Plasma and Fusion Research

Volume 14, 3401073 (2019)

Regular Articles

A Note on Stability Analysis by the Second Variations of Lagrangian and Hamiltonian for Ideal Incompressible Plasmas
Keisuke ARAKI
Okayama University of Science, Okayama 700-0005, Japan
(Received 10 January 2019 / Accepted 26 February 2019 / Published 3 June 2019)


From the viewpoint of the differential geometrical approach to the Lagrangian mechanical variational problem, a proof is presented for a general conservation law that the Lagrangian displacement type perturbation field satisfies around a stationary solution, previously derived by Hirota et al. for the Hall magnetohydrodynamic system. Additionally, this mathematical approach is applied to the Hamiltonian mechanical stability analyses of the magnetohydrodynamic and Hall magnetohydrodynamic systems.


stability analysis, Lagrangian mechanics, Hamiltonian mechanics, second variation, double Beltrami flow

DOI: 10.1585/pfr.14.3401073


  • [1] V.I. Arnold, Mathematical methods of classical mechanics 2nd Ed. (Springer, New York, 1989).
  • [2] J.E. Marsden and T. Ratiu, Introduction to mechanics and symmetry (Springer, New York, 1999).
  • [3] Y. Hattori, J. Phys. A: Math. Gen. 27, L21 (1994).
  • [4] K. Araki, J. Phys. A:Math. Theor. 50, 235501 (2017).
  • [5] K. Araki, submitted to Phys. Plasmas; arXiv:1601.05477v2 (2018).
  • [6] S. Kobayashi and K. Nomizu, Foundations of differential geometry. Volume I (Interscience, New-York, 1963).
  • [7] E. Noether, Gott. Nachr. 1918, 235 (1918); transl. M.A. Tavel, arXiv:physics/0503066.
  • [8] M. Hirota, Z. Yoshida and E. Hameiri, Phys. Plasmas 13, 022107 (2006).
  • [9] S.M. Mahajan and Z. Yoshida, Phys. Rev. Lett. 81, 4863 (1998).