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Based on the Cauchy-Kowalevski theorem for a system of partial differential equations to be integrable, a kind of generalized Birkhoffian systems (GBSs) with local, analytic properties are put forward, whose manifold admits a presymplectic structure described by a closed 2-form which is equivalent to the self-adjointness of the GBSs. Their relations with Birkhoffian systems, generalized Hamiltonian systems are investigated in detail. Analytic, algebraic and geometric properties of GBSs are formulated, together with their integration methods induced from the Birkhoffian systems. As an important example, nonholonomic systems are reduced into GBSs, which gives a new approach to some open problems of nonholonomic mechanics.
[1] Bloch, A.M., Fernandez, O.E., Mestdag, T.: Hamiltonization of nonholonomic systems and the inverse problem of the calculus of variations. Rep. Math. Phys. 63 2009 225–249 DOI 10.1016/S0034-4877(09)90001-5 | MR 2519467 | Zbl 1207.37045
[2] Bloch, A.M., Baillieul, J., Crouch, P., Marsden J.: Nonholonomic Mechanics and Control. Springer, London 2003 MR 1978379 | Zbl 1045.70001
[3] Cortes, J.M.: Geometric, Control and Numerical Aspects of Nonholonomic Systems. Springer, Berlin 2002 MR 1942617 | Zbl 1009.70001
[4] Guo, Y.X., Luo, S.K., Shang, M., Mei, F.X.: Birkhoffian formulation of nonholonomic constrained systems. Rep. Math. Phys. 47 2001 313–322 DOI 10.1016/S0034-4877(01)80046-X | MR 1847630
[5] Hojman, S.: Construction of genotopic transformations for first order systems of differential equations. Hadronic J. 5 1981 174–184 MR 0642608 | Zbl 0515.70022
[6] Ibort, L.A., Solano, J.M.: On the inverse problem of the calculus of variations for a class of coupled dynamical systems. Inverse Problems 7 1991 713–725 DOI 10.1088/0266-5611/7/5/005 | MR 1128637 | Zbl 0756.34019
[7] Krupková, O., Musilová, J.: Non-holonomic variational systems. Rep. Math. Phys. 55 2005 211–220 DOI 10.1016/S0034-4877(05)80028-X
[8] Li, J.B., Zhao, X.H., Liu, Z.R.: Theory of Generalized Hamiltonian Systems and Its Applications. Science Press of China Beijing 2007
[9] Liu, C., Liu, S.X., Guo, Y.X.: Inverse problem for Chaplygin’s nonholonomic. Sci. Chin. G 53 2010 (to appear)
[10] Massa, E., Pagani, E.: Jet bundle geometry, dynamical connections, and the inverse problem of Lagrangian mechanics. Ann. Inst. Henri Poincaré: Phys. Theor. 61 1994 17–62 MR 1303184 | Zbl 0813.70004
[11] Mei, F.X., Shi, R.C., Zhang, Y.F., Wu, H.B.: Dynamics of Birkhoffian systems. Press of Beijing Institute of Technology Beijing 1996 (in Chinese)
[12] Morando, P., Vignolo, S.: A geometric approach to constrained mechanical systems, symmetries and inverse problems. J. Phys. A: Math. Gen. 31 1998 8233–8245 DOI 10.1088/0305-4470/31/40/015 | MR 1651497 | Zbl 0940.70008
[13] Santilli, R.M.: Foundations of Theoretical Mechanics I. Springer-Verlag, New York 1978 MR 0514210 | Zbl 0401.70015
[14] Santilli, R.M.: Foundations of Theoretical Mechanics II. Springer-Verlag, New York 1983 MR 0681293 | Zbl 0536.70001
[15] Sarlet, W.: The Helmholtz conditions revisited. A new approach to the inverse problem of Lagrangian dynamics. J. Phys. A: Math. Gen. 15 1982 1503–1517 DOI 10.1088/0305-4470/15/5/013 | MR 0656831 | Zbl 0537.70018
[16] Sarlet, W., Cantrijn, F., Saunders, D.J.: A differential geometric setting for mixed first- and second-order ordinary differential equations. J. Phys. A: Math. Gen. 30 1997 4031–4052 DOI 10.1088/0305-4470/30/11/029 | MR 1457421 | Zbl 0932.37040
[17] Sarlet, W., Cantrijn, F., Saunders, D.J.: A geometrical framework for the study of non-holonomic Lagrangian systems. J. Phys. A: Math. Gen. 28 1995 3253–3268 DOI 10.1088/0305-4470/28/11/022 | MR 1344117 | Zbl 0858.70013
[18] Sarlet, W., Thompson, G., Prince, G.E.: The inverse problem in the calculus of variations: the use of geometrical calculus in Douglas’s analysis. Trans. Amer. Math. Soc. 354 2002 2897–2919 DOI 10.1090/S0002-9947-02-02994-X | MR 1895208
[19] Saunders, D.J., Sarlet, W., Cantrijn, F.: A geometrical framework for the study of non-holonomic Lagrangian systems II. J. Phys. A: Math. Gen. 29 1996 4265–4274 DOI 10.1088/0305-4470/29/14/042 | MR 1406933 | Zbl 0900.70196
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