Abstract
An open issue in classical relativistic mechanics is the consistent treatment of the dynamics of classical N-body systems of mutually interacting particles. This refers, in particular, to charged particles subject to EM interactions, including both binary interactions and self-interactions (EM-interacting N-body systems). The correct solution to the question represents an overriding prerequisite for the consistency between classical and quantum mechanics. In this paper it is shown that such a description can be consistently obtained in the context of classical electrodynamics, for the case of a N-body system of classical finite-size charged particles. A variational formulation of the problem is presented, based on the N -body hybrid synchronous Hamilton variational principle. Covariant Lagrangian and Hamiltonian equations of motion for the dynamics of the interacting N-body system are derived, which are proved to be delay-type ODEs. Then, a representation in both standard Lagrangian and Hamiltonian forms is proved to hold, the latter expressed by means of classical Poisson Brackets. The theory developed retains both the covariance with respect to the Lorentz group and the exact Hamiltonian structure of the problem, which is shown to be intrinsically non-local. Different applications of the theory are investigated. The first one concerns the development of a suitable Hamiltonian approximation of the exact equations that retains finite delay-time effects characteristic of the binary interactions and self-EM-interactions. Second, basic consequences concerning the validity of Dirac generator formalism are pointed out, with particular reference to the instant-form representation of Poincaré generators. Finally, a discussion is presented both on the validity and possible extension of the Dirac generator formalism as well as the failure of the so-called Currie “no-interaction” theorem for the non-local Hamiltonian system considered here.
Similar content being viewed by others
References
J.D. Jackson, Classical Electrodynamics (John Wiley and Sons, 1975).
H. Goldstein, Classical Mechanics, 2nd edition (Addison-Wesley, 1980).
P.A.M. Dirac, Rev. Mod. Phys. 21, 392 (1949).
P.A.M. Dirac, Proc. Roy. Soc. London A 167, 148 (1938).
W. Pauli, Theory of Relativity (Pergamon, N.Y., 1958).
R. Feynman, Lectures on Physics, Vol. 2 (Addison-Wesley Publishing Company, Reading, MA, USA, 1970) special reprint (1988).
C. Cremaschini, M. Tessarotto, EPJ Plus 126, 42 (2011).
C. Cremaschini, M. Tessarotto, EPJ Plus 126, 63 (2011).
M. Dorigo, M. Tessarotto, P. Nicolini, A. Beklemishev, AIP Conf. Proc. 1084, 152 (2008).
H.A. Lorentz, Arch. Néderl. Sci. Exactes Nat. 25, 363 (1892).
M. Abraham, Theorie der Elektrizität: Elektromagnetische Strahlung, Vol. II (Teubner, Leiptzig, 1905).
L.D. Landau, E.M. Lifschitz, Field Theory, Theoretical Physics, Vol. 2 (Addison-Wesley, N.Y., 1957).
D.G. Currie, J. Math. Phys. 4, 1470 (1963).
D.G. Currie, T.F. Jordan, E.C.G. Sudarshan, Rev. Mod. Phys. 35, 350 (1963).
C. Fronsdal, Phys. Rev. D 4, 1689 (1971).
A. Komar, Phys. Rev. D 18, 1881 (1978).
A. Komar, Phys. Rev. D 18, 1887 (1978).
A. Komar, Phys. Rev. D 18, 3617 (1978).
A. Komar, Phys. Rev. D 19, 2908 (1979).
J.S. Nodvik, Ann. Phys. 28, 225 (1964).
M. Tessarotto, C. Cremaschini, P. Nicolini, A. Beklemishev, Proceedings of the 25th RGD International Symposium on Rarefied Gas Dynamics, St. Petersburg, Russia, 2006, edited by M.S. Ivanov, A.K. Rebrov (Novosibirsk Publishing House of the Siberian Branch of the Russian Academy of Sciences, 2007).
M. Tessarotto, C. Cremaschini, M. Dorigo, P. Nicolini, A. Beklemishev, AIP Conf. Proc. 1084, 158 (2008).
A.N. Beard, R. Fong, Phys. Rev. 182, 1397 (1969).
A.F. Kracklauer, J. Math. Phys. 17, 693 (1976).
J. Martin, J.L. Sanz, J. Math. Phys. 19, 780 (1978).
N. Mukunda, E.C.G. Sudarshan, Phys. Rev. D 23, 2210 (1981).
A.P. Balachandran, D. Dominici, G. Marmo, N. Mukunda, J. Nilsson, J. Samuel, E.C.G. Sudarshan, F. Zaccaria, Phys. Rev. D 26, 3492 (1982).
J.T. Cannon, T.F. Jordan, J. Math. Phys. 5, 299 (1964).
E.C.G. Sudarshan, N. Mukunda, Found. Phys. 13, 385 (1983).
G. Marmo, N. Mukunda, E.C.G. Sudarshan, Phys. Rev. D 30, 2110 (1984).
S. De Bièvre, J. Math. Phys. 27, 7 (1986).
F.-B. Li, J. Math. Phys. 31, 1395 (1990).
I.T. Todorov, Ann. Inst. H. Poincaré A 28, 207 (1978).
Ph. Droz-Vincent, Phys. Scr. 2, 129 (1970).
Ph. Droz-Vincent, Ann. Inst. H. Poincaré A 27, 407 (1977).
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Cremaschini, C., Tessarotto, M. Hamiltonian structure of classical N-body systems of finite-size particles subject to EM interactions. Eur. Phys. J. Plus 127, 4 (2012). https://doi.org/10.1140/epjp/i2012-12004-4
Received:
Accepted:
Published:
DOI: https://doi.org/10.1140/epjp/i2012-12004-4