Euler and mathematical methods in mechanics (on the 300th anniversary of the birth of Leonhard Euler)

V. V. Kozlov

Trudy Matematicheskogo Instituta imeni V.A. Steklova, Tome 62 (2007) no. 4, pp. 639-661

Résumé

This article concerns the life of Leonhard Euler and his achievements in theoretical mechanics. A number of topics are discussed related to the development of Euler's ideas and methods: divergent series and asymptotics of solutions of non-linear differential equations; the hydrodynamics of a perfect fluid and Hamiltonian systems; vortex theory for systems on Lie groups with left-invariant kinetic energy; energy criteria of stability; Euler's problem of two gravitating centres in curved spaces.

Bibliographie

[1] A. N. Krylov, “Leonard Eiler”, Sobranie trudov akademika A. N. Krylova. T. 1. Ch. 2. Nauchno-populyarnye stati. Biograficheskie kharakteristiki, Izd-vo AN SSSR, M.–L., 1951, 192–217

[2] G. Khardi, Raskhodyaschiesya ryady, IL, M., 1951 ; G. H. Hardy, Divergent series, Clarendon, Oxford, 1949 | MR | MR | Zbl

[3] A. N. Kuznetsov, “Differentsiruemye resheniya vyrozhdayuschikhsya sistem obyknovennykh uravnenii”, Funkts. analiz i ego prilozh., 6:2 (1972), 41–51 | MR | Zbl

[4] A. N. Kuznetsov, “O suschestvovanii vkhodyaschikh v osobuyu tochku reshenii avtonomnoi sistemy, obladayuschei formalnym resheniem”, Funkts. analiz i ego prilozh., 23:4 (1989), 63–74 | MR | Zbl

[5] Zh.-P. Ramis, Raskhodyaschiesya ryady i asimptoticheskaya teoriya, In-t kompyuternykh issledovanii, Moskva–Izhevsk, 2002; J.-P. Ramis, Séries divergentes et théories asymptotiques, Panoramas et Synthèses, 121, Soc. Math. France, Paris, 1993 | MR | Zbl

[6] V. V. Kozlov, “Asimptoticheskie dvizheniya i problema obrascheniya teoremy Lagranzha–Dirikhle”, PMM, 50:6 (1986), 928–937 | MR | Zbl

[7] V. V. Kozlov, V. P. Palamodov, “Ob asimptoticheskikh resheniyakh uravnenii klassicheskoi mekhaniki”, Dokl. AN SSSR, 263:2 (1982), 285–289 | MR | Zbl

[8] V. V. Kozlov, S. D. Furta, Asimptotiki reshenii silno nelineinykh sistem differentsialnykh uravnenii, Izd-vo MGU, M., 1996 | MR | Zbl

[9] K. G. Ya. Yakobi, Lektsii po analiticheskoi mekhanike, In-t kompyuternykh issledovanii, Moskva–Izhevsk, 2006; C. G. J. Jacobi, Vorlesungen über analytische Mechanik, Dokumente Gesch. Math., 8, Deutsche Mathematiker Vereinigung, Freiburg, 1996 | MR | Zbl

[10] V. I. Arnold, “O topologii trekhmernykh statsionarnykh techenii idealnoi zhidkosti”, PMM, 30:1 (1966), 183–185 | MR | Zbl

[11] V. V. Kozlov, “Zamechaniya o statsionarnykh dvizheniyakh sploshnoi sredy”, PMM, 47:2 (1983), 341–342 | MR | Zbl

[12] S. V. Bolotin, “Neintegriruemost zadachi $n$ tsentrov pri $n>2$”, Vestn. Mosk. un-ta. Ser. 1. Matem., mekh., 1984, no. 3, 65–68 | MR | Zbl

[13] V. V. Kozlov, “Gidrodinamika gamiltonovykh sistem”, Vestn. Mosk. un-ta. Ser. 1. Matem., mekh., 1983, no. 6, 10–22 | MR | Zbl

[14] V. V. Kozlov, “Ob odnom obobschenii metoda Gamiltona–Yakobi”, PMM, 60:6 (1996), 929–939 | MR | Zbl

