Calculus of variations in the large: birth, formation, applications

A. V. Boeva ; R. R. Mukhin

Čebyševskij sbornik, Tome 24 (2023) no. 3, pp. 263-288.

Résumé

The work is devoted to the evolution of the concepts and methods of the calculus of variations in the large, a branch of mathematics that is a little over a century old. The subject of this area is the study of qualitative characteristics of variational problems. In the development of the calculus of variations in the large several periods can be distinguished with features inherent in each of them. The first period is defined from the end of the 19th century. until the end of the 1940s, when the theory was born and formed, which was formed from two main parts - the Morse theory and the theory of Lyusternik-Shnirelman categories. Here, the features of traditional mathematics are still noticeable. In the next period - from the end of the 1940s to the end of the 1970s. the calculus of variation in the large was formed as a separate area of mathematics, and it acquired its modern form, based on the concepts and methods of algebraic topology. Ample opportunities opened up for solving new problems in mathematics, and a number of impressive results were obtained. The modern period can be defined from the late 1970s. until now. Its main feature is the unprecedented convergence of mathematics and the field of its applications, especially with physics. It has not always been possible to indicate a distinguishable boundary between the two fields of science; even the term "physical mathematics" has appeared. The calculus of variations in the large is included in the qualitative theory, which represents a significant part of modern mathematics and finds wide applications. But its place is even more significant, it is one of the foundations that forms our worldview.

Détail

Keywords: critical point, extremal, geodesic line, topology, category.

Bibliographie

[1] Campbell L., Garnett W., The life of James Clerk Maxwell, MacMillan and Co, L., 1882, 342 pp.

[2] Maxwell J.C., “On the transformations of surfaces by bending”, The scientific papers of James Clerk Maxwell, v. 1, ed. W.D. Niven, Dover Publ., N.Y., 1965, 80–114 | MR

[3] Maxwell J.C., “Faraday”, The scientific papers of James Clerk Maxwell, v. 2, ed. W.D. Niven, Dover Publ., N.Y., 1965, 355–360 | MR

[4] Maxwell J.C., “On Hills and Dales”, The scientific papers of James Clerk Maxwell, v. 2, ed. W.D. Niven, Dover Publ., N.Y., 1965, 233–240 | MR

[5] Cayley A., “On Contour and Slope Lines”, Phil. Mag., XVIII (1859), 264–268 | DOI

[6] Mukhin R.R., “On the Poincaré-Birkhoff theorem as the most important result of the theory of dynamical systems”, Chebyshev collection, 23:1 (2022), 209–222 (in Russian) | DOI | MR | Zbl

[7] Poincaré H., “Sur les lignes géodésiques des surfaces convexes”, Trans. AMS, 6, 237–274 | MR

[8] Jacobi C.G., “Note von der geodätischen Linie auf einem Ellipsoid und der verschiedenen Anwendungen einer merkwürdigen Substitution”, Crelles J., 19 (1839), 309–313 | MR

[9] Darboux G., Leçon sur sur la théorie des surfaces, v. 3, Gauthier-Villars, Paris, 1894 | MR

[10] Hadamard J., “Les surfaces à courbures opposeés et leurs lignes géodésiques”, J. Math. pures et appl., 4 (1898), 27–73

[11] “Briefe Jacobi's an Bessel”, C.G.J. Jacobi's Gesammelte Werke, v. 7, Druck und Verlag von Georg Reiner, Berlin, 1891, 377–386

[12] Poincaré H., “Memoire sur les courbes définies par une équations differentielle”, J. math. pures et appl. Sér. 3, 7 (1881), 375–422 ; 8 (1882), 251-296; J. math. pures et appl. Sér. 4, 1 (1885), 167–244; 2 (1886), 151–217; Puankare A., O krivykh, opredelyaemykh differentsialnymi uravneniyami, GITTL, M., 1947, 392 pp. | MR

[13] Morse M., The calculus of variations in the large, AMS, N.Y., 1934, 360 pp. | MR | Zbl

[14] Pogrebyssky I.B., “On geodesic lines on convex surfaces”, Henri Poincaré. Collect. works, v. 2, Nauka, M., 1972, 982–983 (in Russian)

[15] Birkhoff G.D., “An extension of Poincare's last geometric theorem”, Acta. Math., 47 (1926), 297–311 | DOI | MR

[16] Birkhoff G.D., “Dynamical systems with two degrees of freedom”, Trans. AMS, 18 (1917), 199–300 ; G.D. Birkhoff, Coll. math. papers, v. 2, AMS, N.Y., 1950, 1–102 | DOI | MR

[17] Birkhoff G.D., Dynamical Systems, Providence, Rhode Island, 1927, ix+295 pp. ; Dzh. Birkgof, Dinamicheskie sistemy, Per. s angl., RKhD, Izhevsk, 1999, 408 pp. | MR

