Classical dynamics of non-holonomic systems : a geometric approach
Annales de l'I.H.P. Physique théorique, Volume 55 (1991) no. 1, p. 511-544
@article{AIHPA_1991__55_1_511_0,
     author = {Massa, Enrico and Pagani, Enrico},
     title = {Classical dynamics of non-holonomic systems : a geometric approach},
     journal = {Annales de l'I.H.P. Physique th\'eorique},
     publisher = {Gauthier-Villars},
     volume = {55},
     number = {1},
     year = {1991},
     pages = {511-544},
     zbl = {0731.70012},
     mrnumber = {1130215},
     language = {en},
     url = {http://www.numdam.org/item/AIHPA_1991__55_1_511_0}
}
Massa, Enrico; Pagani, Enrico. Classical dynamics of non-holonomic systems : a geometric approach. Annales de l'I.H.P. Physique théorique, Volume 55 (1991) no. 1, pp. 511-544. http://www.numdam.org/item/AIHPA_1991__55_1_511_0/

[1] J.L. Lagrange, Mécanique Analytique, in Œuvres de Lagrange, Paris, 1788.

[2] T. Levi-Civita and U. Amaldi, Lezioni di Meccanica Razionale, Zanichelli, Bologna, 1984, first edition 1922, second edition 1929.

[3] N.G. Chetaev, O printsipe Gaussa (On the Gauss Principle), Izd. Fiz.-Matem. Obshch. pri Kazansk. Univ., Ser. 3, Vol. 6, 1932-1933, pp. 323-326, Izvestia of the Physico-Mathematical Society of Kazan University, Series 3, Vol. 6, 1932-1933, pp. 62-67.

[4] N.G. Chetaev, Papers on Analytical Mechanics. On the Gauss Principle, Moscow, Izd. Akad. Nauk S.S.S.R., 1962.

[5] V.V. Rumiantsev, Certain Variational Principles of Mechanics, in Adv. Theor. Appl. Mech., A. ISHLINSKY and F. CHERNOUSKO Eds., English translation, MIR Publishers, Moscow, 1981. | Zbl 0547.70022

[6] Y. Pironneau, Sur les liaisons non holonomes, non linéaires, déplacements virtuels à travail nul, conditions de Chetaev, Proceedings of the I.U.T.A.M.-I.S.I.M.M. Symposium on Modern Developments in Analytical Mechanics,Academy of Sciences of Turin, Turin, June 7-11, 1982, Atti della Accademia delle Scienze di Torino, Supplemento al Vol. 117, 1983, pp. 671-686. | MR 773517

[7] K. Gauss, Uber ein neues allgemeines Grundgesetz der Mechanik, Crelle J. reine Math., Vol. 4, 1829, pp. 232.

[8] G. Hamel, Theoretische Mechanik, Eine einheitliche Einfuhrung in die gesamte Mechanik, Springer-Verlag, Berlin, 1949. | MR 34137 | Zbl 0036.24301

[9] E.T. Whittaker, A Treatise on the Analytical Dynamics of Particles and Rigid Bodies, 4th ed., Cambridge University Press, Cambridge, 1959. | MR 103613 | Zbl 0049.40801

[10] J.F. Pommaret, Systems of Partial Differential Equations and Lie Pseudogroups, Gordon and Breach, New York, 1978. | MR 517402 | Zbl 0401.58006 | Zbl 0418.35028

[11] P. Libermann and C.M. Marle, Differential Geometry and Analytical Mechanics, D. Reidel Publ. Comp, Dordrecht, Holland, 1987. | MR 882548

[12] M. Ferrers, Extension of Lagrange's Equations, Quart. J. Math., Vol. 12, 1873, pp. 1-4. | JFM 04.0457.01

[13] C. Neumann, Uber die rollende Bewegung eines Korpers auf einer gegebenen Horizontaleben unter dem Einfluss der Schwere, Ber. Verh. Konigl. Sachs. Ges. Wiss., Leipzig Math. Phys. Cl. 37, 1885, pp. 352-378. | JFM 17.0893.02

[14] G.A. Maggi, Di alcune nuove forme delle equazioni della dinamica applicabili ai sistemi anolonomi, Atti Accad. Naz. Lincei Rend. Cl. Fis. Mat. Nat., Vol. 10, 1901, pp. 287-291. | JFM 32.0714.02

[15] P. Appel, Sur les liaisons exprimées par des relations non linéaires entre les vitesses, C.R. Acad. Sci. Paris, T. 152, 1911, pp. 1197-1199. | JFM 42.0755.02

[16] P. Appel, Exemple de mouvement d'un point assujetti à une liaison exprimée par une relation non linéaire entre les composantes de la vitesse, Rend. Circ. Mat. Palermo, Vol. 32, 1911, pp. 48-50. | JFM 42.0756.01

[17] P. Appel, Traité de Mécanique Rationnelle, Gauthier-Villars, Paris, 1903. | JFM 40.0751.01

[18] P.V. Voronets, On Equations of Motion for Nonholonomic Systems, Math. Sb., Vol. 22, 1901, pp. 659-686 (in russian).

