Two charges in an external electromagnetic field : a generalized covariant hamiltonian formulation

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1. [1] J.L. Sanz and J. Martin, Ann. Inst. H. Poincaré, 24 A, 1976, p. 347. 
2. [2] The proof of this theorem made by Ph. Droz-Vincent (Il Nuovo Cimento, 12 B, 1972, p. 1) is incorrect, but the theorem can be proven to be correct (see Section II). 
3. [4] D.G. Currie, Phys. Rev., t. 142, 1966, p. 817. MR198924
4. [5] R.N. Hill, J. Math. Phys., t. 8, 1967, p. 201. 
5. [6] L. Bel, Ann. Inst. H. Poincaré, t. 12, 1970, p. 307. MR266567
6. [7] L. Bel, A. Salas and J.M. Sánchez, Phys. Rev., 7 D, 1973, p. 1099. 
7. [8] A. Salas and J.M. Sánchez, Il Nuovo Cimento, 20 B, 1974, p. 209. 
8. [9] L. Bel and J. Martin, Phys. Rev., 8 D, 1973, p. 4347. 
9. [10] R. Lapiedra and L. Mas, Phys. Rev., 13 D, 1976, p. 2805. 
10. [11] L. Bel and J. Martin, Phys. Rev., 9 D, 1974, p. 2760. 
11. [12] L. Bel and X. Fustero, Ann. Inst. H. Poincaré, 25, 1976, p. 411. Zbl0347.35070
12. [13] Ph. Droz-Vincent, Lett. Il Nuovo Cimento, t. 1, 1969, p. 839. 
13. [14] J. Wray, Phys. Rev., 1 D, 1969, p. 2212. 
14. [15] L. Bel, Ann. Inst. H. Poincaré, t. 14, 1971, p. 189. MR286379
15. [16] L. Bel, Doctoral course, Departamento de Fisica Teórica, Universidad de Barcelona, 1976. 
16. [19] R. Abraham, Foundations of Mechanics, W. A. Benjamin, 1967. Zbl0158.42901
17. [20] C. Godbillon, Géométrie diferentielle et Mécanique Analitique, Hermann, 1969. Zbl0174.24602MR242081
18. [21] L. Bel, Doctoral Course, Universidad Autónoma de Madrid, 1972, unpublished. 
19. [22] J. Martin and J.L. Sanz, to appear in J. Math. Phys., Section 2. 
20. [23] Let us consider a symplectic form on (TM4)N with local expression Ω = 1/2ΩABdyA dyB (A, B = 0, ... 8N - 1; yα = xα1, ..., y4(N - 1) + α = xαN, y4N+α = πα1, ..., y4(2N-1)+α = παN) where ΩAB are skewsymmetric functions on (TM4)N. The Poisson bracket of two functions f and g on (TM4)N is defined by [f, g] = - Ω-1AB∂f ∂yA∂g∂yB where Ω-1AB is the inverse matrix of ΩAB (i. e., Ω-1ABΩBC = δAC). As is well-known in the literature (see, for example, L. Bel, Ann. Inst. H. Poincaré, t. 18 A, 1973, p. 57 ; H.P. Kunzle, Symposia Mathematica, t. 14, 1974, p. 53 ; J. Math. Phys., t. 15, 1974, p. 1033) condition (22) can be equivalently written in the form [xαa, xβb] = 0 ([, ] being the Poisson bracket relative to Ω), which is the classical form of expressing the canonical character of the position variables xαa. 
21. [24] J.L. Sanz, to be published in J. Math. Phys. 
22. [25] D.G. Currie, T.F. Jordan and E.C.G. Sudarshan, Rev. Mod. Phys., t. 35, 1963, p. 350. MR151138
23. [26] R.N. Hill, J. Math. Phys., t. 8, 1967, p. 1756. 
24. [27] J. Martin and J.L. Sanz, J. Math. Phys., t. 19, 1978, p. 780. 
25. [28] D. Hirondel, J. Math. Phys., t. 15, 1974, p. 1689. 
26. [29] F.J. Kennedy, J. Math. Phys., t. 16, 1975, p. 1844. 
27. [30] L. Bel and J. Martin, Ann. Inst. H. Poincaré, 22 A, 1957, p. 173. MR378697
28. [32] Y. Choquet-Bruhat, Géométrie différentielle et systèmes extérieures, Dunod, 1968. Zbl0164.22001MR236824
29. [33] J.L. Sanz, Tesis Doctoral, Universidad Autónoma de Madrid, 1976.