×

zbMATH — the first resource for mathematics

Historical and critical review of the development of nonholonomic mechanics: the classical period. (English, Russian) Zbl 1352.37172
Regul. Chaotic Dyn. 21, No. 4, 455-476 (2016); translation in Nelineĭn. Din. 12, No. 3, 385-411 (2016).
Summary: In this historical review we describe in detail the main stages of the development of nonholonomic mechanics starting from the work of Earnshaw and Ferrers to the monograph of Yu. I. Neimark and N. A. Fufaev. In the appendix to this review we discuss the d’Alembert-Lagrange principle in nonholonomic mechanics and permutation relations.

MSC:
37J60 Nonholonomic dynamical systems
01A05 General histories, source books
70F25 Nonholonomic systems related to the dynamics of a system of particles
70H45 Constrained dynamics, Dirac’s theory of constraints
PDF BibTeX XML Cite
Full Text: DOI MNR
References:
[1] Albouy, A., There Is a projective dynamics, Eur. Math. Soc. Newsl., 89, 37-43, (2013) · Zbl 1364.37169
[2] Appell, P., Traité de mécanique rationelle: Vol. 2, Paris: Gauthier-Villars, 1896.
[3] Appell, P., Exemple de mouvement d’un assujetti, a une exprimée par une relation non linéaire entre LES composantes de la vitesse, Rendiconti del circolo matematico di Palermo, 32, 48-50, (1911) · JFM 42.0756.01
[4] Arnol’d, V. I.; Givental’, A. B.; Arnold, V. I. (ed.); Novikov, S.P. (ed.), Symplectic geometry, 1-138, (2001) · Zbl 0780.58016
[5] Arnol’d, V. I., Kozlov, V.V., and Neishtadt, A. I., Mathematical Aspects of Classical and Celestial Mechanics, 3rd ed., Encyclopaedia Math. Sci., vol. 3, Berlin: Springer, 2006. · Zbl 1105.70002
[6] Béghin, H., Sur LES conditions d’application des équations de Lagrange à un système non holonome, Bull. Soc. Math. France, 57, 118-124, (1929) · JFM 55.1094.04
[7] Bilimovitch, A.D., La pendule nonholonome, Mat. Sb., 29, 234-240, (1914)
[8] Bizyaev, I.A.; Borisov, A.V.; Kazakov, A.O., Dynamics of the Suslov problem in a gravitational field: reversal and strange attractors, Regul. Chaotic Dyn., 20, 605-626, (2015) · Zbl 1344.37073
[9] Bizyaev, I.A., Borisov, A.V., and Mamaev, I. S., The Hojman Construction and Hamiltonization of Nonholonomic Systems, SIGMA Symmetry Integrability Geom. Methods Appl., 2016, vol. 12, Paper 012, 19 pp. · Zbl 1331.37087
[10] Blackall, C. J., On volume integral invariants of non-holonomic dynamical systems, Amer. J. Math., 63, 155-168, (1941) · JFM 67.0770.01
[11] Bloch, A., Nonholonomic Mechanics and Control, New York: Springer, 2003. · Zbl 1045.70001
[12] Borisov, A.V.; Kazakov, A.O.; Sataev, I.R., The reversal and chaotic attractor in the nonholonomic model of chaplygin’s top, Regul. Chaotic Dyn., 19, 718-733, (2014) · Zbl 1358.70006
[13] Borisov, A. V.; Kilin, A.A.; Mamaev, I. S., On the Hadamard-Hamel problem and the dynamics of wheeled vehicles, Regul. Chaotic Dyn., 20, 752-766, (2015) · Zbl 1367.70034
