Perturbative methods in Celestial Mechanics and the roots of Quantum Mechanics: a historical note

 Access to full text

 Full (PDF)

 Full (DjVu)

1. AITCHISON, I., MACMANUS, D.A., and SNYDER, T.M., 2004, Understanding Heisenberg's `magical' paper of July 1925: a new look at the calculational details, American J. Phys.72, 1370. Zbl1219.81004MR2093526DOI10.1119/1.1775243
2. https://www.aip.org/history/heisenberg/p07.htm 
3. ALI, M.K., 1985, The quantum normal form and its equivalents, J. Math. Phys.26, 2565. MR803802DOI10.1063/1.526775
4. ARNOLD, V.I., 1963, Small denominators and the problem of stability of motion in classical and celestial mechanics, Russ. Math. Surveys, 18, 9. MR170705
5. BERRY, M., 1989, Quantum chaology, not quantum chaos, Phys. Scr.40, 335. Zbl1063.81572MR1014859DOI10.1088/0031-8949/40/3/013
6. BIRKHOFF, G., 1927, Dynamical Systems. American Mathematical Society. Providence. Zbl53.0732.01MR209095
7. BOCCALETTI, D., and PUCACCO, G., 1998Theory of Orbits Vol II. Perturbative and Geometric Methods, Springer, Berlin. Zbl0927.70002MR1391960DOI10.1007/978-3-662-03319-7
8. BORN, M., 1935, Atomic Physics, Blackie & Son, Glasgow. 
9. BORN, M., and JORDAN, P., 1925, Zur Quantenmechanik, Zeitschrift für Physik34, 858; Reprinted in translation in B. L. van der Waerden (Ed.), Sources of Quantum Mechanics, North-Holland, Amsterdam, 1967. 
10. BORN, M., HEISENBERG, W., and JORDAN, P., 1926, Zur Quantenmechanik II, Zeitschrift für Physik35, 557; Reprinted in translation in B. L. van der Waerden (Ed.), Sources of Quantum Mechanics, North-Holland, Amsterdam, 1967. 
11. BUCHER, M., 2008, The rise and premature fall of the old quantum theory, ArXiv. 0802.1366. 
12. CARATI, A., and GALGANI, L., Fondamenti della meccanica quantistica: uno studio storico critico. electronic page http:/www.mat.unimi.it/users/galgani 
13. CHRISTODOULIDI, H., and EFTHYMIOPOULOS, C., 2013, Low-dimensional q-Tori in FPU Lattices: Dynamics and Localization Properties, Physica D261, 92. Zbl1336.37058MR3144013DOI10.1016/j.physd.2013.07.007
14. CONTOPOULOS, G., 1960, A third integral of motion in a galaxy, Z. Astrophys., 49, 273. Zbl0099.45007MR114635
15. CONTOPOULOS, G., 1966, Tables of the third integral, Astrophys. J. Suppl., 13, 503. 
16. CONTOPOULOS, G., 2002, Order and Chaos in Dynamical Astronomy, Springer, Berlin. Zbl1041.85001MR1988785DOI10.1007/978-3-662-04917-4
17. CREHAN, P,1990, The proper quantum analogue of the Birkhoff-Gustavson method of normal forms, J. Phys. A Math. Gen.23, 5815. Zbl0728.58033MR1090015
18. DEPRIT, A., 1969, Canonical Transformations Depending on a Small Parameter, Cel. Mech., 1, 12. Zbl0172.26002MR253611DOI10.1007/BF01230629
19. DIRAC, P.A.M., 1926, On the theory of Quantum Mechanics, Proc. Roy. Soc. A109, 642. Zbl52.0975.01MR23198
20. DIRAC, P.A.M., 1930, The Principles of Quantum Mechanics, Oxford Science Publications. Zbl56.0745.05MR23198
21. DYSON, F.J., 1949, The Radiation Theories of Tomonaga, Schwinger, and Feynman, Phys. Rev.75, 486. Zbl0032.23702MR28203
22. DYSON, F.J., 1952, Divergence of Perturbation Theory in Quantum Electrodynamics, Phys. Rev.85, 631. Zbl0046.21501MR46927
23. Attempts to make quantum electrodynamics into a completely solvable theoryhttp://www.webofstories.com/play/freeman.dyson/92 
24. ECKHARDT, B., 1986, Birkhoff-Gustavson normal form in classical and quantum mechanics, J. Phys. A Math. Gen.19, 2961. Zbl0622.70015MR861216
