Nineteen fifty-four: Kolmogorov’s new “metrical approach” to Hamiltonian dynamics. (English) Zbl 07920602
Summary: We review Kolmogorov’s 1954 fundamental paper On the persistence of conditionally periodic motions under a small change in the Hamilton function (Dokl. akad. nauk SSSR, 1954, vol. 98, pp. 527-530), both from the historical and the mathematical point of view. In particular, we discuss Theorem 2 (which deals with the measure in phase space of persistent tori), the proof of which is not discussed at all by Kolmogorov, notwithstanding its centrality in his program in classical mechanics. In Appendix, an interview (May 28, 2021) to Ya. Sinai on Kolmogorov’s legacy in classical mechanics is reported.
MSC:
| 01A60 | History of mathematics in the 20th century |
| 37J40 | Perturbations of finite-dimensional Hamiltonian systems, normal forms, small divisors, KAM theory, Arnol’d diffusion |
| 37J05 | Relations of dynamical systems with symplectic geometry and topology (MSC2010) |
| 37J25 | Stability problems for finite-dimensional Hamiltonian and Lagrangian systems |
| 70H08 | Nearly integrable Hamiltonian systems, KAM theory |
Keywords:
Kolmogorov’s theorem on invariant tori; KAM theory; history of dynamical systems; small divisors; Hamiltonian systems; perturbation theory; symplectic transformations; nearly-integrable systems; measure of invariant toriReferences:
| [1] | Lindstedt, A., Beitrag zur Integration der Differentialgleichungen der Störungstheorie, Mém. Acad. Sci. St.-Pet., 31, 4, 1-20, 1883 · JFM 15.0983.02 |
| [2] | Poincaré, H., Les méthodes nouvelles de la mécanique céleste: In 3 Vols., 1892, Paris: Gauthier-Villars, Paris |
| [3] | Proceedings of the International Congress of Mathematicians (Cambridge, Mass., 1950): In 2 Vols., Providence, R.I.: AMS, 1952. · Zbl 0049.00102 |
| [4] | Kolmogorov, A. N., On Dynamical Systems with an Integral Invariant on the Torus, Selected Works of A. N. Kolmogorov: Vol. 1. Mathematics and Mechanics, 344-348, 1991, Dordrecht: Kluwer, Dordrecht |
| [5] | Kolmogorov, A. N.; Casati, G.; Ford, J., Preservation of Conditionally Periodic Movements with Small Change in the Hamilton Function, Stochastic Behaviour in Classical and Quantum Hamiltonian Systems, 349-354, 1991, Dordrecht: Kluwer, Dordrecht |
| [6] | Proceedings of the International Congress of Mathematicians (Amsterdam, 1954): Vol. 1, Groningen/Amsterdam: Noordhoff/North-Holland, 1957. · Zbl 1372.00106 |
| [7] | Kolmogorov, A. N., Théorie générale des systèmes dynamiques et mécanique classique, Proc. of the Internat. Congr. of Mathematicians (Amsterdam, 1954): Vol. 1, 355-374, 1991, Dordrecht: Kluwer, Dordrecht |
| [8] | Kolmogorov, A. N., Théorie générale des systèmes dynamiques et mécanique classique, Séminaire Janet. Mécanique analytique et mécanique céleste, 1, 1-20, 1957 |
| [9] | Moser, J. K., Review of “Théorie générale des systèmes dynamiques et mécanique classique” (1957), Math. Rev., Providence,R.I.: AMS, 1959. |
| [10] | Moser, J., A New Technique for the Construction of Solutions of Nonlinear Differential Equations, Proc. Natl. Acad. Sci. USA, 47, 11, 1824-1831, 1961 · Zbl 0104.30503 · doi:10.1073/pnas.47.11.1824 |
| [11] | Moser, J., On Invariant Curves of Area-Preserving Mappings of an Annulus, Nachr. Akad. Wiss. Göttingen. Math.-Phys. Kl. IIa, 1962, 1, 1-20, 1962 · Zbl 0107.29301 |
| [12] | Arnol’d, V. I., Proof of a Theorem of A. N. Kolmogorov on the Invariance of Quasi-Periodic Motions under Small Perturbations of the Hamiltonian, Russian Math. Surveys, 18, 5, 9-36, 1963 · Zbl 0129.16606 · doi:10.1070/RM1963v018n05ABEH004130 |
| [13] | Moser, J., Convergent Series Expansions for Quasi-Periodic Motions, Math. Ann., 169, 1, 136-176, 1976 · Zbl 0149.29903 · doi:10.1007/BF01399536 |
| [14] | Truesdell, C., History of Classical Mechanics: Part 2. The 19th and 20th Centuries, Naturwissenschaften, 63, 119-130, 1976 · doi:10.1007/BF00600486 |