[14] Borisov, A. V.; Mamaev, I. S., The rolling motion of a rigid body on a plane and a sphere: hierarchy of dynamics, Regul. Chaotic Dyn., 7, 177-200, (2002) · Zbl 1058.70009
[15] Borisov, A. V.; Mamaev, I. S., On the history of the development of the nonholonomic dynamics, Regul. Chaotic Dyn., 7, 43-47, (2002) · Zbl 1011.70002
[16] Borisov, A.V.; Mamaev, I. S., Conservation laws, hierarchy of dynamics and explicit integration of nonholonomic systems, Regul. Chaotic Dyn., 13, 443-490, (2008) · Zbl 1229.70038
[17] Borisov, A. V.; Mamaev, I. S., Symmetries and reduction in nonholonomic mechanics, Regul. Chaotic Dyn., 20, 553-604, (2015) · Zbl 1342.37001
[18] Borisov, A. V.; Mamaev, I. S.; Bizyaev, I. A., The hierarchy of dynamics of a rigid body rolling without slipping and spinning on a plane and a sphere, Regul. Chaotic Dyn., 18, 277-328, (2013) · Zbl 1367.70007
[19] Borisov, A. V.; Mamaev, I. S.; Bizyaev, I. A., The Jacobi integral in nonholonomic mechanics, Regul. Chaotic Dyn., 20, 383-400, (2015) · Zbl 1367.70036
[20] Borisov, A. V.; Mamaev, I. S.; Kilin, A.A., Dynamics of rolling disk, Regul. Chaotic Dyn., 8, 201-212, (2003) · Zbl 1112.37320
[21] Borisov, A. V.; Mamaev, I. S.; Kilin, A.A.; Bizyaev, I.A., Qualitative analysis of the dynamics of a wheeled vehicle, Regul. Chaotic Dyn., 20, 739-751, (2015) · Zbl 1367.70035
[22] Bottema, O., On the small vibrations of nonholonomic systems, Indag. Math., 11, 296-298, (1949)
[23] Bottema, O., Die bewegung eines einfachen wagenmodells, Z. Angew. Math. Mech., 44, 585-593, (1964) · Zbl 0152.42505
[24] Bourlet, C., Étude théorique sur la bicyclette: 1, Bull. Soc. Math. France, 27, 47-67, (1899) · JFM 30.0668.02
[25] Boussinesq, J., Aperçu sur la théorie de la bicyclette, J. Math. Pure Appl., 5, 117-135, (1899) · JFM 30.0669.02
[26] Boussinesq, J., Complément à une étude récente concernant la théorie de la bicyclette: influence, sur l’équilibre, des mouvements latéraux spontanés du cavalier, J. Math. Pure Appl., 5, 217-232, (1899) · JFM 30.0670.03
[27] Bravo-Doddoli, A.; García-Naranjo, L.C., The dynamics of an articulated n-trailer vehicle, Regul. Chaotic Dyn., 20, 497-517, (2015) · Zbl 1342.37064
[28] Brendelev, V. N., On commutation relations in nonholonomic mechanics, Mosc. Univ. Mech. Bull., 33, 30-35, (1978) · Zbl 0421.70027
[29] Brill, A., Vorlesungen zur Einführung in die Mechanik raumerfüllender Massen, Leipzig: Teubner, 1909. · JFM 40.0737.01
[30] Capon, R. S., Hamilton’s principle in relation to nonholonomic mechanical systems, Quart. J. Mech. Appl. Math., 5, 472-480, (1952) · Zbl 0048.17502
[31] Carathéodory, C.; Der, S., No article title, Z. Angew. Math. Mech., 13, 71-76, (1933) · Zbl 0006.37301
[32] Cartan, É., Lessons on Integral Invariants, Paris: Hermann, 1922.
