# American Institute of Mathematical Sciences

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
##### References:
 [1] A. Agrachev, The curvature and hyperbolicity of Hamiltonian systems,, Proceed. Steklov Math. Inst., 256 (2007), 26.  doi: 10.1134/S0081543807010026.  Google Scholar [2] A. Agrachev and R. Gamkrelidze, Feedback-invariant optimal control theory and differential geometry, I. Regular extremals,, J. Dynamical and Control Systems, 3 (1997), 343.  doi: 10.1007/BF02463256.  Google Scholar [3] A. Agrachev, N. Chtcherbakova and I. Zelenko, On curvatures and focal points of dynamical Lagrangian distributions and their reductions by first integrals,, J. of Dynamical and Control Syst., 11 (2005), 297.  doi: 10.1007/s10883-005-6581-4.  Google Scholar [4] C. Boehmer and T. Harko, Nonlinear stability analysis of the Emden-Fowler equation,, J. Nonlinear Math. Phys., 17 (2010), 503.  doi: 10.1142/S1402925110001100.  Google Scholar [5] R. Bryant, Two exotic holonomies in dimension four, path geometries, and twistor theory,, Proc. Sympos. Pure Math., 53 (1991), 33.   Google Scholar [6] I. Bucataru, Linear connections for systems of higher order differential equations,, Houston Journal of Mathematics, 31 (2005), 315.   Google Scholar [7] I. Bucataru, O. A. Constantinescu and M. F. Dahl, A geometric setting for systems of ordinary differential equations,, Int. J. Geom. Methods Mod. Phys., 8 (2011), 1291.  doi: 10.1142/S0219887811005701.  Google Scholar [8] F. Bullo and A. D. Lewis, "Geometric Control of Mechanical Systems,", Springer Verlag, (2004).   Google Scholar [9] E. Cartan, Sur les variétés a connexion projective,, Bull. Soc. Math. France, 52 (1924), 205.   Google Scholar [10] E. Cartan, Observations sur le mémoir précédent,, Math. Z., 37 (1933), 619.  doi: 10.1007/BF01474603.  Google Scholar [11] S.-S. Chern, The geometry of the differential equation $y'''=F(x,y,y',y'')$,, Sci. Rep. Nat. Tsing Hua Univ., 4 (1940), 97.   Google Scholar [12] S.-S. Chern, Sur la géométrie d'un systéme d'équations différentialles du second ordre,, Bull. Sci. Math. 63 (1939), 63 (1939), 206.   Google Scholar [13] S.-S. Chern, The geometry of higher path-spaces,, Journal of the Chinese Mathematical Society, 2 (1940), 247.   Google Scholar [14] M. Crampin, G. Prince and G. Thompson, A geometrical version of the Helmholtz conditions in time-dependent Lagrangian dynamics,, J. Phys. A: Math. Gen., 17 (1984).   Google Scholar [15] M. Crampin, E. Martinez and W. Sarlet, Linear connections for systems of second-order ordinary differential equations,, Ann. Inst. Henri Poincare, 65 (1996), 223.   Google Scholar [16] M. Crampin and D. Saunders, On the geometry of higher-order ordinary differential equations and the Wuenschmann invariant,, Groups, 29 (2006), 79.   Google Scholar [17] B. Doubrov, B. Komrakov and T. Morimoto, Equivalence of holonomic differential equations,, Lobachevskii Journal of Math., 3 (1999), 39.   Google Scholar [18] M. Dunajski and P. Tod, Paraconformal geometry of n-th order ODEs, and exotic holonomy in dimension four,, J. Geom. Phys., 56 (2006), 1790.  doi: 10.1016/j.geomphys.2005.10.007.  Google Scholar [19] M. E. Fels, The equivalence problem for systems of second order ordinary differential equations,, Proc. London Math. Soc., 71 (1995), 221.  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] I. M. Gelfand and I. Zakharevich, Webs, Veronese curves, and bi-Hamiltonian systems,, Journal of Functional Analysis, 99 (1991), 150.  doi: 10.1016/0022-1236(91)90057-C.  Google Scholar [22] M. Godliński and P. Nurowski, $GL(2,R)$ geometry of ODEs,, J. Geom. Phys., 60 (2010), 991.  doi: 10.1016/j.geomphys.2010.03.003.  Google Scholar [23] J. Grifone, Structure presque-tangente et connexions I,, Ann. Inst. Fourier, 22 (1972), 287.   Google Scholar [24] B. Jakubczyk, Curvatures of single-input control systems,, Control and Cybernetics, 38 (2009), 1375.   Google Scholar [25] B. Jakubczyk and W. Kryński, Relative curvatures of vector fields and their conjugate points,, in preparation., ().   Google Scholar [26] B. Jakubczyk and W. Kryński, Vector fields with distributions and invariants of ODEs., Preprint 728, (2010).   Google Scholar [27] S. Kobayashi, "Transformation Groups in Differential Geometry,", Springer-Verlag, (1972).   Google Scholar [28] D. Kosambi, System of differential equations of second order,, Quart. J. Math. Oxford Ser., 6 (1935), 1.   Google Scholar [29] D. Kosambi, Path spaces of higher order,, Quart. J. Math. Oxford Ser., 7 (1936), 97.   Google Scholar [30] W. Kryński, "Equivalence Problems for Tangent Distributions and Ordinary Differential Equations,", PhD thesis, (2008).   Google Scholar [31] W. Kryński, Paraconformal structures and differential equations,, Differential Geometry and its Applications, 28 (2010), 523.  doi: 10.1016/j.difgeo.2010.05.003.  Google Scholar [32] W. Kryński, Geometry of isotypic Kronecker webs,, Central European Journal of Mathematics, 10 (2012), 1872.  doi: 10.2478/s11533-012-0081-z.  Google Scholar [33] T. Mestdag and M. Crampin, Involutive distributions and dynamical systems of second-order type,, Diff. Geom. Appl., 29 (2011), 747.  doi: 10.1016/j.difgeo.2011.08.003.  Google Scholar [34] R. Miron, "The Geometry of Higher-Order Lagrange Spaces,", Kluwer Academic Publishers, (1997).   Google Scholar [35] W. Respondek and S. Ricardo, When is a control system mechanical?,, J. Geometric Mechanics, 2 (2010), 265.  doi: 10.3934/jgm.2010.2.265.  Google Scholar [36] Z. Shen, "Lectures on Finsler Geometry,", World Scientific, (2001).  doi: 10.1142/9789812811622.  Google Scholar [37] S. Sternberg, "Lectures on Differential Geometry,", Prentice-Hall, (1964).   Google Scholar [38] F.-J. Turiel, $C^\infty$-equivalence entre tissus de Veronese et structures bihamiltoniennes,, Comptes Rendus de l'Acad. des Sciences. Ser. I. Math., 328 (1999), 891.  doi: 10.1016/S0764-4442(99)80292-4.  Google Scholar [39] F.-J. Turiel, $C^\infty$-classification des germes de tissus de Veronese,, Comptes Rendus de l'Acad. des Sciences. Ser. I. Math., 329 (1999), 425.  doi: 10.1016/S0764-4442(00)88618-8.  Google Scholar [40] E. Wilczynski, "Projective Differential Geometry of Curves and Rules Surfaces,", Teubner, (1906).   Google Scholar [41] T. Yajima and H. Nagahama, KCC-theory and geometry of the Rikitake system,, Journal of Physics A - Mathematical and Theoretical, 40 (2007), 2755.  doi: 10.1088/1751-8113/40/11/011.  Google Scholar [42] I. Zakharevich, Kronecker webs, bihamiltonian structures, and the method of argument translation,, Transform. Groups, 6 (2001), 267.  doi: 10.1007/BF01263093.  Google Scholar

