Abstract
The aim of this paper was to show that the Lagrange–d’Alembert and its equivalent the Gauss and Appel principle are not the only way to deduce the equations of motion of the nonholonomic systems. Instead of them we consider the generalization of the Hamiltonian principle for nonholonomic systems with non-zero transpositional relations. We apply this variational principle, which takes into the account transpositional relations different from the classical ones, and we deduce the equations of motion for the nonholonomic systems with constraints that in general are nonlinear in the velocity. These equations of motion coincide, except perhaps in a zero Lebesgue measure set, with the classical differential equations deduced with the d’Alembert–Lagrange principle. We provide a new point of view on the transpositional relations for the constrained mechanical systems: the virtual variations can produce zero or non-zero transpositional relations. In particular, the independent virtual variations can produce non-zero transpositional relations. For the unconstrained mechanical systems, the virtual variations always produce zero transpositional relations. We conjecture that the existence of the nonlinear constraints in the velocity must be sought outside of the Newtonian mechanics. We illustrate our results with examples.
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References
Alekseev, V.M., Tikhomirov, V.M., Fomin, S.V.: Optimal Control. Contemporary Soviet Mathematics. Consultants Bureau, New York (1987). ISBN:0-306-10996-4
Arnold, V.I., Kozlov, V.V., Neishtadt, A.I.: Mathematical aspects of classical mechanics. Dynamical Systems III, Third Edition, Encyclopaedia of Mathematical Sciences 3. Springer-Verlag, Berlin (2006). ISBN 978-3-540-28246-4
Appell, P.: Exemple de mouvement d’un point assujetti à une liaison exprimé par une relation non linéaire entre les composantes de la vitesse. Rend. Circ. Mat. Palermo 32, 48–50 (1911)
Birkhoff, G.D.: Dynamical Systems, with an Addendum by Jurgen Moser. American Mathematical Society Colloquium Publications. American Mathematical Society, Providence (1966)
Bloch, A.M.: Nonholonomic mechanics and control, with the collaboration of J. Baillieul, P. Crouch and J. Marsden, with scientific input from P. S. Krishnaprasad, R. M. Murray and D. Zenkov. Interdisciplinary Applied Mathematics 24, Systems and Control. Springer-Verlag, New York (2003). ISBN: 0-387-95535-6
Cantrijn, F., Cortés, J., de León, M., de Diego, D.M.: On the geometry of generalized Chaplygin systems. Math. Proc. Cambr. Philos. Soc. 132, 323–351 (2002)
Chaplygin, S.A.: On the theory of motion of nonholonomic systems. Theorems on the reducing multiplier. Mat. Sb. 28, 303–314 (1911). (in Russian)
Chetaev, N.G.: Izv. Fiz. Mat. Obshch. Kazan 6, 68–71 (1932). (in Russian)
Chetaev, N.G.: On Gauss principle Izv. Fiz. Mat. Obshch. Kazan 6, 323–326 (1941). (in Russian)
Favretti, M.: Equivalence of dynamics for nonholonomic systems with transverse constraints. J. Dynam. Differ. Equ. 10, 511–536 (1998)
Ferrers, N.M.: Extension of Lagrange’s equations, Quart. J. Pure Appl. Math. 12, 1–5 (1872)
Gantmacher, F.R.: Lektsi po analitisheskoi mechanic, Ed. Nauka, Moscow, 1966 (in Russian)
Gelfand, I.M., Fomin, S.V.: Calculus of Variations. Prentice-Hall Inc, Englewood Cliffs (1963)
Gracia, X., Marín-Solano, J., Muñoz-Lecanda, M.: Some geometric aspects of variational calculus in constrained systems. Rep. Math. Phys. 51, 127–148 (2003)
Griffiths, P.A.: Exterior differential systems and the calculus of variations. Progress in Mathematics 25, Birkhäuser. Springer, Boston (1983). ISBN 0-7643-3103-8
Hamel, G.: Teoretische Mechanik, Eine einheitliche Einführung in die gesamte Mechanik. Springer-Verlag, Berlin-New York (1978). Corrected reprint of the 1949 edition, ISBN: 3-540-03816-7
Hertz, H.: Die Prinzipien der Mechanik in neuem Zusammenhaange dargestellt. Ges, Werke, Leipzig, Barth (1910)
Hölder, O.: Ueber die prinzipien von Hamilton und Maupertius, Nachtichten Kön. Ges. Wissenschaften zu Gottingen Math.-Phys. Kl. (1896), 122–157.
