Frontiers of Physics

, 13:135203 | Cite as

Geometric field theory and weak Euler–Lagrange equation for classical relativistic particle-field systems

  • Peifeng Fan
  • Hong Qin
  • Jian Liu
  • Nong Xiang
  • Zhi Yu
Research Article
  • 9 Downloads

Abstract

A manifestly covariant, or geometric, field theory of relativistic classical particle-field systems is developed. The connection between the space-time symmetry and energy-momentum conservation laws of the system is established geometrically without splitting the space and time coordinates; i.e., space-time is treated as one entity without choosing a coordinate system. To achieve this goal, we need to overcome two difficulties. The first difficulty arises from the fact that the particles and the field reside on different manifolds. As a result, the geometric Lagrangian density of the system is a function of the 4-potential of the electromagnetic fields and also a functional of the particles’ world lines. The other difficulty associated with the geometric setting results from the mass-shell constraint. The standard Euler–Lagrange (EL) equation for a particle is generalized into the geometric EL equation when the mass-shell constraint is imposed. For the particle-field system, the geometric EL equation is further generalized into a weak geometric EL equation for particles. With the EL equation for the field and the geometric weak EL equation for particles, the symmetries and conservation laws can be established geometrically. A geometric expression for the particle energy-momentum tensor is derived for the first time, which recovers the non-geometric form in the literature for a chosen coordinate system.

Keywords

relativistic particle-field system different manifolds mass-shell constraint geometric weak Euler–Lagrange equation symmetry conservation laws 

Notes

Acknowledgements

This research was supported by the National Magnetic Confinement Fusion Energy Research Project (Grant Nos. 2015GB111003 and 2014GB124005), the National Natural Science Foundation of China (Grant Nos. NSFC- 11575185, 11575186, and 11305171), JSPS-NRF-NSFC A3 Foresight Program (Grant No. 11261140328), the Key Research Program of Frontier Sciences CAS (QYZDB-SSW-SYS004), Geo- Algorithmic Plasma Simulator (GAPS) Project, and the National Magnetic Confinement Fusion Energy Research Project (Grant No. 2013GB111002B).

