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.
Similar content being viewed by others
References
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)
P. J. Olver, Applications of Lie Groups to Differential Equations, New York: Springer-Verlag, 1993, pp 242–283
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)
R. M. Wald, General Relativity, Chicago and London: The University of Chicago Press, 1984, pp 23–27
H. Qin, R. H. Cohen, W. M. Nevins, and X. Q. Xu, Geometric gyrokinetic theory for edge plasmas, Phys. Plasmas 14(5), 056110 (2007)
L. D. Landau and E. M. Lifshitz, The Classical Theory of Fields, Oxford: Butterworth-Heinemann, 1975, pp 46–89
T. D. Brennan and S. E. Gralla, On the magnetosphere of an accelerated pulsar, Phys. Rev. D 89(10), 103013 (2014)
F. Carrasco and O. Reula, Covariant hyperbolization of force-free electrodynamics, Phys. Rev. D 93(8), 085013 (2016)
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)
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)
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)
M. Fathi, A dynamical approach to the exterior geometry of a perfect fluid as a relativistic star, Chin. Phys. C 37(2), 025101 (2013)
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)
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 65–66
R. Hakim, Remarks on relativistic statistical mechanics (I), J. Math. Phys. 8(6), 1315 (1967)
R. Hakim, Remarks on relativistic statistical mechanics (II): Hierarchies for the Reduced Densities, J. Math. Phys. 8(7), 1379 (1967)
M. Gedalin, Covariant relativistic hydrodynamics of multispecies plasma and generalized Ohm’s law, Phys. Rev. Lett. 76(18), 3340 (1996)
G. Hornig, The covariant transport of electromagnetic fields and its relation to magnetohydrodynamics, Phys. Plasmas 4(3), 646 (1997)
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)
C. Tian, Manifestly covariant classical correlation dynamics (I): General theory, Ann. Phys. 18(10–11), 783 (2009)
C. Tian, Manifestly covariant classical correlation dynamics (II): Transport equations and Hakim equilibrium conjecture, Ann. Phys. 19(1–2), 75 (2010)
E. D’Avignon, P. J. Morrison, and F. Pegoraro, Action principle for relativistic magnetohydrodynamics, Phys. Rev. D 91(8), 084050 (2015)
S. Yang and X. Wang, On Lorentz invariants in relativistic magnetic reconnection, Phys. Plasmas 23(8), 082903 (2016)
Y. Wang, J. Liu, and H. Qin, Lorentz covariant canonical symplectic algorithms for dynamics of charged particles, Phys. Plasmas 23(12), 122513 (2016)
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)
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)
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)
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)
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)
J. Squire, H. Qin, and W. M. Tang, Gauge properties of the guiding center variational symplectic integrator, Phys. Plasmas 19(5), 052501 (2012)
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)
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)
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).
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Fan, P., Qin, H., Liu, J. et al. Geometric field theory and weak Euler–Lagrange equation for classical relativistic particle-field systems. Front. Phys. 13, 135203 (2018). https://doi.org/10.1007/s11467-018-0793-z
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s11467-018-0793-z