[33] Carvallo, E., Théorie du mouvement du monocycle et de la bicyclette, Paris: Gauthier-Villars, 1899. · JFM 32.0739.01
[34] Chaplygin, S.A., On a motion of a heavy body of revolution on a horizontal plane, Regul. Chaotic Dyn., 7, 119-130, (2002) · Zbl 1058.70002
[35] Chaplygin, S. A., On a ball’s rolling on a horizontal plane, Regul. Chaotic Dyn., 7, 131-148, (2002) · Zbl 1058.70003
[36] Chaplygin, S. A., On the theory of motion of nonholonomic systems. the reducing-multiplier theorem, Regul. Chaotic Dyn., 13, 369-376, (2008) · Zbl 1229.37082
[37] Chetaev, N. G., On the Gauss principle, 323-326, (1932)
[38] Chow, W. L., Über systeme von linearen partiellen differentialgleichungen erster ordnung, Math. Ann., 117, 98-105, (1940) · JFM 65.0398.01
[39] Crescini, E., Sur moto di una sfera che rotola su di un plano fisso, Rendiconti Accad. dei Lincei, 5, 204-209, (1889) · JFM 21.0942.01
[40] Dautheville, S., Sur LES systèmes non holonomes, Bull. Soc. Math. France, 37, 120-132, (1909) · JFM 40.0777.03
[41] Delassus, E., Sur la réalisation matérielle des liaisons, C. R. Acad. Sci. Paris, 152, 1739-1743, (1911) · JFM 42.0756.02
[42] Duistermaat, J. J., Chaplygin’s Sphere, arXiv:math/0409019v1 (2004).
[43] Earnshaw, S., Dynamics, or An Elementary Treatise on Motion, 3rd ed., Cambridge: Deighton, 1844.
[44] Eden, R. J., The Hamiltonian dynamics of non-holonomic systems, Proc. Roy. Soc. London. Ser. A, 205, 564-583, (1951) · Zbl 0042.21701
[45] Efimov, M. I., On caplygin’s equations of nonholonomic mechanical systems, Prikl. Mat. Mekh., 17, 748-750, (1953)
[46] Ehlers, K.; Koiller, J.; Montgomery, R.; Rios, P.M., Nonholonomic systems via moving frames: Cartan equivalence and Chaplygin hamiltonization, The Breadth of Symplectic and Poisson Geometry, 232, 75-120, (2005) · Zbl 1095.70007
[47] Essén, H., On the geometry of nonholonomic dynamics, Trans. ASME J. Appl. Mech., 61, 689-694, (1994) · Zbl 0812.70009
[48] Ferrers, N.M., Extension of lagrange’s equations, Quart. J. Pure Appl. Math., 12, 1-5, (1872) · JFM 04.0457.01
[49] Frobenius, G., Über das pfaffsche problem, J. Reine Angew. Math., 1877, 230-315, (1877) · JFM 09.0249.03
[50] Gibbs, J.W., On the fundamental formulae of dynamics, Amer. J. Math., 2, 49-64, (1879) · JFM 11.0643.01
[51] Hadamard, J., Sur LES mouvements de roulement, Mémoires de la Société des sciences physiques et naturelles de Bordeaux, sér. 4, 5, 397-417, (1895) · JFM 26.0849.02
[52] Halmos, P.R., How towrite mathematical texts, Uspekhi Mat. Nauk, 26, 243-269, (1971) · Zbl 0216.28301
[53] Hamel, G., Die Lagrange-eulerschen gleichungen der mechanik, Z. Math. u. Phys., 50, 1-57, (1904) · JFM 35.0748.08
[54] Hamel, G., Theoretische Mechanik: Eine einheitliche Einführung in die gesamte Mechanik, 2nd ed., Berlin: Springer, 1978. · Zbl 0388.70001
[55] Hawkins, Th., Frobenius, Cartan, and the problem of Pfaff, Arch. Hist. Exact Sci., 59, 381-436, (2005) · Zbl 1078.01013
[56] Hertz, H., Gesammelte Werke: Vol. 3. Die Prinzipien der Mechanik, Leipzig: Barth, 1894. · JFM 67.0970.03
[57] Holm, D.D., Geometric Mechanics: P. 1. Dynamics and Symmetry, 2nd ed., London: Imperial College Press, 2011. · Zbl 1227.70001