[19] P.E. Jourdain, Note on an Analogue of the Gauss Principle of Least Constraint, The Quarterly J. Pure Appl. Math., 40, 1909, pp. 153-157. | JFM 40.0780.01

[20] V. Volterra, Sopra una classe di equazioni dinamiche, Atti Acad. Sci. cl. Fis. Mat. Nat., Torino, Vol. 33, 1887. | JFM 29.0604.03

[21] S.A. Chaplygin, On the Theory of Motion of Nonholonomic Systems. Theorems on the Reducing Multiplier, Math. Sb., Vol. 28, 1911, pp. 303-314, (in russian).

[22] L.A. Pars, A Treatise on Analytical Dynamics, Heinemaun, London, 1965. | Zbl 0125.12004

[23] F.R. Gantmacher, Lezioni di Meccanica Analitica, Editori Riuniti, Roma, 1980.

[24] Iu.I. Neimark and N.A. Fufaev Dynamics of Nonholonomic Systems, Translations of Mathematical Monographs, Vol. 33, American Mathematical Society, Providence, Rhode Island, 1972. | Zbl 0245.70011

[25] I.L. Khmelevskii, On a Problem of Chaplygin, P.M.M., Vol. 26, 1962, pp. 289-304. | MR 142223 | Zbl 0114.15001

[26] I. Iliev, On the Conditions for the Existence of the Reducing Chaplygin Factor, P.M.M., Vol. 49, 1985, pp. 295-301. | MR 855811 | Zbl 0596.70011

[27] V.S. Novoselov, Applications of Nonlinear Nonholonomic Coordinates in Analytical Mechanics, Leningrad, Gos. Univ., Ucenye Zap. 217, Ser. Mat. Nauk, Vol. 31, 1957, pp. 50-83 (in russian). | Zbl 0085.17703

[28] V.S. Novoselov, Extended Equations of Motion of Nonlinear Nonholonomic Systems, Leningrad, Gos. Univ. , Ucenye Zap. 217, Ser. Mat. Nauk, Vol. 31, 1957, pp. 84-89 (in russian). | MR 90976 | Zbl 0085.17704

[29] V.S. Novoselov, Application of Helmholtz's Method to the Study of the Motion of the Chaplygin Systems, Vestn. L.G.U., Ser. Mat. Mekh. Astron., Vol. 13, 1958, pp. 102- 111. | MR 99128 | Zbl 0080.17303 | Zbl 0095.17703

[30] V.S. Novoselov, Application of Helmholtz's Method to the Investigation of the Motion of Nonholonomic Systems, Vestn. L.G.U., Ser. Mat. Mekh. Astron., Vol. 13, 1958, pp. 80-87. | MR 99129 | Zbl 0080.17303

[31] V.S. Novoselov, Variational Methods in Mechanics, Leningrad, Izd. Leningrad. Gos. Univ., 1966. | MR 210354 | Zbl 0141.41703

[32] A.S. Sumbatov, Reduction of the Differential Equations of Nonholonomic Mechanics to Lagrange Form, P.M.M. Vol. 36, 1972, pp. 194-201.

[33] A.S. Sumbatov, On the Linear Integrals of the Chaplygin Equations, P.M.M., Vol. 45, 1981, pp. 339-342. | Zbl 0499.70022

[34] N.K. Moshchuk, Reducing the Equations of Motion of Certain Non-Holonomic Chaplygin Systems to Lagrangian and Hamiltonian Form, P.M.M., Vol. 51, 1987, pp. 172-177. | MR 946561 | Zbl 0653.70015

[35] N.K. Moshchuk, On the Motion of the Chaplygin's Sledge, P.M.M., Vol. 51, 1987, pp. 426-430. | MR 971656 | Zbl 0664.70019

