Acta Mechanica

, Volume 230, Issue 6, pp 2267–2277 | Cite as

Hamiltonian description of nonlinear curl forces from cofactor systems

  • A. Ghose-ChoudhuryEmail author
  • Partha Guha
Original Paper


Recently, Berry and Shukla presented (J Phys A 45:305201, 2012; J Phys A 46:422001, 2013; Proc R Soc A 471:20150002, 2015) a fundamental new dynamics concerning forces (accelerations) depending only on position, i.e. without velocity-dependent dissipation, which was partly anticipated in the papers of cofactor systems introduced by the Linköping school (Rauch-Wojciechowski et al. in J Math Phys 40:6366–6398, 1999; Lundmark in Stud Appl Math 110(3):257–296, 2003; Lundmark in Integrable nonconservative Newton systems with quadratic integrals of motion. Linköping Studies in Science and Technology. Thesis No. 756, Linköping Univ., Linköping, 1999). In this paper, we extend their results to nonlinear curl forces, where the nonlinearity is with respect to the coordinate dependence of the forces, and study the Hamiltonians for homogeneous quadratic and cubic cases presenting the conditions for existence of Hamiltonian curl forces. In particular, we examine the existence and expressions of the Hamiltonian curl forces for planar systems when the accelerations are given by general (both homogeneous and inhomogeneous) second-order and homogeneous cubic polynomials, and also associate the cubic case with an example from optics.


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  1. 1.
    Berry, M.V., Shukla, P.: Classical dynamics with curl forces, and motion driven by time-dependent flux. J. Phys. A 45, 305201 (2012)CrossRefzbMATHGoogle Scholar
  2. 2.
    Berry, M.V., Shukla, P.: Physical curl forces: dipole dynamics near optical vortices. J. Phys. A 46, 422001 (2013)CrossRefzbMATHGoogle Scholar
  3. 3.
    Berry, M.V., Shukla, P.: Hamiltonian curl forces. Proc. R. Soc. A 471, 20150002 (2015)CrossRefzbMATHGoogle Scholar
  4. 4.
    Rauch-Wojciechowski, S., Marciniak, K., Lundmark, H.: Quasi-Lagrangian systems of Newton equations. J. Math. Phys. 40, 6366–6398 (1999)CrossRefzbMATHGoogle Scholar
  5. 5.
    Lundmark, H.: Higher-dimensional integrable Newton systems with quadratic integrals of motion. Stud. Appl. Math. 110(3), 257–296 (2003)CrossRefzbMATHGoogle Scholar
  6. 6.
    Lundmark, H.: Integrable nonconservative Newton systems with quadratic integrals of motion, Linköping Studies in Science and Technology, Thesis No. 756, Linköping Univ., Linköping (1999)Google Scholar
  7. 7.
    Gutzwiller, M.C.: The anistropic Kepler problem in two dimensions. J. Math. Phys. 14, 139–152 (1973)CrossRefGoogle Scholar
  8. 8.
    Devaney, R.L.: Nonregularizability of the anisotropic Kepler problem. J. Differ. Eqns 29, 253 (1978)CrossRefzbMATHGoogle Scholar
  9. 9.
    Chaumet, P.C., Nieto-Vesperinas, M.: Time-averaged total force on a dipolar sphere in an electromagnetic field. Opt. Lett. 25, 1065–1067 (2000)CrossRefGoogle Scholar
  10. 10.
    Shimizu, Y., Sasada, H.: Mechanical force in laser cooling and trapping. Am. J. Phys. 66, 960–967 (1998)CrossRefGoogle Scholar
  11. 11.
    Albaladejo, S., Marqués, M.I., Laroche, M., Sáenz, J.J.: Scattering forces from the curl of the spin angular momentum. Phys. Rev. Lett 102, 113602 (2009). CrossRefGoogle Scholar
  12. 12.
    Roberts, J.A.G., Quispel, G.R.W.: Chaos and time-reversal symmetry. Order and chaos in reversible dynamical systems. Phys. Rep. 216, 63–177 (1992)CrossRefGoogle Scholar
  13. 13.
    Wu, P., Huang, R., Tischer, C., Jonas, A., Florin, E.L.: Direct measurement of the nonconservative force field generated by optical tweezers. Phys. Rev. Lett. 103, 108101 (2009)CrossRefGoogle Scholar
  14. 14.
    Wedemann, R.S., Plastino, A.R., Tsallis, C.: Curl forces and the nonlinear Fokker–Planck equation. Phys. Rev. E 94, 062105 (2016)CrossRefGoogle Scholar
  15. 15.
    Ghose-Choudhury, A., Guha, P., Paliathanasis, A., Leach, P.G.L.: Noetherian symmetries of noncentral forces with drag term. Int. J. Geom. Methods Mod. Phys. 14, 1750018 (2017)CrossRefzbMATHGoogle Scholar
  16. 16.
    Douglas, J.: Solution of the inverse problem of the calculus of variations. Trans. Am. Math. Soc. 50, 71–128 (1941)CrossRefzbMATHGoogle Scholar
  17. 17.
    Daicu, F.: Near-collision dynamics for particle systems with quasihomogeneous poten- tials. J. Differ. Eqn. 128, 58–77 (1996)CrossRefGoogle Scholar
  18. 18.
    Wojtkowski, M.P.: Integrability via Reversibility (2015). arXiv:1502.03074 [math.DS]

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© Springer-Verlag GmbH Austria, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of PhysicsDiamond Harbour Women’s UniversitySarishaIndia
  2. 2.SN Bose National Centre for Basic SciencesKolkataIndia

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