[58] Holm, D.D., Geometric Mechanics: P. 2. Rotating, Translating and Rolling, 2nd ed., London: Imperial College Press, 2011. · Zbl 1381.70001
[59] Isénoff, I., Sur LES équations générales du mouvement des systèmes matériels non holonomes, J. Math. Pures Appl., sér. 8, 3, 245-264, (1920) · JFM 47.0719.01
[60] Jacobi, C.G. J., Vorlesungen über Dynamik, 2nd ed., Berlin: Reimer, 1884. · JFM 16.0028.01
[61] Karapetyan, A.V. and Rumyantsev, V.V., Stability of Conservative and Dissipative Systems, Itogi Nauki Tekh. Ser. Obshch. Mekh., vol. 6, Moscow: VINITI, 1983 (Russian). · Zbl 0596.70024
[62] Koiller, J., Reduction of some classical non-holonomic systems with symmetry, Arch. Rational Mech. Anal., 118, 113-148, (1992) · Zbl 0753.70009
[63] Kooijman, J.D.G.; Meijaard, J.P.; Papadopoulos, J.M.; Ruina, A.; Schwab, A. L., A bicycle can be self-stable without gyroscopic or caster effects (supplementary material available online), Science, 332, 339-342, (2011) · Zbl 1226.70003
[64] Korteweg, D., Extrait d’une lettre à M. Appel, Rendiconti del circolo matematico di Palermo, 14, 7-8, (1900) · JFM 31.0693.03
[65] Korteweg, D., Über eine ziemlich verbrietete unrichtige behandlungswiese eines problemes der rolleden bewegung und insbesondere über kleine rollende schwingungen um eine gleichgewichtslage, Nieuw Archief voor Wiskunde, 4, 130-155, (1899) · JFM 30.0639.02
[66] Kozlov, V.V., The dynamics of systems with servoconstaints: 1, Regul. Chaotic Dyn., 20, 205-224, (2015) · Zbl 1353.70036
[67] Kozlov, V.V., The dynamics of systems with servoconstaints: 2, Regul. Chaotic Dyn., 20, 401-427, (2015) · Zbl 1353.70012
[68] Kozlov, V.V., Euler and mathematical methods in mechanics: on the 300th anniversary of the birth of leonhard Euler, Russian Math. Surveys, 62, 639-661, (2007) · Zbl 1140.01017
[69] Kozlov, V.V., On the theory of integration of the equations of nonholonomic mechanics, Regul. Chaotic Dyn., 7, 191-176, (2002) · Zbl 1006.37040
[70] Kozlov, V.V., On equilibria of nonholonomic systems, Mosc. Univ. Mech. Bull., 49, 22-29, (1994) · Zbl 0897.70011
[71] Kozlov, V.V., Stability of equilibria of nonholonomic systems, Sov. Math. Dokl., 33, 654-656, (1986) · Zbl 0622.58019
[72] Kozlov, V.V., Realization of nonintegrable constraints in classical mechanics, Sov. Phys. Dokl., 28, 735-737, (1983) · Zbl 0579.70014
[73] Lagrange, J.L., Méchanique analytique, Sceaux: Gabay, 1989.
[74] Laurent-Gengoux, C., Pichereau, A., and Vanhaecke, P., Poisson Structures, Grundlehren Math. Wiss., vol. 347. Heidelberg: Springer, 2013. · Zbl 1284.53001
[75] León, M., A historical review on nonholomic mechanics, Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Math. RACSAM, 106, 191-224, (2012) · Zbl 1264.37017
[76] Lie, S., Theorie der Transformationsgruppen: Vol. 1, Leipzig: Teubner, 1888. · JFM 21.0356.02
[77] Lie, S., Theorie der Transformationsgruppen: Vol. 2, Leipzig: Teubner, 1890. · JFM 22.0372.01
[78] Lie, S., Theorie der Transformationsgruppen: Vol. 3, Leipzig: Teubner, 1893. · JFM 25.0623.01
[79] Lindelöf, E., Sur le mouvement d’un corps de révolution roulant sur un plan horizontal, Acta Societ. Scient. Fennicae, 1895, vol. 20, no. 10, 18 pp. · JFM 26.0849.03
