PT phase transition in a (2 + 1)-d relativistic system. Ann. Phys. 2015, 363, 185–193. [Google Scholar] [CrossRef]
- Saha, A.; Talukdar, B. On the non-standard Lagrangian equations. arXiv 2013, arXiv:1301.2667. [Google Scholar]
- Cariñena, J.F.C.; Rañada, M.F.R.; Santander, M. Lagrangian formalism for nonlinear second-order Riccati systems: One-dimensional integrability and two-dimensional superintegrability. J. Math. Phys. 2005, 46, 062703. [Google Scholar] [CrossRef]
- Cariñena, J.F.C.; Nunez, J.F. Geometric approach to dynamics obtained by deformation of Lagrangians. Nonlinear Dyn. 2016, 83, 457–461. [Google Scholar] [CrossRef]
- Cariñena, J.F.C.; Guha, P.; Rañada, M.F.R. A geometric approach to higher-order Riccati chain: Darboux polynomials and constants of the motion. J. Phys. Conf. Ser. 2009, 175, 012009. [Google Scholar] [CrossRef]
- Dirac, P.A.M. Generalized Hamiltonian dynamics. Proc. R. Soc. Lond. A 1958, 246, 326–332. [Google Scholar] [CrossRef]
- Goldstein, H.; Poole, C.; Safko, J. Classical Mechanics, 3rd ed.; Addison-Wesley: Glenview, IL, USA, 2001. [Google Scholar]
- de León, M.; Laínz, M. A review on contact Hamiltonian and Lagrangian systems. Rev. Acad. Canar. Cienc. 2019, 31, 1. [Google Scholar]
- Curtright, T.; Zachos, C. Evolution profiles and functional equations. J. Phys. A Math. Theor. 2009, 42, 485208. [Google Scholar] [CrossRef]
- Curtright, T.L.; Zachos, C.K. Chaotic maps, Hamiltonian flows and holographic methods. J. Phys. A Math. Theor. 2010, 43, 445101. [Google Scholar] [CrossRef]
- Curtright, T.; Veitia, A. Logistic map potentials. Phys. Lett. A 2011, 375, 276–282. [Google Scholar] [CrossRef]
- Curtright, T. Potentials Unbounded Below. SIGMA 2011, 7, 042. [Google Scholar] [CrossRef]
- Curtright, T.L.; Zachos, C.K. Branched Hamiltonians and supersymmetry. J. Phys. A Math. Theor. 2014, 47, 145201. [Google Scholar] [CrossRef]
- Curtright, T. The BASICs of Branched Hamiltonians. Bulg. J. Phys. 2018, 45, 102–113. [Google Scholar]
- Bagchi, B.; Kamil, S.M.; Tummuru, T.R.; Semorádová, I.; Znojil, M. Branched Hamiltonians for a Class of Velocity Dependent Potentials. J. Phys. Conf. Ser. 2017, 839, 012011. [Google Scholar] [CrossRef]
- Choudhury, A.G.; Guha, P. Branched Hamiltonians and time translation symmetry breaking in equations of the Liénard type. Mod. Phys. Lett. A 2019, 34, 1950263. [Google Scholar] [CrossRef]
- Bagarello, F.; Gazeau, J.P.; Szafraniec, F.H.; Znojil, M. (Eds.) Non-Selfadjoint Operators in Quantum Physics: Mathematical Aspects; John Wiley & Sons: Hoboken, NJ, USA, 2015. [Google Scholar]
- Znojil, M. PT-symmetric model with an interplay between kinematical and dynamical non-localities. J. Phys. A Math. Theor. 2015, 48, 195303. [Google Scholar] [CrossRef]
- Bagchi, B.; Choudhury, A.G.; Guha, P. On quantized Liénard oscillator and momentum dependent mass. J. Math. Phys. 2015, 56, 012105. [Google Scholar] [CrossRef]
- Bagchi, B.; Ghosh, R.; Goswami, P. Generalized Uncertainty Principle and Momentum-Dependent Effective Mass Schrödinger Equation. J. Phys. Conf. Ser. 2020, 1540, 012004. [Google Scholar] [CrossRef]
- Chandrasekar, V.K.; Senthilvelan, M.; Lakshmanan, M. Unusual Liénard-type nonlinear oscillator. Phys. Rev. E 2005, 72, 066203. [Google Scholar] [CrossRef]
- Bagchi, B.; Ghosh, D.; Tummuru, T.R. Branched Hamiltonians for a quadratic type Liénard oscillator. J. Nonlinear Evol. Equ. Appl. 2020, 2018, 101–106. [Google Scholar]