[18] Morse M., “Relations between the critical points of a real functiond of n independent variables”, Trans. AMS, 22 (1925), 84–110 | DOI | MR

[19] Morse M., “The foundations of a theory of the calculus of variations in the large”, Trans. AMS, 30 (1928), 213–274 | DOI | MR

[20] Lefschetz S., “Continuous transformations of manifolds”, Proc. National Acad. Sci., 11 (1925), 290–292 | DOI

[21] Hopf H., “Vectorfelder in n-dimensionalen Mannigfoltigkeiten”, Math. Ann., 96 (1926), 225–251 | DOI | MR

[22] Morse M., “The calculus of variations in the large”, Mat. Congr. Zurick, 1932, 173–188 | MR | Zbl

[23] Lusternik L., “Sur quelque méthodes topologiques dans le géométrie differentielle”, Acti. Congr. Inter. Mat. Bologna, 4 (1928), 291–296

[24] Lusternik L., Schnirelmann L., “Sur un principe topologique en analyse”, Comp. Ren., 188 (1929), 295–298

[25] Lusternik L., Schnirelmann L., “Existence des trois géodésiques fermées sur tout surfaces de genre 0”, Comp. Ren., 188 (1929), 269–271

[26] Lyusternik L.A., Shnirelman L.G., Topological methods in variational problems, Gosizdat, M., 1930, 68 pp. (in Russian)

[27] Lyusternik L.A., Shnirelman L.G., “Topological methods in variational problems and their applications to differential geometry of surfaces”, Russian Mat. Survey, 2:1(17) (1947), 166–217 (in Russian) | MR | Zbl

[28] Shvarts A.S., “Topology of spaces of closed curves”, Proc. Moscow Math. Soc., 9, 1960, 3–44 (in Russian) | Zbl

[29] Klein F., Elementarmathematik vom hoheren Standpunkte aus, v. I, Springer, Berlin, 1968, 614 pp. | MR | MR

[30] Thom R., “Sur une partition en cellules associée a une function sur une variété”, Comp. Ren., 228 (1949), 973–975 | MR | Zbl

[31] Tu L.V., “The life and works of Raoul Bott”, The founders of index theory, ed. S.-T. Yau, Int. Press, Sommerville, MA, 2003, 85–112 | MR

[32] Bott R., “Morse theory indominable”, Publ. Math. IHES, 68 (1988), 99–114 | DOI | MR | Zbl

[33] Milnor J., Morse theory, Princeton Univ.Press, Princeton, N.J., 1963, 154 pp. | MR | Zbl

[34] Bott R., “An application of the Morse theory to the topology of Lie-groops”, Bull. SMF, 84 (1956), 251–281 | MR | Zbl

[35] Bott R., “The stable homotopy of the classical groops”, Ann. Math., 70:2 (1959), 313–337 | DOI | MR | Zbl

[36] Atyah M., “Obituary Raoul Bott”, Bull. London Math. Soc., 42 (2010), 170–180 | DOI | MR

[37] Andronov A.A., Pontryagin L.S., “Rough systems”, Rep. Acad. Sci. USSR, 14 (1937), 247–250 (in Russian) | MR | Zbl

[38] Smale S., “Finding a Horseshoe on the Beaches of Rio”, Chaos Avant-Gard, World Sci, Singapore, 2000, 7–22 | MR | Zbl

[39] Smale S., “Morse inequalities for the dynamical system”, Bull. AMS, 66 (1960), 43–49 ; Smeil S., “Neravenstva Morsa dlya dinamicheskikh sistem”, Matematika, 11:4 (1967), 79–87 | DOI | MR | Zbl

[40] Andronov A.A., “Mathematical problems in self-oscillation theory”, A.A. Andronov. Coll. Works, Acad. Sci. USSR, M., 1956, 32–71 (in Russian)

[41] Smale S., “Generalized Poincare conjecture in a higher dimensions”, Bull. AMS, 66 (1960), 373–375 | DOI | MR | Zbl

[42] Smale S., “Generalized Poincare conjecture in dimensions greater than four”, Ann. Math., 74 (1961), 391–406 | DOI | MR | Zbl

[43] Palis J., “On Morse-Smale dynamical systems”, Topology, 8:4 (1969), 385–404 | DOI | MR

[44] Palis J., Smale S., “Structural stability theorems”, Global Analysis, Proc. Simp. Pure Math., 14, Providence, RI, 1970, 223–231 | DOI | MR | Zbl

[45] Smale S., “On gradient dynamical systems”, Ann. Math., 74:1 (1961), 199–206 | DOI | MR | Zbl

[46] Smale S., “A structurally stable differential homomorphysm with an infinite number of periodic points”, Proc. Int. Symp. Nonlin. Oscillations (Kyiv, 1961), AN Ukranian SSR, Kyiv, 1963, 365–366 | MR

