No data available.
Please log in to see this content.
You have no subscription access to this content.
No metrics data to plot.
The attempt to load metrics for this article has failed.
The attempt to plot a graph for these metrics has failed.
Hamiltonian description of relativistically interacting two‐particle systems
1.D. G. Currie, T. F. Jordan, and E. C. G. Sudarshan, Rev. Mod. Phys. 35, 350 (1963).
2.D. G. Currie, J. Math. Phys. 4, 1470 (1963).
3.J. T. Cannon and T. F. Jordan, J. Math. Phys. 5, 299 (1964).
4.H. Leutwyler, Nuovo Cimento 37, 556 (1965).
5.J. A. Wheeler and R. P. Feynman, Rev. Mod. Phys. 17, 157 (1945).
6.L. Bel, Ann. Inst. Henri Poincaré 14A, 189 (1971).
7.For an excellent summary see D. J. Simms, Z. Naturforsch. 28a, 538 (1973).
8.J. M. Souriau, Structure des systèmes dynamiques (Dunod, Paris, 1970).
9.B. Kostant, in Lectures in Modern Analysis and Applications III, edited by C. T. Taam (Springer, Berlin, 1970).
10.J. M. Souriau, C. R. Acad. Sci. Paris 263B, 1191 (1966)
10.and J. M. Souriau, C. R. Acad. Sci. Paris 265B, 165 (1967).
11.P. Renouard, “Variétés symplectiques et quantification,” Thése, Orsay (1969).
12.R. Arens, Commun. Math. Phys. 21, 139 (1971)
12.and R. Arens, J. Math. Phys. 12, 2415 (1971).
13.J. Rawnsley, “Some applications of quantisation,” Thesis, Oxford (1972).
14.R. Hermann, Lie Groups for Physicists (Benjamin, New York, 1966).
15.J. M. Souriau, “Sur la variété de Kepler,” J. Elhadad, “Sur l’interpretation en géometrie symplectique des états quantiques de l’atome d’hydrogeâne,” and D. J. Simms, “Geometric Quantization of Energy Levels in the Kepler Problem,” in Symposia Mathematica (Academic, New York) (to be published);
15.F. A. E. Pirani, “Once more the Kepler problem,” preprint.
16.P. Havas and J. Plebanski, Bull. Amer. Phys. Soc. 5, 433 (1961).
17.D. G. Currie, Phys. Rev. 142, 817 (1966).
18.R. N. Hill, J. Math. Phys. 8, 201 (1967).
19.D. G. Currie and T. F. Jordan, Phys. Rev. 167, 1178 (1968).
20.R. Arens and D. G. Babbitt, Pacific J. Math. 28, 243 (1969).
21.P. Droz‐Vincent, Physica Scripta 2, 129 (1970).
22.L. Bel, Ann. Inst. Henri Poincaré 12A, 307 (1970).
23.R. Arens, Arch. Rat. Mech. 47, 255 (1972).
24.H. P. Künzle, “Galilei and Lorentz invariance of classical particle interactions,” in Symposia Mathematica (Academic, New York) (to be published).
25.E. H. Kerner, J. Math. Phys. 6, 1218 (1965).
26.R. N. Hill and E. H. Kerner, Phys. Rev. Lett. 17, 1156 (1966).
27.R. N. Hill, J. Math. Phys. 8, 1756 (1967).
28.A. N. Beard and R. Fong, Phys. Rev. 182, 1397 (1969).
29.P. Droz‐Vincent, Nuovo Cimento B 12, 1 (1972).
30.A. Peres, Phys. Rev. Lett. 27, 1666 (1971)
30.and A. Peres, 28, 392 (1972)., Phys. Rev. Lett.
31.H. W. Woodcock and P. Havas, Phys. Rev. D 6, 3422 (1972).
32.L. Bel, C. R. Acad. Sci. Paris 273A, 405 (1971)
32.and L. Bel, Ann. Inst. Henri Poincaré 18A, 57 (1973).
33.For a survey see, for example, P. Havas in Problems in the Foundations of Physics, edited by M. Bunge (Springer, New York, 1971).
34. is a fiber coordinate system of (α,β,…range from 0 to 3, A,B,… from 1 to 3; with unless otherwise specified).
35.P. Droz‐Vincent, C. R. Acad. Sci. Paris 275A, 1263 (1972)
35.and “Local existence for finitely predictive two‐body interactions,” preprint (1973).
36. will be used to lower and raise indices.
37.R. S. Palais, A Global Formulation of the Lie Theory of Transformation Groups (American Mathematical Society, Providence, 1957).
38.However, this distinction is less clear than for E. For example, the imbedding map does not appear to induce an almost tangent structure on Σ except in the most special cases like, e.g.,
39.If the second order system E is not invariant under a time translation group, i.e., if the force law is explicitly time dependent, this construction obviously fails. However it can be argued that every isolated system must be at least invariant under time translations and that systems with time dependent forces can always be considered as part of a larger system that would have some invariance group to make this construction possible.
40.H. P. Künzle, Ann. Inst. Henri Poincaré 17A, 337 (1972).
41.Alternatively, one might instead of let (cf. Droz‐Vincent, Ref. 35).
42.Dropping all the bars for quantities on Σ from now on.
43.Explicit solutions of equivalent systems (not formulated on Σ) have been obtained for one‐dimensional motion by R. N. Hill, J. Math. Phys. 11, 1918 (1970);
43.R. A. Rudd and R. N. Hill, J. Math. Phys. 11, 2740 (1970);
43.C. S. Shukre and T. F. Jordan, J. Math. Phys. 13, 868 (1972).
43.L. Bel, A. Salas, and J. M. Sanchez have solved the system formally by expanding in a power series in the coupling constant.
44.A. Staruszkiewicz, Ann. Inst. Henri Poincaré 14, 69 (1971)
44.and very recently B. Bruhns, Phys. Rev. D 8, 2370 (1973) have introduced a Fokker‐type Lagrangian on Σ for a special interaction (essentially the same as our example in Sec. 4) and obtained some and all, respectively, of these first integrals.
45.For a Galilei invariant one‐particle‐system there is no invariant θ such that defines a second order system (cf., for example, Ref. 24). Hence there is no such θ for the noninteracting two‐particle‐system and therefore none for at least a weakly interacting system (cf. Refs. 8 and 24).
46.This cannot be said of all the quantities some of which correspond to an (origin dependent) angular momentum vector, the others to the “center of mass”.
47.R. Abraham and J. E. Marsden, Foundations of Mechanics (Benjamin, New York, 1967).
48.J. Marsden and A. Weinstein, “Reduction of symplectic manifolds with symmetry,” preprint (1972).
49.See Ref. 32 or 24. The condition can be stated somewhat more elegantly in the terminology of almost tangent structures [cf. J. Klein and A. Voutier, Ann. Inst. Fourier 18, 241 (1968)].
50.In the restricted sense in which we use this term here this would consgist in showing that there exists a function and coordinates complementary to the such that [cf. (P2.38)] and
51.See, for example, P. Havas, Suppl. Nuovo Cimento Ser. 10, 5, 364 (1957).
52.As was done already by J. L. Synge, Proc. Roy. Soc. A 177, 118 (1940)
52.(who, however, used only retarded interactions). The present example in advanced—retarded form was apparently first considered by A. D. Fokker, Physica 9, 33 (1929).
53.Also Staruskiewicz and Bruhns (Ref. 44) have chosen a simple Lagrangian, rather than the simple equations of motion.
Article metrics loading...
Full text loading...