| [15] | Smale, S., The Mathematics of Time: Essays on Dynamical Systems, Economic Processes, and Related Topics, 1980, New York: Springer, New York · Zbl 0451.58001 · doi:10.1007/978-1-4613-8101-3 |
| [16] | Kolmogorov, A. N., Selected Works: Vol. 1. Mathematics and Mechanics, 1991, Dordrecht: Kluwer, Dordrecht · Zbl 0732.01045 |
| [17] | Dahan-Dalmedico, A.; Chabert, J.-L.; Chemla, K., Chaos et déterminisme, 1-416, 2016, Paris: Seuil, Paris |
| [18] | Lehto, O., Mathematics without Borders: A History of the International Mathematical Union, 1998, New York: Springer, New York · Zbl 0889.01021 · doi:10.1007/978-1-4612-0613-2 |
| [19] | Arnold, V. I.; Khesin, B. A.; Tabachnikov, S. L., From Hilbert’s Superposition Problem to Dynamical Systems, The Arnoldfest: Proceedings of a Conference in Honour of V. I. Arnold for His Sixtieth Birthday, 11-29, 2014, Providence, R.I.: AMS, Providence, R.I. |
| [20] | Shiryaev, A. N., Andrei Nikolaevich Kolmogorov (April 25, 1903 to October 20, 1987): A Biographical Sketch of His Life and Creative Paths, Kolmogorov in Perspective, 1-87, 2000, Providence, R.I.: AMS, Providence, R.I. |
| [21] | Sinai, Ya. G., Remembrances of A. N. Kolmogorov, Kolmogorov in Perspective, 117-120, 2000, Providence, R.I.: AMS, Providence, R.I. · Zbl 1062.01017 |
| [22] | Moser, J., Remark on the Paper “On Invariant Curves of Area-Preserving Mappings of an Annulus”, Regul. Chaotic Dyn., 6, 3, 337-338, 2001 · Zbl 0992.37053 · doi:10.1070/RD2001v006n03ABEH000181 |
| [23] | Aubin, D.; Dahan Dalmedico, A., Writing the History of Dynamical Systems and Chaos: Longue durée and Revolution, Disciplines and Cultures, Historia Math., 29, 3, 273-339, 2002 · Zbl 1026.01014 · doi:10.1006/hmat.2002.2351 |
| [24] | Broer, H. W., KAM Theory: The Legacy of A. N. Kolmogorov’s 1954 Paper, Bull. Amer. Math. Soc. (N. S.), 41, 4, 507-521, 2004 · Zbl 1050.37030 · doi:10.1090/S0273-0979-04-01009-2 |
| [25] | Grattan-Guinness, I., Landmark Writings in Western Mathematics 1640 - 1940, 2005, Amsterdam: Elsevier, Amsterdam |
| [26] | Chierchia, L., A. N. Kolmogorov’s 1954 Paper on Nearly-Integrable Hamiltonian Systems, Regul. Chaotic Dyn., 13, 2, 130-139, 2008 · Zbl 1229.37077 · doi:10.1134/S1560354708020056 |
| [27] | Curbera, G. P., Mathematicians of the World, Unite!: The International Congress of Mathematicians — A Human Endeavor, 2009, Wellesley, Mass.: Peters/CRC, Wellesley, Mass. · Zbl 1166.01001 · doi:10.1201/b10584 |
| [28] | Demidov, S. S., Les relations mathématiques franco-russes entre les deux guerres mondiales, Rev. Hist. Sci., 62, 1, 119-142, 2009 · Zbl 1220.01006 · doi:10.3917/rhs.621.0119 |
| [29] | Sevryuk, M. B., Translation of the V. I. Arnold Paper “From Superpositions to KAM Theory”, Regul. Chaotic Dyn., 19, 6, 734-744, 2014 · Zbl 1348.70002 · doi:10.1134/S1560354714060100 |
| [30] | Dumas, H. S., The KAM Story: A Friendly Introduction to the Content, History, and Significance of Classical Kolmogorov - Arnol’d - Moser Theory, 2014, Singapore: World Sci., Singapore · Zbl 1362.37004 · doi:10.1142/8955 |
| [31] | Chierchia, L.; Koudjinan, C., V. I. Arnold’s “Pointwise” KAM Theorem, Regul. Chaotic Dyn., 24, 6, 583-606, 2019 · Zbl 1435.37085 · doi:10.1134/S1560354719060017 |
| [32] | Mazliak, L., Andrei Nikolaevitch Kolmogorov’s Visits to France. Resuming the Scientific Relationship between France and Soviet Union after Stalin’s Death, in 26th Internat. Congr. of History of Science and Technology (Prague, Czech Republic, Jul 2021), 13 pp. |
| [33] | Fascitiello, I., Thirty Years After: Insights on the Cultural Origins of Kolmogorov’s 1954 Invariant Tori Theorem from a Short Conversation with Arnold, https://arxiv.org/abs/2212.06030 (2022). |
| [34] | Fascitiello, I., Andrej N. Kolmogorov’s 1954 Theorem on the Persistence of Invariant Tori: A Historical Perspective on Its Cultural Roots and Its Meaning in the History of Classical Mechanics, 2023, Roma: Università Roma Tre, Roma |
| [35] | Biasco, L. and Chierchia, L., Singular KAM Theory, https://arxiv.org/abs/2309.17041 (2023). |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
