Two charges in an external electromagnetic field : a generalized covariant hamiltonian formulation
Sanz, J. L.
Annales de l'I.H.P. Physique théorique, Tome 31 (1979), p. 115-139 / Harvested from Numdam
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     author = {Sanz, J. L.},
     title = {Two charges in an external electromagnetic field : a generalized covariant hamiltonian formulation},
     journal = {Annales de l'I.H.P. Physique th\'eorique},
     volume = {31},
     year = {1979},
     pages = {115-139},
     mrnumber = {561918},
     language = {en},
     url = {http://dml.mathdoc.fr/item/AIHPA_1979__31_2_115_0}
}

Sanz, J. L. Two charges in an external electromagnetic field : a generalized covariant hamiltonian formulation. Annales de l'I.H.P. Physique théorique, Tome 31 (1979) pp. 115-139. http://gdmltest.u-ga.fr/item/AIHPA_1979__31_2_115_0/

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[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.

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