March  2013, 5(1): 85-129. doi: 10.3934/jgm.2013.5.85

Vector fields with distributions and invariants of ODEs

1. 

Institute of Mathematics, Polish Academy of Sciences, ul. Śniadeckich 8, 00-956 Warszawa, Poland, Poland

Received  July 2012 Revised  February 2013 Published  April 2013

We study dynamic pairs $(X,V)$ where $X$ is a vector field on a smooth manifold $M$ and $V\subset TM$ is a vector distribution, both satisfying certain regularity conditions. We construct basic invariants of such objects and solve the equivalence problem. In particular, we assign to $(X,V)$ a canonical connection and a canonical frame on a certain frame bundle. We compute the curvature and torsion. The results are applied to the problem of time scale preserving equivalence of ordinary differential equations and of Veronese webs. The framework of dynamic pairs $(X,V)$ is shown to include sprays, control-affine systems, mechanical control systems, Veronese webs and other structures.
Citation: BronisŁaw Jakubczyk, Wojciech Kryński. Vector fields with distributions and invariants of ODEs. Journal of Geometric Mechanics, 2013, 5 (1) : 85-129. doi: 10.3934/jgm.2013.5.85
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show all references

References:
[1]

Proceed. Steklov Math. Inst., 256 (2007), 26-46. doi: 10.1134/S0081543807010026.  Google Scholar

[2]

J. Dynamical and Control Systems, 3 (1997), 343-389. doi: 10.1007/BF02463256.  Google Scholar

[3]

J. of Dynamical and Control Syst., 11 (2005), 297-327. doi: 10.1007/s10883-005-6581-4.  Google Scholar

[4]

J. Nonlinear Math. Phys., 17 (2010), 503-516. doi: 10.1142/S1402925110001100.  Google Scholar

[5]

Proc. Sympos. Pure Math., 53 (1991), 33-88.  Google Scholar

[6]

Houston Journal of Mathematics, 31 (2005), 315-332.  Google Scholar

[7]

Int. J. Geom. Methods Mod. Phys., 8 (2011), 1291-1327. doi: 10.1142/S0219887811005701.  Google Scholar

[8]

Springer Verlag, New York, 2004.  Google Scholar

[9]

Bull. Soc. Math. France, 52 (1924), 205-241.  Google Scholar

[10]

Math. Z., 37 (1933), 619-622. doi: 10.1007/BF01474603.  Google Scholar

[11]

Sci. Rep. Nat. Tsing Hua Univ., 4 (1940), 97-111.  Google Scholar

[12]

Bull. Sci. Math. 63 (1939), 206-212.  Google Scholar

[13]

Journal of the Chinese Mathematical Society, 2 (1940), 247-276.  Google Scholar

[14]

J. Phys. A: Math. Gen., 17 (1984).  Google Scholar

[15]

Ann. Inst. Henri Poincare, 65 (1996), 223-249.  Google Scholar

[16]

Groups, geometry and physics, Monogr. Real Acad. Ci. Exact. Fs.-Qum. Nat. Zaragoza, 29 (2006), 79-92.  Google Scholar

[17]

Lobachevskii Journal of Math., 3 (1999), 39-71.  Google Scholar

[18]

J. Geom. Phys., 56 (2006), 1790-1809. doi: 10.1016/j.geomphys.2005.10.007.  Google Scholar

[19]

Proc. London Math. Soc., 71 (1995), 221-240. doi: 10.1112/plms/s3-71.1.221.  Google Scholar

[20]

R. V. Gamkrelidze (Ed.), "Geometry I,", Encyclopaedia of Math. Sciences, 28 ().   Google Scholar

[21]

Journal of Functional Analysis, 99 (1991), 150-178. doi: 10.1016/0022-1236(91)90057-C.  Google Scholar

[22]

J. Geom. Phys., 60 (2010), 991-1027. doi: 10.1016/j.geomphys.2010.03.003.  Google Scholar

[23]

Ann. Inst. Fourier, Grenoble, 22 (1972), 287-334.  Google Scholar

[24]

Control and Cybernetics, 38 (2009), 1375-1391.  Google Scholar

[25]

B. Jakubczyk and W. Kryński, Relative curvatures of vector fields and their conjugate points,, in preparation., ().   Google Scholar

[26]

Preprint 728, Institute of Mathematics, Polish Academy of Sciences, Warsaw (2010). Google Scholar

[27]

Springer-Verlag, 1972.  Google Scholar

[28]

Quart. J. Math. Oxford Ser., 6 (1935), 1-12. Google Scholar

[29]

Quart. J. Math. Oxford Ser., 7 (1936), 97-104. Google Scholar

[30]

PhD thesis, Institute of Mathematics, Polish Academy of Sciences, 2008 (in Polish). Google Scholar

[31]

Differential Geometry and its Applications, 28 (2010), 523-531. doi: 10.1016/j.difgeo.2010.05.003.  Google Scholar

[32]

Central European Journal of Mathematics, 10 (2012), 1872-1888. doi: 10.2478/s11533-012-0081-z.  Google Scholar

[33]

Diff. Geom. Appl., 29 (2011), 747-757. doi: 10.1016/j.difgeo.2011.08.003.  Google Scholar

[34]

Kluwer Academic Publishers, 1997.  Google Scholar

[35]

J. Geometric Mechanics, 2 (2010), 265-302. doi: 10.3934/jgm.2010.2.265.  Google Scholar

[36]

World Scientific, Singapore, 2001. doi: 10.1142/9789812811622.  Google Scholar

[37]

Prentice-Hall, Englewood Cliffs, N. J., 1964.  Google Scholar

[38]

Comptes Rendus de l'Acad. des Sciences. Ser. I. Math., 328 (1999), 891-894. doi: 10.1016/S0764-4442(99)80292-4.  Google Scholar

[39]

Comptes Rendus de l'Acad. des Sciences. Ser. I. Math., 329 (1999), 425-428. doi: 10.1016/S0764-4442(00)88618-8.  Google Scholar

[40]

Teubner, 1906. Google Scholar

[41]

Journal of Physics A - Mathematical and Theoretical, 40 (2007), 2755-2772. doi: 10.1088/1751-8113/40/11/011.  Google Scholar

[42]

Transform. Groups, 6 (2001), 267-300. doi: 10.1007/BF01263093.  Google Scholar

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