[47] Smale S., “A structurally stable systems are not dense”, Am. J. Math., 88:2 (1966), 491–496 ; Smeil S., “Grubye sistemy ne plotny”, Matematika, 11:4 (1967), 107–112 | DOI | MR | Zbl

[48] Dirac P.A. M., “Quantized singularities in the electromagnetic field”, Proc. Roy. Soc., 31 (1931), 60–72

[49] Smale S., “Topology and mechanics”, Invent. Math., 1970, no. 10, 305–351 | DOI | MR | MR

[50] Palmore J.L., “Classifying relative equilibria”, Bull. AMS, 79 (1973), 904–908 ; 81 (1975), 488-491 | DOI | MR | Zbl | DOI

[51] Kozlov V.V., “Calculus of variations in the large and classical mechanics”, Russian Mat. Survey, 40:2(242) (1985), 33–60 (in Russian) | MR | Zbl

[52] Kozlov V.V., “Integrability and non-integrability in Hamiltonian mechanics”, Russian Mat. Survey, 38:1 (1983), 3–67 (in Russian) | MR | Zbl

[53] Kozlov V.V., “Topological obstacles to the integrability of natural mechanical systems”, Rep. Acad. Sci. USSR, 249:6 (1979), 1299–1302 (in Russian) | MR | Zbl

[54] Arnold V.I., Mathematical methods of classical mechanics, Springer-Verlag, N.Y., 1989, 536 pp. | MR | MR

[55] Novikov S.P., “Multivalued functions and functionals. An analogue of the Morse theory”, Rep. Acad. Sci. USSR, 260:1 (1981), 31–34 (in Russian) | MR

[56] Novikov S.P., Shmeltser I., “Periodic Solutions of the Kirchhoff Equations for Free Motion of a Rigid Body in a Fluid and the Extended Lyusternik-Shnirelman-Morse (LSM) Theory”, Funct. analysis and its applications, 15:3 (1981), 54–66 (in Russian) | MR

[57] Novikov S.P., “Variational Methods and Periodic Solutions of Kirchhoff Type Equations”, Funct. analysis and its applications, 15:4 (1981), 37–52 (in Russian) | MR

[58] Kirchhoff G., Vorlesungen über mechanic, B. G. Teubner, Leipzig, 1897, 475 pp.

[59] Borisov A.V., Mamaev I.S., Poisson structures and Lie algebras in Hamiltonian mechanics, Ed. House “Udmurt. Univ.”, Izhevsk, 1999, 464 pp. (in Russian) | MR

[60] Novikov S.P., “Hamiltonian formalism and a many-valued analogue of Morse theory”, Russian Mat. Survey, 37:5(227) (1982), 3–49 (in Russian) | DOI | MR | Zbl

[61] Michel L., Mozrzymas B., “Application of Morse theory to the symmetry breaking in the Landau theory of second order phase transition”, Group theoretical methods in physics, Lecture notes in physics, Springer-Verlag, Berlin, 1978, 447–461 | DOI

[62] Belavin A.A., Polyakov A.M., Schwarz A.S., Tyupkin Y.S., “Pseudopartical solutions of the Yang-Mills equations”, Phys. Lett., 59 (1975), 85–87 | DOI | MR

[63] Polyakov A. M., “The spectrum of particles in quantum field theory”, JETP Letters, 20:6 (1974), 430–433 (in Russian)

[64] Atyah M., “Geometry of Yang-Mills fields. Fermi lecture”, Michel Atyah. Collected works, v. 5, Clarendon Press, Oxford, 1988, 75–174 | MR

[65] Atyah M., Bott R., “The Yang-Mills equations over Riemann surfaces”, Phil. Trans. Soc. Lond., A 308 (1982), 523–615 | MR

[66] Witten E., “Supersymmetry and MORSE Theory”, J. Dif. Geometry, 17:4 (1982), 661–692 | MR | Zbl

[67] Donaldson S., “Self-dual connections and the topology of smooth 4-manifolds”, Bull. AMS, 8 (1983), 81–83 | DOI | MR | Zbl

[68] Donaldson S., Kronheimer P., The Geometry of Four-Manifolds, OUP, Oxford, 1990, 440 pp. | MR

[69] Bourbaki N., Elements of the history of mathematics, Springer-Verlag, N.Y., 1994, 301 pp. | MR | Zbl

[70] Alvarez-Gaume L., “Supersymmetry and the Atiyah-Singer Index Theorem”, Comm. Math. Phys., 90 (1983), 161–173 | DOI | MR | Zbl

[71] Monastyrsky M.I., Modern Mathematics in the Light of the Fields Medals, Janus-K, M., 2000, 200 pp. (in Russian)