[80] Llibre, J.; Ramirez, R.; Sadovskaia, N., A new approach to the vakonomic mechanics, Nonlinear Dynam., 78, 2219-2247, (2014) · Zbl 1345.70020
[81] Maggi, G., Di alcune nuove forme delle equazioni Della dinamica, applicabili ai sistemi anolonomi, Atti della R. Acc. nazionale dei Lincei, 5, 287-292, (1901) · JFM 32.0714.02
[82] Maruskin, J.M.; Bloch, A. M.; Marsden, J.E.; Zenkov, D. V., A fiber bundle approach to the transpositional relations in nonholonomic mechanics, J. Nonlinear Sci., 22, 431-461, (2012) · Zbl 1251.70016
[83] Meijaard, J.P.; Papadopoulos, J.M.; Ruina, A.; Schwab, A. L., Linearized dynamics equations for the balance and steer of a bicycle: A benchmark and review, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 463, 1955-1982, (2007) · Zbl 1161.70006
[84] Molenbrock, P., Over de zuiver rollende beweging Van een lichaam over een willekeurig oppervlak, Nieuw Archief voor Wiskunde, 17, 130-157, (1890) · JFM 22.0927.02
[85] Monforte, J.C., Geometric, Control and Numerical Aspects of Nonholonomic Systems, Lecture Notes in Math., vol. 1793, Berlin: Springer, 2002. · Zbl 1009.70001
[86] Moser, J. and Zehnder, E. J., Notes on Dynamical Systems, Courant Lect. Notes Math., vol. 12, Providence,R.I.: AMS, 2005. · Zbl 1087.37001
[87] Neimark, Ju. I. and Fufaev, N.A., Dynamics of Nonholonomic Systems, Trans. Math. Monogr., vol. 33, Providence,R.I.: AMS, 1972. · Zbl 0245.70011
[88] Neumann, C. G., Beiträge zur analytischenmechanik: 1, Berichte über die Verhandlungen der Königlich Sächsischen Gesellschaft der Wissenschaften zu Leipzig, Mathematisch-Physische Classe, 51, 371-444, (1899)
[89] Neumann, C., Über die rollende bewegung eines Körpers auf einer gegebenen horizontalebene unter dem einfluss der schwere, Math. Ann., 27, 478-501, (1886)
[90] Ohsawa, T.; Fernandez, O. E.; Bloch, A.M.; Zenkov, D.V., Nonholonomic Hamilton — Jacobi theory via Chaplygin hamiltonization, J. Geom. Phys., 61, 1263-1291, (2011) · Zbl 1339.70035
[91] Pavon, M., Hamilton — Jacobi equations for nonholonomic dynamics, J. Math. Phys., 46, 032902, (2005) · Zbl 1067.70018
[92] Poincaré, H., Sur une forme nouvelle des équations de la Mécanique, C. R. Acad. Sci. Paris, 132, 369-371, (1901) · JFM 32.0715.01
[93] Poincaré, H., LES idées de Hertz sur la mécanique, Revue Générale des Sciences, 8, 734-743, (1897) · JFM 28.0054.04
[94] Posch, H. A.; Hoover, W. G.; Vesely, F. J., Canonical dynamics of the nosé oscillator: stability, order, and chaos, Phys. Rev. A(3), 33, 4253-4265, (1986)
[95] Pöschl, Th.M. F., Sur LES équations canoniques des systèmes non holonomes, C. R. Acad. Sci. Paris, 156, 1829-1831, (1913) · JFM 44.0816.01
[96] Quanjel, J., LES équations générales de la mécanique dans le cas des liaisons non-holonomes, Rendiconti del circolo matematico di Palermo, 22, 263-273, (1906) · JFM 37.0722.02
[97] Rashevsky, P.K., Any two points of a totally nonholonomic space may be connected by an admissible line, uch. zap. ped. inst. im. liebknechta, Ser. Phys. Math., 3, 83-94, (1938)
[98] Rocard, Y., L’instabilité en mécanique: Automobiles, avions, ponts suspendus, Paris: Masson, 1954.