- Ruby, V.C.; Senthilvelan, M.; Lakshmanan, M. Exact quantization of a PT-symmetric (reversible) Liénard-type nonlinear oscillator. J. Phys. A Math. Theor. 2012, 45, 382002. [Google Scholar] [CrossRef]
- Bagchi, B.; Ghosh, D.; Modak, S.; Panigrahi, P.K. Nonstandard Lagrangians and branching: The case of some nonlinear Liénard systems. Mod. Phys. Lett. A 2019, 34, 1950110. [Google Scholar] [CrossRef]
- von Roos, O. Position-dependent effective masses in semiconductor theory. Phys. Rev. B 1983, 27, 7547. [Google Scholar] [CrossRef]
- van der Pol, B. LXXXVIII. On “relaxation-oscillations”. Philos. Mag. 1926, 2, 978–992. [Google Scholar] [CrossRef]
- Strogatz, S.H. Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry, and Engineering, 2nd ed.; CRC Press: Boca Raton, FL, USA, 2014. [Google Scholar]
- Mickens, R.E. Truly Nonlinear Oscillations; World Scientific: Hackensack, NJ, USA, 2009. [Google Scholar]
- Demina, M.V. Liouvillian integrability of the generalized Duffing oscillators. Anal. Math. Phys. 2021, 11, 25. [Google Scholar] [CrossRef]
- Whittaker, E.T. A Treatise on the Analytical Dynamics of Particles and Rigid Bodies; Cambridge University Press: Cambridge, UK, 1988. [Google Scholar]
- Yan, C.C. Construction of Lagrangians and Hamiltonians from the equation of motion. Am. J. Phys. 1978, 46, 671–675. [Google Scholar] [CrossRef]
- Nucci, M.C.; Leach, P.G.L. The Jacobi Last Multiplier and its applications in mechanics. Phys. Scr. 2008, 78, 065011. [Google Scholar] [CrossRef]
- Nucci, M.C.; Leach, P.G.L. An Old Method of Jacobi to Find Lagrangians. J. Nonlinear Math. Phys. 2009, 16, 431–441. [Google Scholar] [CrossRef]
- Nucci, M.C.; Tamizhmani, K.M. Lagrangians for Dissipative Nonlinear Oscillators: The Method of Jacobi Last Multiplier. J. Nonlinear Math. Phys. 2010, 17, 167–178. [Google Scholar] [CrossRef]
- Mitra, S.; Ghose-Choudhury, A.; Poddar, S.; Garai, S.; Guha, P. The Jacobi Last Multiplier, Lagrangian and Hamiltonian for Levinson–Smith type equations. Phys. Scr. 2024, 99, 015237. [Google Scholar] [CrossRef]
- Cariñena, J.F.; Fernández–Núñez, J. Jacobi Multipliers in Integrability and the Inverse Problem of Mechanics. Symmetry 2021, 13, 1413. [Google Scholar] [CrossRef]
- Cariñena, J.F.; Guha, P. Non-standard Hamiltonian structures of Liénard equation and contact geometry. Int. J. Geom. Methods Mod. Phys. 2019, 16, 1940001. [Google Scholar] [CrossRef]
- Cariñena, J.F.; Guha, P. Geometry of non-standard Hamiltonian structures of Liénard equations and contact structure. Int. J. Geom. Methods Mod. Phys. 2024, 21, 2440005. [Google Scholar] [CrossRef]
- Fernández, F.M.; Guardiola, R.; Ros, J.; Znojil, M. Strong-coupling expansions for the PT-symmetric oscillators V(x) = aix + b(ix)2 + c(ix)3. J. Phys. A: Math. Gen. 1998, 31, 10105. [Google Scholar] [CrossRef]
- Armendáriz-Picón, C.; Damour, T.; Mukhanov, V. k-Inflation. Phys. Lett. B 1999, 458, 209–218. [Google Scholar] [CrossRef]
- Arkani-Hamed, N.; Cheng, H.-C.; Luty, M.; Mukohyama, S. Ghost condensation and a consistent infrared modification of gravity. J. High Energy Phys. 2004, 0405, 074. [Google Scholar] [CrossRef]
- Teitelboim, C.; Zanelli, J. Dimensionally continued topological gravitation theory in Hamiltonian form. Class. Quantum Gravity 1987, 4, L125. [Google Scholar] [CrossRef]
- Witten, E. Dynamical breaking of supersymmetry. Nucl. Phys. B 1981, 188, 513–554. [Google Scholar] [CrossRef]