[36] V.V. Rumiantsev, On Hamilton's Principle for Nonholonomic Systems, P.M.M., Vol. 42, 1978, pp. 407-419. | Zbl 0419.70015

[37] V.V. Rumiantsev, On the Lagrange and Jacobi Principles for Nonholonomic Systems, P.M.M., Vol. 43, 1979, pp. 625-632. | MR 620568 | Zbl 0443.70012

[38] V.V. Rumiantsev, On Integral Principles for Nonholonomic Systems, P.M.M., Vol. 46, 1982, pp. 1-8. | MR 685650 | Zbl 0514.70016

[39] V.V. Rumiantsev, On Some Problems of Analytical Dynamics of Nonholonomic Systems, Proceedings of the I.U.T.A.M.-I.S.I.M.M. Symposium on Modern Developments in Analytical Mechanics, Academy of Sciences of Turin, Turin, June 7-11, 1982, Atti dell'Accademia delle Scienze di Torino, Supplemento al Vol. 117, 1983, pp. 697-716. | MR 773520

[40] I.L. Khmelevskii, On Hamilton's Principle for Nonholonomic Systems, P.M.M., Vol. 24, 1960, pp. 1177-1182. | Zbl 0102.39501

[41] E. Kh. Naziev, On the Hamilton-Jacobi Method for Nonholonomic Systems, P.M.M., Vol. 36, 1972, pp. 1038-1045. | MR 334625

[42] A. Pignedoli, On some Investigations of the Italian School about Holonomic and Non-Holonomic Systems, Proceedings of the I.U.T.A.M.-I.S.I.M.M. Symposium on Modern Developments in Analytical Mechanics, Academy of Sciences of Turin, Turin, June 7-11, 1982, Atti della Accademia delle Scienze di Torino, Supplemento al Vol. 117, 1983, pp. 645-670. | MR 773516

[43] C. Cattaneo, Sulla struttura locale delle equazioni dinamiche di un sistema anolonomo, Rend. Sci. Fis. Mat. Nat. Accad. Lincei, Vol. 34, 1963, pp. 396-402. | MR 160344 | Zbl 0168.21902

[44] E. Clauser, Geometrizzazione della dinamica dei sistemi a vincoli mobili, Rend. Sc. Fis. Mat. Nat. Accad. Lincei, Vol. 19, 1955, pp. 33-39. | MR 78795

[45] G. Ferrarese, Sulle equazioni di moto di un sistema soggetto ad un vicolo anolonomo mobile, Rendiconti di Matematica Univ. Roma, Vol. 22, 1963, pp. 351-370. | MR 168156 | Zbl 0126.19503

[46] E. Delassus, Sur les liaisons non linéaires, C.R. Acad. Sci. Paris, T. 153, 1911, p. 626-628. | JFM 42.0757.01

[47] E. Delassus, Sur les liaisons non linéaires et les mouvements étudiés par M. Appell, C.R. Acad. Sci. Paris, T. 153, 1911, pp. 707-710. | JFM 42.0757.02

[48] E. Delassus, Leçons sur la dynamique des systèmes matériels, Hermann, Paris, 1913. | JFM 44.0764.01

[49] V. Valcovici, Une extension des liaisons non holonomes, C.R. Acad. Sci. Paris, T. 243, 1956, pp. 1012-1014. | MR 81623 | Zbl 0071.18003

[50] V. Valcovici, Une extension des liaisons non holonomes, C.R. Acad. Sci. Paris, T. 243, 1956, pp. 1096-1098. | MR 81055 | Zbl 0071.18004 | Zbl 0071.18003

[51] V. Valcovici, Une extension des liaisons non holonomes et des principes variationnels, Ber. Verh. Sachs. Akad. Wiss. Leipzig, Math.-Nat. Kl., Vol. 102, 1958, pp. 1-39. | MR 90218 | Zbl 0078.37505

[52] Q.K. Ghori, Equations of Motion of a System with Nonlinear Nonholonomic Constraint, Arch. Rat. Mech. Anal., Vol. 11, 1962, pp. 114-116. | MR 150999 | Zbl 0131.39103

[53] Duo San, Equations of Motion of Systems with Second-order Nonlinear Nonholonomic Constraints, P.M.M., Vol. 37, 1973, pp. 329-334. | Zbl 0293.70014

[54] V.I. Kirgetov, On "Possible Displacements" of Material Systems, P.M.M., Vol. 23, 1959, pp. 956-962. | MR 113302 | Zbl 0091.17002