Kharlamov, P.V.: A critique of some mathematical models of mechanical systems with differential constraints. J. Appl. Math. Mech. 56, 584–594 (1992)
Kirgetov, V.I.: On the ranspositional relations in mechanics. J. Appl. Math. Mech. 22, 682–693 (1958)
Korteweg, D.J.: Über eine ziemlich verbreitete unrichtige Behand-lungsweise eines. Probl. der Rollenden Beweg., Nieuw Archiv. voor Wiskd. 4, 130–155 (1899)
Kozlov, V.V.: The dynamics of systems with non-integrable restrictions I. Vestn. Mosk. Univ. Ser. I Mat. Mekh. 3, 92–100 (1982)
Kozlov, V.V.: The dynamics of systems with non-integrable restrictions II. Vestn. Mosk. Univ. Ser. I Mat. Mekh. 4, 70–76 (1982)
Kozlov, V.V.: Dynamics of systems with non-integrable restrictions III. Vestn. Mosk. Univ., Ser. 3 Mat. Mekh. 3, 102–111 (1983)
Kozlov, V.V.: Realization of nonintegrable constraints in classical mechanics. Dokl. Akad. Nauk SSSR 272, 550–554 (1983)
Kozlov, V.V.: On the theory of integration of the equations of nonholonomic mechanics. Adv. Mech. 8, 85–107 (1985)
Kozlov, V.V.: Gauss principle and realization of the constraints. Regular Chaotic Dyn. 13, 431–434 (2008)
Kupka, I., Oliva, W.M.: The nonholonomic mechanics. J. Differ. Equ. 169, 169–189 (2001)
Lewis, A.D., Murray, R.M.: Variational principle for constrained mechanical systems: theory and experiments. Int. J. Non-linear Mech. 30, 793–815 (1995)
Llibre, J., Ramírez, R., Sadovskaia, N.: Integrability of the constrained rigid body. Nonlinear Dyn. 73, 2273–2290 (2013)
Llibre, J., Ramírez, R., Sadovskaia, N.: Inverse problems in ordinary differential equations, preprint, (2012)
Lurie, A.I.: Analytical Dynamics. Foundations of Engineering Mechanics. Springer-Verlag, Berlin (2002). ISBN 3-540-42982-4
Marle, C.M.: Various approaches to conservative and nonconservative nonholonomic systems. Rep. Math. Phys. 42, 211–229 (1998)
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). doi:10.1007/s00332-012-9144-3
Neimark, J., Fufaev, N.A.: Dynamics of Nonholonomic Systems. American Mathematical Society, Rhode Island (1972). ISBN 082183617X / 0-8218-3617-X
Novoselov, V.S.: Example of a nonlinear nonholonomic constraints that is not of the type of N.G. Chetaev, Vestnik Leningrad Univ., 12 (1957)
Oliva, W.M.: Geometric Mechanics, Lecture Notes in Mathematics, vol. 1798. Springer-Verlag, Berlin (2002)
Pars, L.A.: A Treatise on Analytical Dynamics. John Wiley & Sons Inc, New York (1965)
Poincaré, H.: Hertz’s ideas in mechanics, in addition to H. Hertz, Die Prizipien der Mechanik in neum Zusammemhauge dargestellt (1894)
Polak, L.S.: Variation principle of mechanic, Ed. Fisico-matematicheskoi literature, 1960 (in Russian)
Ramirez, R., Sadovskaia, N.: On the dynamics of nonholonomic systems. Rep. Math. Phys. 60, 427–451 (2007). doi:10.1016/S0034-4877(08)00005-0
Ramirez, R.: Dynamics of nonholonomic systems, Publisher VINITI 3878 (1985) (in Russian)
Rubanovskii, V.N., Samsonov, V.A.: Stability of steady motions, in examples and problems, M. Nauka 1998 (in Russian)
Rumyantsev, V.V.: On the forms of the Hamilton principle in quasicoordinates. J. Appl. Math. Mech. 63, 165–171 (1999). doi:10.1016/S0021-8928(99)00024-6
Sadovskaia, N.: Inverse problem in theory of ordinary differential equations, Thesis Ph. D., Univ. Politécnica de Cataluña, 2002 (in Spanish)
Synge, J.L.: On the geometry of dynamics, Phil. Trans. Roy. Soc. Lond. Ser. A 226, 31–106 (1927)
Sumbatov, A.S.: Nonholonomic systems. Regular Chaotic Dyn. 7, 221–238 (2002)
Suslov, G.K.: On a particular variant of d’Alembert principle. Math. Sb. 22, 687–691 (1901). (in Russian)
Zampieri, G.: Nonholonomic versus vakonomic dynamics. J. Differential Equations 163, 335–347 (2000). doi:10.1006/jdeq.1999.3727
Vershik, A.M., Faddev, L.D.: Differential geometry and Lagrangian mechanics with constraints, Soviet Physics-Doklady 17, (1972) (in Russian)
Vierkandt, A.: Über gleitende und rollende. Beweg. Monatshefte Mathh. Phys. III, 31–54 (1982)
Volterra, V.: Sopra una classe di equazione dinamiche. Atti Accad. Sci. Torino 33, 451–475 (1898)
Voronets, P.: On the equations of motion for nonholonomic systems. Math. Sb. 22, 659–686 (1901). (in Russian)
Acknowledgments
The first author is partially supported by a MINECO/FEDER Grant MTM2008–03437, an AGAUR Grant number 2014SGR-568, an ICREA Academia, FP7–PEOPLE–2012–IRSES–316338 and 318999, and FEDER-UNAB10-4E-378. The second author was partly supported by the Spanish Ministry of Education through projects TSI2007-65406-C03-01 “E-AEGIS” and Consolider CSD2007-00004 “ARES”.
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Llibre, J., Ramírez, R. & Sadovskaia, N. A new approach to the vakonomic mechanics. Nonlinear Dyn 78, 2219–2247 (2014). https://doi.org/10.1007/s11071-014-1554-3
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DOI: https://doi.org/10.1007/s11071-014-1554-3