[15] I. S. Arzhanykh, Pole impulsov, Nauka, Tashkent, 1965 ; I. S. Arzhanykh, Momentum fields, Nat. Lending Lib., Boston Spa, Yorkshire, 1971 | MR | Zbl | Zbl

[16] V. V. Kozlov, Obschaya teoriya vikhrei, Udmurt. un-t, Izhevsk, 1998 ; V. V. Kozlov, Dynamical systems. X. General theory of vortices, Encyclopaedia Math. Sci., 67, Springer-Verlag, Berlin, 2003 | MR | Zbl | MR | Zbl

[17] V. V. Kozlov, “Vikhrevaya teoriya volchka”, Vestn. Mosk. un-ta. Ser. 1. Matem., mekh., 1990, no. 4, 56–62 | MR | Zbl

[18] H. Poincaré, “Sur une forme nouvelle des équations de la méchanique”, C. R. Acad. Sci. Paris, 132 (1901), 369–371 | Zbl

[19] V. V. Kozlov, V. A. Yaroschuk, “Ob invariantnykh merakh uravnenii Eilera–Puankare na unimodulyarnykh gruppakh”, Vestn. Mosk. un-ta. Ser. 1. Matem., mekh., 1993, no. 2, 91–95 | MR | Zbl

[20] J.-J. Moreau, “Une méthode de “cinématique fonctionnelle” en hydrodynamique”, C. R. Acad. Sci. Paris, 249 (1959), 2156–2158 | MR | Zbl

[21] V. I. Yudovich, “Ploskie nestatsionarnye dvizheniya idealnoi neszhimaemoi zhidkosti”, Dokl. AN SSSR, 136:3 (1961), 564–567 | Zbl

[22] V. I. Arnol'd, “Sur un principe variationnel pour les écoulements stationnaires des liquides parfaits et ses applications aux problèmes de stabilité non linéaires”, J. Mécanique, 5:1 (1966), 29–43 | Zbl

[23] M. V. Deryabin, Yu. N. Fedorov, “O reduktsiyakh na gruppu geodezicheskikh potokov s (levo-) pravoinvariantnoi metrikoi i ikh polyakh simmetrii”, Dokl. RAN, 391:4 (2003), 439–442 | MR | Zbl

[24] V. V. Kozlov, “Lineinye sistemy s kvadratichnym integralom”, PMM, 56:6 (1992), 900–906 | MR | Zbl

[25] V. V. Kozlov, “O stepeni neustoichivosti”, PMM, 57:5 (1993), 14–19 | MR | Zbl

[26] A. Ostrowski, H. Schneider, “Some theorems on the inertia of general matrices”, J. Math. Anal. Appl., 4:1 (1962), 72–84 | DOI | MR | Zbl

[27] V. V. Kozlov, “Zamechaniya o sobstvennykh chislakh veschestvennykh matrits”, Dokl. RAN, 403:5 (2005), 589–592 | MR | Zbl

[28] V. V. Kozlov, A. A. Karapetyan, “O stepeni ustoichivosti”, Differents. uravneniya, 41:2 (2005), 186–192 | MR | Zbl

[29] H. K. Wimmer, “Inertia theorems for matricies, controllability, and linear vibrations”, Linear Algebra Appl., 8:4 (1974), 337–343 | DOI | MR | Zbl

[30] P. Lancaster, M. Tismenetsky, “Inertia characteristics of selfajoint matrix polynomials”, Linear Algebra Appl., 52–53 (1983), 479–496 | DOI | MR | Zbl

[31] A. A. Shkalikov, “Operator pencils arising in elasticity and hydrodynamics: the instability index formula”, Recent developments in operator theory and its applications (Winnipeg, Canada, 1994), Oper. Theory Adv. Appl., 87, Birkhäuser, Basel, 1996, 358–385 | MR | Zbl

[32] A. V. Borisov, I. S. Mamaev (red.), Klassicheskaya dinamika v neevklidovykh prostranstvakh, In-t kompyuternykh issledovanii, Moskva–Izhevsk, 2004

[33] V. V. Kozlov, A. O. Harin, “Kepler's problem in constant curvature spaces”, Celestial Mech. Dynam. Astronom., 54:4 (1992), 393–399 | DOI | MR | Zbl