[55] V.I. Kirgetov, On the Relaxation of Material System, P.M.M., Vol. 24, 1960, pp. 48-58. | MR 119482 | Zbl 0096.38301

[56] B.G. Kuznetsov, Generalized Virtual Displacements, P.M.M., Vol. 23, 1959, pp. 963-974. | MR 113303 | Zbl 0091.16501

[57] G.H. Pozharitskii, On the Equations of Motion for Systems with Nonideal Constraints, P.M.M., Vol. 24, 1960, pp. 669-675. | MR 127575 | Zbl 0102.39703

[58] L.N. Semenova, On Routh Theorem for Nonholonomic Systems, P.M.M., Vol. 29, 1965, pp. 167-169. | MR 201108 | Zbl 0142.23101

[59] L. Salvadori, Sulla stabilita dei moti merostatici di particolari sistemi anolonomi, Rend. Accad. Sci. Fis. Mat. Napoli, Vol. (IV), 20, 1953, pp. 66-78. | MR 72582 | Zbl 0058.17802

[60] L. Salvadori, Influenza di una sollecitazione dissipativa sulla stabilita' dei moti merostatici di una classe di sistemi anolonomi, Rend. Accad. Sci. Fis. Mat. Napoli, Vol. (IV), 33, 1966, pp. 191-198. | Zbl 0237.70025

[61] V.V. Rumiantsev, On Stability of Motion of Nonholonomic System, P.M.M., Vol. 31, 1967, pp. 282-293. | Zbl 0159.26101

[62] A.V. Karapetyan, On the Stability of Steady Motions of Chaplygin's Nonholonomic Systems, P.M.M., Vol. 42, 1978, pp. 863-870. | Zbl 0438.70018

[63] A.V. Karapetyan, On the Problem of Steady Motion Stability of Nonholonomic Systems, P.M.M., Vol. 44, 1980, pp. 295-300. | Zbl 0464.70009

[64] A.V. Karapetyan, On the Realization of Non-holonomic Constraints and the Stability of Celtic Stones, P.M.M., Vol. 45, 1981, pp. 30-36. | Zbl 0493.70008

[65] A.V. Karapetyan, On the Stability of the Stationary Motions of Systems with Friction, P.M.M., Vol. 51, 1987, pp. 431-436. | MR 971657 | Zbl 0664.70017

[66] G. Fusco and M. Oliva, Dissipative Systems with Constraints, J. Diff. Eq., Vol. 63, 1986, pp. 362-388. | MR 848274 | Zbl 0594.58037

[67] N.N. Polyakhov, S.A. Zegzhda and M.P. Yushkov, Generalization of Gauss's Principle to Nonholonomic Systems of Higher Orders, Sov. Phys. Dokl., Vol. 28, 1983, pp. 330-332. | Zbl 0536.70021

[68] A.M. Vershik and L.D. Faddeev, Differential Geometry and Lagrangian Mechanics with Constraints, Sov. Phys. Dokl., Vol. 17, 1972, pp. 34-36. | Zbl 0243.70014

[69] R.W. Weber, Hamiltonian Systems with Constraints and their Meaning in Mechanics, Arch. Rat. Mech. Anal., Vol. 91, 1985, pp. 309-335. | MR 807817 | Zbl 0606.58024

[70] N.A. Fufaev, A Sphere Rolling on a Horizontal Rotating Plane, P.M.M., Vol. 47, 1984, pp. 27-29. | Zbl 0534.70008

[71] N.A. Fufaev, Theory of the Motion of Systems with Rolling, P.M.M., Vol. 49, 1985, pp. 41-49. | Zbl 0593.70021

[72] Fam Guen, On the Theory of Ignorable Displacements for Generalized Poincare'-Chetaev Equation, P.M.M., Vol. 45, 1981, pp. 343-350. | Zbl 0508.70011

[73] S. Benenti, Geometrical Aspects of the Dynamics of Non-Holonomic Systems, Conférence « Journées Relativistes 1987 », Chambery, 14-16 mai 1987.

[74] E.G. Virga, Un'osservazione sui vincoli non perfetti, Riv. Mat. Univ. Parma, Vol. (4), 13, 1987, pp. 379-384. | Zbl 0672.70018

[75] F. Cardin and G. Zanzotto, On Constrained Mechanical Systems: D'Alembert and Gauss' Principles, J. Math. Phys., Vol. 30, 1989, pp. 1473-1479. | MR 1002251 | Zbl 0678.70018