References

  1. 1.
    E. Noether, Invariante Variationsprobleme, Nachr. König. Gesell. Wiss. Göttingen, Math.-Phys. Kl. 235–257 (1918); also available in English at Transport Theory Statist. Phys. 1, 186–207 (1971)CrossRefGoogle Scholar
  2. 2.
    P. J. Olver, Applications of Lie Groups to Differential Equations, New York: Springer-Verlag, 1993, pp 242–283Google Scholar
  3. 3.
    C. Markakis, K. Uryū, E. Gourgoulhon, J. P. Nicolas, N. Andersson, A. Pouri, and V. Witzany, Conservation laws and evolution schemes in geodesic, hydrodynamic, and magnetohydrodynamic flows, Phys. Rev. D 96(6), 064019 (2017)ADSCrossRefGoogle Scholar
  4. 4.
    R. M. Wald, General Relativity, Chicago and London: The University of Chicago Press, 1984, pp 23–27CrossRefGoogle Scholar
  5. 5.
    H. Qin, R. H. Cohen, W. M. Nevins, and X. Q. Xu, Geometric gyrokinetic theory for edge plasmas, Phys. Plasmas 14(5), 056110 (2007)ADSCrossRefGoogle Scholar
  6. 6.
    L. D. Landau and E. M. Lifshitz, The Classical Theory of Fields, Oxford: Butterworth-Heinemann, 1975, pp 46–89Google Scholar
  7. 7.
    T. D. Brennan and S. E. Gralla, On the magnetosphere of an accelerated pulsar, Phys. Rev. D 89(10), 103013 (2014)ADSCrossRefGoogle Scholar
  8. 8.
    F. Carrasco and O. Reula, Covariant hyperbolization of force-free electrodynamics, Phys. Rev. D 93(8), 085013 (2016)ADSMathSciNetCrossRefGoogle Scholar
  9. 9.
    J. Yu, Q. Ma, V. Motto-Ros, W. Lei, X. Wang, and X. Bai, Generation and expansion of laser-induced plasma as a spectroscopic emission source, Front. Phys. 7(6), 649 (2012)CrossRefGoogle Scholar
  10. 10.
    Z. H. Hu, M. D. Chen, and Y. N. Wang, Current neutralization and plasma polarization for intense ion beams propagating through magnetized background plasmas in a two-dimensional slab approximation, Front. Phys. 9(2), 226 (2014)CrossRefGoogle Scholar
  11. 11.
    J. Zhu, K. Zhu, L. Tao, X. Xu, C. Lin, W. Ma, H. Lu, Y. Zhao, Y. Lu, J. Chen, and X. Yan, Distribution uniformity of laser-accelerated proton beams, Chin. Phys. C 41(9), 097001 (2017)ADSCrossRefGoogle Scholar
  12. 12.
    M. Fathi, A dynamical approach to the exterior geometry of a perfect fluid as a relativistic star, Chin. Phys. C 37(2), 025101 (2013)CrossRefGoogle Scholar
  13. 13.
    H. Qin, J. W. Burby, and R. C. Davidson, Field theory and weak Euler-Lagrange equation for classical particlefield systems, Phys. Rev. E 90(4), 043102 (2014)ADSCrossRefGoogle Scholar
  14. 14.
    L. Infeld, Bull. Acad. Pol. Sci. 5, 491 (1957); also available in the book: Asim O. Barut,Electrodynamics and Classical Theory of Fields & Particles, New York: Dover Publication, INC, 1980, pp 65–66Google Scholar
  15. 15.
    R. Hakim, Remarks on relativistic statistical mechanics (I), J. Math. Phys. 8(6), 1315 (1967)ADSCrossRefGoogle Scholar
  16. 16.
    R. Hakim, Remarks on relativistic statistical mechanics (II): Hierarchies for the Reduced Densities, J. Math. Phys. 8(7), 1379 (1967)ADSCrossRefGoogle Scholar
  17. 17.
    M. Gedalin, Covariant relativistic hydrodynamics of multispecies plasma and generalized Ohm’s law, Phys. Rev. Lett. 76(18), 3340 (1996)ADSCrossRefGoogle Scholar
  18. 18.
    G. Hornig, The covariant transport of electromagnetic fields and its relation to magnetohydrodynamics, Phys. Plasmas 4(3), 646 (1997)ADSMathSciNetCrossRefGoogle Scholar
  19. 19.
    K. C. Baral and J. N. Mohanty, Covariant formulation of the Fokker–Planck equation for moderately coupled relativistic magnetoplasma, Phys. Plasmas 7(4), 1103 (2000)ADSCrossRefGoogle Scholar
  20. 20.
    C. Tian, Manifestly covariant classical correlation dynamics (I): General theory, Ann. Phys. 18(10–11), 783 (2009)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    C. Tian, Manifestly covariant classical correlation dynamics (II): Transport equations and Hakim equilibrium conjecture, Ann. Phys. 19(1–2), 75 (2010)MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    E. D’Avignon, P. J. Morrison, and F. Pegoraro, Action principle for relativistic magnetohydrodynamics, Phys. Rev. D 91(8), 084050 (2015)ADSMathSciNetCrossRefGoogle Scholar
  23. 23.
    S. Yang and X. Wang, On Lorentz invariants in relativistic magnetic reconnection, Phys. Plasmas 23(8), 082903 (2016)ADSCrossRefGoogle Scholar
  24. 24.
    Y. Wang, J. Liu, and H. Qin, Lorentz covariant canonical symplectic algorithms for dynamics of charged particles, Phys. Plasmas 23(12), 122513 (2016)ADSCrossRefGoogle Scholar
  25. 25.
    Y. Shi, N. J. Fisch, and H. Qin, Effective-action approach to wave propagation in scalar QED plasmas, Phys. Rev. A 94(1), 012124 (2016)ADSCrossRefGoogle Scholar
  26. 26.
    D. D. Holm, J. E. Marsden, and T. S. Ratiu, The Euler–Poincaré equations and semidirect products with applications to continuum theories, Adv. Math. 137(1), 1 (1998)MathSciNetCrossRefMATHGoogle Scholar
  27. 27.
    J. Squire, H. Qin, W. M. Tang, and C. Chandre, The Hamiltonian structure and Euler-Poincaré formulation of the Vlasov-Maxwell and gyrokinetic systems, Phys. Plasmas 20(2), 022501 (2013)ADSCrossRefGoogle Scholar
  28. 28.
    Y. Zhou, H. Qin, J. W. Burby, and A. Bhattacharjee, Variational integration for ideal magnetohydrodynamics with built-in advection equations, Phys. Plasmas 21(10), 102109 (2014)ADSCrossRefGoogle Scholar
  29. 29.
    Z. Zhou, Y. He, Y. Sun, J. Liu, and H. Qin, Explicit symplectic methods for solving charged particle trajectories, Phys. Plasmas 24(5), 052507 (2017)ADSCrossRefGoogle Scholar
  30. 30.
    J. Squire, H. Qin, and W. M. Tang, Gauge properties of the guiding center variational symplectic integrator, Phys. Plasmas 19(5), 052501 (2012)ADSCrossRefGoogle Scholar
  31. 31.
    J. Xiao, H. Qin, J. Liu, Y. He, R. Zhang, and Y. Sun, Explicit high-order non-canonical symplectic particlein-cell algorithms for Vlasov-Maxwell systems, Phys. Plasmas 22(11), 112504 (2015)ADSCrossRefGoogle Scholar
  32. 32.
    H. Qin, J. Liu, J. Xiao, R. Zhang, Y. He, Y. Wang, Y. Sun, J. W. Burby, L. Ellison, and Y. Zhou, Canonical symplectic particle-in-cell method for long-term large-scale simulations of the Vlasov–Maxwell equations, Nucl. Fusion 56(1), 014001 (2016)ADSCrossRefGoogle Scholar

Copyright information

© Higher Education Press and Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  • Peifeng Fan
    • 1
    • 2
  • Hong Qin
    • 3
    • 4
    • 5
  • Jian Liu
    • 3
  • Nong Xiang
    • 1
    • 4
  • Zhi Yu
    • 1
    • 4
  1. 1.Institute of Plasma PhysicsChinese Academy of SciencesHefeiChina
  2. 2.Science Island Branch of Graduate SchoolUniversity of Science and Technology of ChinaHefeiChina
  3. 3.Department of Modern PhysicsUniversity of Science and Technology of ChinaHefeiChina
  4. 4.Center for Magnetic Fusion TheoryChinese Academy of SciencesHefeiChina
  5. 5.Plasma Physics LaboratoryPrinceton UniversityPrincetonUSA

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