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Abstract: |
The essential physics of guiding-center (of charged particles) can be well depicted by the Littlejohnˇs Lagrangian in the 4-dimensional configuration space, which was adopted to build up geometric algorithms[1] based on the discrete variational principle[2,3], and later also practiced on other discrete variational approaches[4-6]. The key to success is believed to verify the conditions supposed to be satisfied by the Lagrangian differential forms, which are collectively referred to be (De Rahm) cohomology, as emphasized in Ref.5. Among others it is performed on the Euler-Lagrangian 1-form L E , the canonical 1-form L and the canonical 2-form L :L for both the single particles and the guiding-center. The up-to-date efforts, however, did not see the way for guiding-centers, by which either L E is shown to be closed or L is shown to be conserved within the framework of Lagrangian system unless restricted to the solution space. On the other hand, since L is closed and non-degenerate, the global Hamiltonian vector fields H X are derived on the symplectic manifolds defined by the L along which L is invariant, dictated by the so-called non-canonical Hamiltonian equations. The geometric algorithms based on alternative variational approaches are also presented for guiding-centers; one in attempt to incorporate into the energy conservation by making use of total variational approach[4,6], and the other of using difference variables introduced into the discrete variational principle[5], however, for comparison purposes only. Also discussed is the possibility of extending the geometric algorithm to physical law, e.g. reducing the current conservation to volume preservation of guiding-centers via coordinate transformation.
References
[1] H.Qin, X.Guan, W.M.Tang, Phys. Plasmas 16, 042510 (2009)
[2] J.M.Wendlandt, J.E.Marsden, Physica D 106 223 (1997)
[3] J.E.Marsden & M.West Acta Numerica (2001), pp. 357514
[4] C.Kane, J.E.Marsden , M.Ortiz, JOURNAL OF MATHEMATICAL PHYSICS 40, NUMBER 7 (1999)
[5] H.Y.Guo, Ke Wu, JOURNAL OF MATHEMATICAL PHYSICS, 44 5978 (2003)
[6] Kang Feng, Mengzhao Qin, Symplectic Geometric Algorithms for Hamiltonian Systems, Springer; 1st Edition. edition (December 9, 2010), ch.14. |
