Classical and quantum dynamics of over-damped non linear systems
DOI:
https://doi.org/10.31349/RevMexFis.69.010701Keywords:
Overdamped non linear systems, Lagrangian, HamiltonianAbstract
Overdamping is a regime in which friction is sufficiently large that the motion either decays to its equilibrium position or it crosses the equilibrium position exactly once before returning monotonically towards the equilibrium position. The phenomena of overdamping has been studied classically and quantum mechanically only for the case of the linear damped harmonic oscillator. Here we study the classical and quantum dynamics of a family of over-damped non linear systems. The main objective of this paper is to find a Lagrangian and Hamiltonian framework to study over-damped non linear systems and to show that a quantum mechanical description can be developed in the momentum representation. Our results reduce to the well known solution of the linear damped harmonic oscillator when the non linear part is set to zero.
References
D. G. Zill, Differential equations with boundary-value problems, (Cengage Learning, 2016).
S. Wolff and J. Yang, Overdamped nonlinear systems. IEEE Transactions on Automatic Control, 15 (1970) 699. https://doi.org/10.1109/TAC.1970.1099577
V. K. Chandrasekar, M. Senthilvelan, and M. Lakshmanan, On the lagrangian and hamiltonian description of the damped linear harmonic oscillator. J. Math. Phys. 48 (2007) 032701. https://doi.org/10.1063/1.2711375
V. K. Chandrasekar, M. Senthilvelan, and M. Lakshmanan, On the complete integrability and linearization of certain secondorder nonlinear ordinary differential equations. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 461 (2005) 2451. https://doi.org/10.1098/rspa.2005.1465
M. Jean Prelle and M. F Singer, Elementary first integrals of differential equations. Transactions of the American Mathematical Society, 279 (1983) 215. https://doi.org/10.1145/800206.806368
C. M. Bender, M. Gianfreda, N. Hassanpour, and H. F. Jones, Comment on “on the lagrangian and hamiltonian description of the damped linear harmonic oscillator” [J. Math. Phys. 48 (2007) 032701.] J. Math. Phys. 57 (2016) 084101. https://doi.org/10.1063/1.4960722
G. Gonzalez, Lagrangians and hamiltonians for one- ´ dimensional autonomous systems. Inter. J. Theo. Phys. 43 (2004) 1885. https://doi.org/10.1023/B:IJTP.0000048998.57747.99.
I. S. Murty, B. L. Deekshatulu, and G. Krishna, On an asymptotic method of krylov-bogoliubov for overdamped nonlinear systems. J. the Franklin Institute, 288 (1969) 49. https://doi.org/10.1016/0016-0032(69)00203-1.
M. Razavy, Classical and quantum dissipative systems, (World Scientific, 2005).
G. Lopez and P. Lopez, Velocity quantization approach of the one-dimensional dissipative harmonic oscillator. International J. Theo. Phys. 45 (2006) 734. https://doi.org/10.1007/s10773-006-9064-9.
G. Gonzalez, Quantum bouncer with quadratic dissipation. Rev. Mex. Fis. 54 (2008) 5. http://www.scielo.org.mx/scielo.php?script=sci artttext&pid= S0035-001X2008000100002&lngles&nrm=iso.
J. J. Sakurai and E. D. Commins, Modern quantum mechanics, (revised edition, 1995).
D. Chruscinski, Quantum mechanics of damped systems. J. Math. Phys. 44 (2003) 3718. https://doi.org/10.1063/1.1599074.
Downloads
Published
How to Cite
Issue
Section
License
Copyright (c) 2022 Gabriel Gonzalez Contreras
This work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.
Authors retain copyright and grant the Revista Mexicana de Física right of first publication with the work simultaneously licensed under a CC BY-NC-ND 4.0 that allows others to share the work with an acknowledgement of the work's authorship and initial publication in this journal.