[99] Routh, G.R.R., The motion of a bicycle, The Messenger of Mathematics, 28, 151-169, (1899)
[100] Routh, E. J., The Advanced Part of a Treatise on the Dynamics of a System of Rigid Bodies: Being Part II of a Treatise on the Whole Subject, 6th ed., New York: Dover, 1955. · Zbl 0065.16802
[101] Rumianstev, V.V., On stability of motion of nonholonomic systems, J. Appl. Math. Mech., 31, 282-293, (1967) · Zbl 0159.26101
[102] Rumyantsev, V.V.; Sumbatov, A. S., On the problem of a generalization of the Hamilton — Jacobi method for nonholonomic systems, Z. Angew. Math. Mech., 58, 477-481, (1978) · Zbl 0395.70017
[103] Schouten, G., Over de rollende beweging Van een omwentelingalichaam op een vlak, Verlangen der Konikl. Akad. van Wet. Amsterdam. Proc., 5, 1-10, (1899)
[104] Schouten, J.A., On non-holonomic connexions, Proc. Amsterdam, 31, 291-299, (1928) · JFM 54.0758.01
[105] Slesser, G.M., Notes on rigid dynamics, Quart. J. Math., 4, 65-77, (1861)
[106] Stückler, B., Über die differentialgleichungen für die bewegung eines idealisierten kraftwagens, Arch. Appl. Mech., 20, 337-356, (1952)
[107] Stückler, B., Über die berechnung der an rollenden fahrzeugen wirkenden haftreibungen, Arch. Appl. Mech., 23, 279-287, (1955)
[108] Suslov, G. K., Theoretical Mechanics, Moscow: Gostekhizdat, 1946 (Russian).
[109] Suslov, G. K., The Foundations of Analytical Mechanics: Vol. 1, Kiev: Imp. Univ., 1900.
[110] Vagner, V.V., A geometric interpretation of nonholonomic dynamical systems, Tr. Semin. Vectorn. Tenzorn. Anal., 5, 301-327, (1941) · Zbl 0063.07917
[111] Dooren, R., Second form of the generalized Hamilton — Jacobi method for nonholonomic dynamical systems, Z. Angew. Math. Phys., 29, 828-834, (1978) · Zbl 0392.70019
[112] Vershik, A.M.; Faddeev, L.D., Differential geometry and Lagrangian mechanics with constraints, Soviet Phys. Dokl., 17, 34-36, (1972) · Zbl 0243.70014
[113] Vierkandt, A., Über gleitende und rollende bewegung, Monatsh. Math. Phys., 3, 31-38, (1892) · JFM 24.0893.01
[114] Volterra, V., Sopra una classe di equazioni dinamiche, Atti della R. Accad. Sci. di Torino, 33, 471-475, (1898) · JFM 29.0604.03
[115] Voronets, P.V., Sur LES équations du mouvement pour LES systèmes non holonomes, Mat. Sb., 22, 659-686, (1901)
[116] Vranceanu, G., LES espaces non holonomes et leurs applications mécaniques, Mém. Sci. Math., 76, 1-70, (1936) · Zbl 0013.28105
[117] Walker, G.T., On a curious dynamical property of celts, Proc. Cambridge Phil. Soc., 8, 305-306, (1895)
[118] Weber, R.W., Hamiltonian systems with constraints and their meaning in mechanics, Arch. Rational Mech. Anal., 91, 309-335, (1986) · Zbl 0606.58024
[119] Whittaker, E.T., A Treatise on the Analytical Dynamics of Particles and Rigid Bodies, 4th ed., New York: Cambridge Univ. Press, 1989.
[120] Woronetz, P., Über die bewegung eines starren Körpers, der ohne gleitung auf einer beliebigen fläche rollt, Math. Ann., 70, 410-453, (1911) · JFM 42.0772.01
[121] Woronetz, P., Über das problem der bewegung von vier massenpunkten unter dem einflusse von inneren kräften, Math. Ann., 63, 387-412, (1907) · JFM 38.0725.03
[122] Zegzhda, S.A., Soltakhanov Sh.Kh., and Yushkov, M.P., The Equations of Motion of Nonholonomic Systems and Variational Principles of Mechanics. The New Class of Control Problems, Moscow: Fizmatlit, 2005 (Russian). · Zbl 1162.70301
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.