Skip Navigation Links
Discontinuity, Nonlinearity, and Complexity

Dimitry Volchenkov (editor), Dumitru Baleanu (editor)

Dimitry Volchenkov(editor)

Mathematics & Statistics, Texas Tech University, 1108 Memorial Circle, Lubbock, TX 79409, USA


Dumitru Baleanu (editor)

Cankaya University, Ankara, Turkey; Institute of Space Sciences, Magurele-Bucharest, Romania


Invariants in 3D for Classical Superintegrable Systems in Complex Phase Space

Discontinuity, Nonlinearity, and Complexity 1(4) (2012) 399--407 | DOI:10.5890/DNC.2012.07.004

Jasvinder Singh Virdi$^{1}$; S.C. Mishra$^{2}$

$^{1}$ Department of Physics, Panjab University, Chandigarh-160014, INDIA

$^{2}$ Department of Physics, Kurukshetra University, Kurukshetra-136119, INDIA

Download Full Text PDF



Physical dynamical systems in higher dimensions are always interesting. In this context we present here the possibility of its three-dimensional complex dynamical invariant in extended complex phase space(ECPS). Lie algebraic method is used to study three-dimensional classical superintegrable system on the extended complex phase space. Such complex invariants play an important role in the analysis of complex trajectories, also study of non-hermitian Hamiltonian systems.


  1. [1]  Giacomini, H.J., Repetto, C.E. and Zandron, P. (1991), Integrals of motion for three-dimensional non- Hamiltonian dynamical systems, J. Phys. A: Math. Gen., 24, 4567-4574.
  2. [2]  Hietarinta, J. (1987), Direct methods for the search of the second invariant, Phys. Rep., 147, 87-154.
  3. [3]  Evans, N.W. (1990), Superintegrability in classical mechanics, Phys. Rev. A., 41, 5666-5676.
  4. [4]  Kibler, M. and Winternitz, P.(1990), Periodicity and quasiperiodicity for superintegrable Hamiltonian systems, Phys. Lett. A., 147, 338-342.
  5. [5]  Whiteman, K.J. (1977), Invariants and stability in classical mechanics, Rep. Prog. Phys., 40, 1033-1069.
  6. [6]  Hollowood, T.J.(1992), Solitons in affine Toda field theories, Nucl. Phys. B., 386, 166.
  7. [7]  Nelson, D.R. and Snerb, N.M. (1998), Winding numbers, complex currents, and non-Hermitian localization, Phys. Rev. E., 80, 23.
  8. [8]  Kausal, R.S. (2009), Classical and quantum mechanics of complex hamiltonian systems: An extended complex phase space approach, Pramana J. Phy., 73, 2.
  9. [9]  Virdi, J.S., Chand, F., Kumar, C.N. and Mishra, S.C. (2012), Complex dynamical invariants for twodimensional nonhermitian Hamiltonian systems, Can. J. Phys., 90, 2, 151-157.
  10. [10]  Bender C.M. and Boettcher, S. (1998), Real spectra in non-Hermitian Hamiltonians having PT symmetry, Phys. Rev. Lett., 80, 5243-5246.
  11. [11]  Xavier, Jr.A.L. and de Aguiar, M.A.M. (1996), A new form of path integral for the coherent states representation and its semiclassical limit,Ann. Phys. (N.Y.), 252, 458.
  12. [12]  Kaushal, R.S. and Mishra, S. C. (1993), Dynamical algebraic approach and invariants for time-dependent Hamiltonian systems in two dimensions, J. Math. Phys., 34, 5843.
  13. [13]  Virdi, J.S. and Mishra, S. C.(2012), Exact complex integrals in two dimensions for shifted harmonic oscillators, Pramana, J. Phys., 79, 1.
  14. [14]  Virdi, J.S., Chand, F., Kumar, C. N. and Mishra, S. C. (2012), Complex dynamical inv ariants for tw o-dimensional complex potentials, Pramana, J. Phys., 79, 2.
  15. [15]  Virdi, J.S. and Mishra, S. C. (2012), Complex invariant in two-dimension for coupled oscillator system, Int. J. Appl. Math. Comp., 4, 1.
  16. [16]  Strukmeier, J. and Riedel, C. (2000), Exact Invariants for a Class of three-dimensional time-dependent classical Hamiltonians, Phys. Rev. Lett., 85, 3830; Phys. Rev. E., 64, 26503.
  17. [17]  Kovacic, I. (2010), Invariants and approximate solutions for certain non-linear oscillators by means of the field method, Applied Math. and Comp., 215, 3482-3487.
  18. [18]  Abdalla, M.S. and Leach, P.G.L. (2011), Lie algebraic approach and quantum treatment of an anisotropic charged particle via the quadratic invariant, J. Math. Phys., 52, 083504.
  19. [19]  Hietarienta, J. (1983), A search for integrable two-dimensional Hamiltonian systems with polynomial potential, Phy. Letts., 96A, 273-278.
  20. [20]  Kim, S.P. and Page, D. N.(2001), Classical and quantum action-phase variables for time-dependent oscillators, Phys. Rev. A, 64, 012104.
  21. [21]  Kusenko, A. and Shrock, R. (1994), General determination of phases in leptonic mass matrices, Phy. Lett. B., 323, 1824; Phys. Rev. D., 50, R30-R33.
  22. [22]  Gunion, J.F. and Haber, H.E. (2005), Conditions for CP violation in the general two-Higgs-doublet model, Phys. Rev. D., 72, 095002.