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World‐line invariance in predictive mechanics
1.P. Havas and J. Plebański, Bull. Am. Phys. Soc. 5, 433 (1960).
2.L. Bel, Ann. Inst. H. Poincaré 14, 189 (1971);
2.the expression “finitely predictive”, which is more general than “predictive” since it only says that the number of initial data which determine the dynamically possible trajectories is finite, was used by Ph. Droz‐Vincent, Phys. Scr. 2, 129 (1970).
3.Two reviews, with numerous references, are those of P. Havas, in Statistical Mechanics of Equilibrium and Non‐Equilibrium, edited by J. Meixner (North‐Holland, Amsterdam, 1965);
3.and of D. G. Currie and T. F. Jordan, in Lectures in Theoretical Physics, edited by W. E. Brittin and A. O. Barut (Gordon and Breach, New York, 1968), Vol. X‐A, p. 91.
4.D. G. Currie, Phys. Rev. 142, 817 (1966).
5.R. N. Hill, J. Math. Phys. 8, 201 (1967).
6.L. Bel, Ann. Inst. H. Poincaré 12, 307 (1970).
7.The treatment in this paper is only local, even if this is not stated explicitly. All functions considered are assumed to be analytic.
8.The essential points of this subsection were obtained by Ph. Droz‐Vincent, Lett. Nuovo Cimento 1, 839 (1969),
8.and Ref. 2. In particular, he derived the integrability conditions (3.29). However, he considered only relativistic invariance and consequently he parametrized the trajectories by the proper times rather than the coordinate time differences (3.17). Some years before, P. Havas and J. Plebański (unpublished, 1960) had followed an approach similar to the one of this paper, but they stopped short of obtaining Eq. (3.29).
9.See, for example, L. P. Eisenhart, Continuous Groups of Transformations (Dover, New York, 1961), Chap. I.
10.This is essential. If the action of the group is given on the derivation of the necessary and sufficient conditions (4.16) is trivial. This is what is done in R. N. Hill and E. H. Kerner, Phys. Rev. Lett. 17, 1156 (1966);
10.and in R. N. Hill, J. Math. Phys. 8, 1756 (1966). The equations in those papers equivalent to our Eq. (4.16) are Eqs. (2) and (17), respectively.
11.A proof, which is an adaptation of the proof of Theorem [29.1] in Eisenhart (Ref. 9), is given in A. Salas, “Systems of classical interacting point particles with Newtonian canonicity: world‐line invariance and canonicity,” Temple University Ph.D. thesis (1978), Appendix 3.
12.For the case of Poincaré invariance and it has been proved geometrically that there are such solutions: R. Arens, Arch. Rat. Mech. Anal. 47, 255 (1972);
12.H. P. Künzle, Symp. Math. 14, 53 (1974) [cf. the well‐known no‐interaction theorems (see Ref. 3), which in addition to world‐line invariance require the transformations of the Poincaré group to be canonical, and the physical position coordinates to be canonical].
13.V. Fock, The Theory of Space Time and Gravitation (Pergamon, New York, 1964), Appendix A.
14.At first sight it might appear that several results of this paper, notably Theorem 1, are a direct consequence of Lie’s fundamental theorems of the theory of continuous groups of transformations, and hence their laborious proof of sufficiency was unnecessary. (For example, the sufficiency of Currie‐Hill’s conditions for Poincaré invariance was assumed even before it was proved rigorously in Ref. 6.) However, this is not the case, as discussed at the end of Ref. 6.
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