- Coffman, M.L. Velocity-Dependent Potentials for Particles Moving in Given Orbits. Am. J. Phys. 1952, 20, 195–199. [Google Scholar] [CrossRef]
- Deriglazov, A.A.; Ramírez, W.G. Recent Progress on the Description of Relativistic Spin: Vector Model of Spinning Particle and Rotating Body with Gravimagnetic Moment in General Relativity. Adv. Math. Phys. 2017, 2017, 7397159. [Google Scholar] [CrossRef]
- Deriglazov, A.A. Rotation Matrix of a Charged Symmetrical Body: One-Parameter Family of Solutions in Elementary Functions. Universe 2024, 10, 250. [Google Scholar] [CrossRef]
- Klauder, J.R. Valid Quantization: The Next Step. J. High Energy Phys. Gravit. Cosmol. 2022, 8, 628–634. [Google Scholar] [CrossRef]
- Graefe, E.-M.; Korsch, H.J.; Rush, A.; Schubert, R. Classical and quantum dynamics in the (non-Hermitian) Swanson oscillator. J. Phys. A Math. Theor. 2015, 48, 055301. [Google Scholar] [CrossRef]
- Novák, R. On the Pseudospectrum of the Harmonic Oscillator with Imaginary Cubic Potential. Int. J. Theor. Phys. 2015, 54, 4142–4153. [Google Scholar] [CrossRef]
- Růžička, F. Hilbert Space Inner Products for PT-symmetric Su-Schrieffer-Heeger Models. Int. J. Theor. Phys. 2015, 54, 4154–4163. [Google Scholar] [CrossRef]
- Kato, T. Perturbation Theory for Linear Operators: Classics in Mathematics; Springer: Berlin/Heidelberg, Germany, 1995. [Google Scholar]
- Heiss, W.D. The physics of exceptional points. J. Phys. A Math. Theor. 2012, 45, 444016. [Google Scholar] [CrossRef]
- Correa, F.; Plyushchay, M.S. Spectral singularities in PT-symmetric periodic finite-gap systems. Phys. Rev. D 2012, 86, 085028. [Google Scholar] [CrossRef]
- Znojil, M. Exceptional points and domains of unitarity for a class of strongly non-Hermitian real-matrix Hamiltonians. J. Math. Phys. 2021, 62, 052103. [Google Scholar] [CrossRef]
- Bagarello, F.; Gargano, F. Model pseudofermionic systems: Connections with exceptional points. Phys. Rev. A 2014, 89, 032113. [Google Scholar] [CrossRef]
- Bagchi, B.; Ghosh, R.; Sen, S. Exceptional point in a coupled Swanson system. EPL 2022, 137, 50004. [Google Scholar] [CrossRef]
- Moiseyev, N. Non-Hermitian Quantum Mechanics; Cambridge University Press: Cambridge, UK, 2011. [Google Scholar]
- Rotter, I. A non-Hermitian Hamilton operator and the physics of open quantum systems. J. Phys. A Math. Theor. 2019, 42, 153001. [Google Scholar] [CrossRef]
- Hashimoto, K.; Kanki, K.; Hayakawa, H.; Petrosky, T. Non-divergent representation of a non-Hermitian operator near the exceptional point with application to a quantum Lorentz gas. Prog. Theor. Exp. Phys. 2015, 2015, 023A02. [Google Scholar] [CrossRef]
- Milburn, T.J.; Doppler, J.; Holmes, C.A.; Portolan, S.; Rotter, S.; Rabl, P. General description of quasiadiabatic dynamical phenomena near exceptional points. Phys. Rev. A 2015, 92, 052124. [Google Scholar] [CrossRef]
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Bagchi, B.; Ghosh, A.; Znojil, M. A Reappraisal of Lagrangians with Non-Quadratic Velocity Dependence and Branched Hamiltonians. Symmetry 2024, 16, 860. https://doi.org/10.3390/sym16070860
Bagchi B, Ghosh A, Znojil M. A Reappraisal of Lagrangians with Non-Quadratic Velocity Dependence and Branched Hamiltonians. Symmetry. 2024; 16(7):860. https://doi.org/10.3390/sym16070860
Chicago/Turabian StyleBagchi, Bijan, Aritra Ghosh, and Miloslav Znojil. 2024. "A Reappraisal of Lagrangians with Non-Quadratic Velocity Dependence and Branched Hamiltonians" Symmetry 16, no. 7: 860. https://doi.org/10.3390/sym16070860