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Discontinuity, Nonlinearity, and Complexity 9(2) (2020) 299--307 | DOI:10.5890/DNC.2020.06.010

Jasvinder Singh Virdi

Department of Physics, Veer Surendra Sai University of Technology, Odisha (India)-768018

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Abstract

For driven time dependent harmonic oscillator a detailed and systematic study of several methods for the building of Integrals is carried out. The central feature of the present discussion is the establishment of Integrals for the dynamical system. In spite of their different procedural details all the approaches lead to the same invariant for the given classical system. Advantage and limitations of different methods are briefly highlighted.

References

1. [1]	Whittaker E. T. (1960) "A Treatise on the Analytical Dynamics of Particle and Rigid Bodies," (Cambridge University press: London).

2. [2]	Strukmeier, J. and Riedel, C. (2000), Exact Invariants for a Class of three-dimensional time-dependent classical Hamiltonians, Phys. Rev. Lett., 85, 3830-3836; (2001), Phys. Rev. E., 64, 26503-26509.

3. [3]	Virdi, J.S, Chand, F., Kumar, C.N., and Mishra, S.C. (2012), Complex dynamical invariants for two-dimensional nonhermitian Hamiltonian systems, Can. Jour. Phys., 90, 151-157 .

4. [4]	Kovacic, I. (2010), Invariants and approximate solutions for certain non-linear oscillators by means of the field method, App. Math. and Comp., 215, 3482-3486.

5. [5]	Virdi, J.S, (2012), Invariants in 3D for Classical Superintegrable Systems in Complex Phase Space, Discontinuity, Nonlinearity, and Complexity, 1(4), 399-406.

6. [6]	Virdi, J.S. and Mishra, S.C. (2012), Exact complex integrals in two dimensions for shifted harmonic oscillators, Pramana, Jour. Phys., 79, 19-40.

7. [7]	Virdi, J.S. (2012), Integrals For Time-Dependent Complex Dynamical System In One Dimension, Rom. Journ. Phys, 57, 1270-1277.

8. [8]

   Bertin, M.C., Pimentel, B.M., and Ramirez, J.A. (2012), Construction of time-dependent dynamical invariants: A new approach, J. Math. Phys., 53, 042104-042106; Ronald, H.Q. and Davidson. C. (2006), Symmetries and invariants of the oscillator and envelope equations with time-dependent frequency, Physical Review, 9, 054001-8.

9. [9]	Virdi, J.S, Ahmed,M., and Srivastva, A.K. (2017), Polynomial integral for square and inverse-square potential systems, AIP Conf. Proc., 1860, 020069, 1-8.

10. [10]	Takayama, K. (1982), Dynamical Invariant For Forced Time Dependent Harmonic Oscillator, Physics letters A, 88(2), 55-62.

11. [11]	Virdi J. S, Chand F, Kumar C. N, andMishra S. C, (2012) Complex dynamical invariants for two-dimensional complex potentials, Pramana, Jour. Phys., 79, 173-179.

12. [12]	Virdi, J.S. (2013), Search of exact invariants for PT and non-PT -symmetric complex Hamiltonian systems, Applied Math. and Comp., 219, 9731-9336.

13. [13]	Heitarienta, J. (1983), A search for integrable two-dimensional Hamiltonian systems with polynomial potential, Phy. Letts. A, 96, 6-11.

14. [14]	Lewis, H.R. (1968), Class of Exact Invariants for Classical and Quantum time dependent Harmonic Oscillators, Jour. of Math. Phy., 9(11), 1976-1982.

15. [15]	Kaushal, R.S. and Korsch, H.J. (1981), Dynamical Noethers invariants for time dependent nonlinear system;, Journal of Mathematical Physics, 22(9), 1904-1910.

16. [16]	Burgan, J.R., Feix,M.R., Fijalkow,M., andMunier, A. (1979), Solution of TheMulti Dimensional Quantum Harmonic Oscillator With Time dependent Frequencies through Fourier, Hermite and Wigner Transforms, Phys. Lett. A., 74, 11- 18.

17. [17]	Lutzky,M. (1978), Noethers Theorem and the time dependent harmonic oscillator, J. Phys. A., 68, 3-6.

18. [18]	Ray, J.R. and Reid, J.L. (1979), Exact time dependent invariants for N Dimensional systems Physical Review A, 74, 23-29.