25. EFTHYMIOPOULOS, C., 2012, Canonical perturbation theory; stability and diffusion in Hamiltonian systems: applications in dynamical astronomy, in P.M. Cincotta, C.M. Giordano, and C. Efthymiopoulos (Eds), Third La Plata International School on Astronomy and Geophysics, Asociación Argentina de Astronomía, Workshop Series, Vol. 3, p.3-146. 
26. EFTHYMIOPOULOS, C., and CONTOPOULOS, G., 2006, Chaos in Bohmian Quantum Mechanics, J. Phys. A. Math Gen39, 1819. Zbl1093.81032MR2209303DOI10.1088/0305-4470/39/8/004
27. ELIASSON, L., 1996, Absolutely Convergent Series Expansions for Quasi Periodic Motions, Math. Phys. Electron. J.2, 1. Zbl0896.34035MR1399458
28. GIORGILLI, A,1998, Small denominators and exponential stability: from Poincaré to the present time, Rend. Sem. Mat. Fis. Milano, LXVIII, 19. Zbl1024.37039MR1927002DOI10.1007/BF02925829
29. GUSTAVSON, F.G., 1966, On Constructing Formal Integrals of a Hamiltonian System. Near an Equilibrium Point, Astron. J.71, 670. 
30. GUTZWILLER, M.C., 1990, Chaos in Classical and Quantum Mechanics, Springer, New York. Zbl0727.70029MR1077246DOI10.1007/978-1-4612-0983-6
31. HAMILTON, W.R., 1834, On a general method of dynamics; by which the study of the motions of all free systems of attracting or repelling points is reduced to the search and differentiation of one central relation, or characteristic function, Phil. Transactions, Part 2, pp. 247-308 (Mathematical Papers 2, 103). 
32. HEISENBERG, W., 1925, Über quantentheoretische Umdeutung kinematischer und mechanischer Beziehungen, Zeitschrift für Physik33, 879. 
33. HÉNON, M., and HEILES, C., 1964, The applicability of the third integral of motion: Some numerical experiments, Astron. J.69, 73. MR158746DOI10.1086/109234
34. HORI, G.I., 1966, Theory of General Perturbation with Unspecified Canonical Variables, Publ. Astron. Soc. Japan18, 287. 
35. JACKSON, J.D., 1962, Classical Electrodynamics, John Wiley and Son. MR436782
36. KOLMOGOROV, A.N., 1954, On the preservation of quasi-periodic motions under a small variation of Hamilton's function, Dokl. Akad. Nauk SSSR, 98, 527. MR68687
37. LICHTENBERG, A.J., and LIEBERMAN, M.A., 1992, Regular and Chaotic Dynamics, Springer, New York. Zbl0748.70001MR1169466DOI10.1007/978-1-4757-2184-3
38. LYNDEN-BELL, D., private communication. 
39. MESSIAH, A., 1961, Quantum Mechanics, Dover Publications. 
40. MOSER, J., 1962, On Invariant Curves of Area-Preserving Mappings of an Annulus, Nachr. Akad. Wiss. Göttingen, Math. Phys. Kl II, 1-20. Zbl0107.29301MR147741
41. NIKOLAEV, A.S., 1996, On the diagonalization of quantum Birkhoff-Gustavson normal form, J. Math. Phys.37, 2643. Zbl0885.47031MR1390227DOI10.1063/1.531534
42. POINCARÉ, H., 1892, Les Méthodes Nouvelles de la Mécanique Céleste, Gauthier-Villars, Paris. MR926907
43. ROBNIK, M., 1984, The algebraic quantisation of the Birkhoff-Gustavson normal form, J. Phys. A Math. Gen.17, 109. Zbl0548.70009MR734112
44. SCHRÖDINGER, E., 1926, An Undulatory Theory of the Mechanics of Atoms and Molecules, Phys. Rev.28, 1049. 
45. SCHRÖDINGER, E., 1926, Über das Verhältnis der Heisenberg Born Jordanischen Quantenmechanik zu der meinen, Annalen der Physik79, 45. 
46. VON ZEIPEL, H., 1916, Recherches sur le mouvement de petites planètes, Arkiv. Mat. Astron. Physik, II, Nos, 1,3,7,9, 12. Zbl46.1379.01
47. WEINBERG, S., 1992, Dreams of a Final Theory, Pantheon Books, New York. 
48. WOOD, W.R., and ALI, M.K., 1987, On the limitations of the Birkhoff-Gustavson normal form approach, J. Phys. A Math. Gen.20, 351. MR874256