show all references

##### References:
 [1] A. Agrachev, The curvature and hyperbolicity of Hamiltonian systems,, Proceed. Steklov Math. Inst., 256 (2007), 26.  doi: 10.1134/S0081543807010026.  Google Scholar [2] A. Agrachev and R. Gamkrelidze, Feedback-invariant optimal control theory and differential geometry, I. Regular extremals,, J. Dynamical and Control Systems, 3 (1997), 343.  doi: 10.1007/BF02463256.  Google Scholar [3] A. Agrachev, N. Chtcherbakova and I. Zelenko, On curvatures and focal points of dynamical Lagrangian distributions and their reductions by first integrals,, J. of Dynamical and Control Syst., 11 (2005), 297.  doi: 10.1007/s10883-005-6581-4.  Google Scholar [4] C. Boehmer and T. Harko, Nonlinear stability analysis of the Emden-Fowler equation,, J. Nonlinear Math. Phys., 17 (2010), 503.  doi: 10.1142/S1402925110001100.  Google Scholar [5] R. Bryant, Two exotic holonomies in dimension four, path geometries, and twistor theory,, Proc. Sympos. Pure Math., 53 (1991), 33.   Google Scholar [6] I. Bucataru, Linear connections for systems of higher order differential equations,, Houston Journal of Mathematics, 31 (2005), 315.   Google Scholar [7] I. Bucataru, O. A. Constantinescu and M. F. Dahl, A geometric setting for systems of ordinary differential equations,, Int. J. Geom. Methods Mod. Phys., 8 (2011), 1291.  doi: 10.1142/S0219887811005701.  Google Scholar [8] F. Bullo and A. D. Lewis, "Geometric Control of Mechanical Systems,", Springer Verlag, (2004).   Google Scholar [9] E. Cartan, Sur les variétés a connexion projective,, Bull. Soc. Math. France, 52 (1924), 205.   Google Scholar [10] E. Cartan, Observations sur le mémoir précédent,, Math. Z., 37 (1933), 619.  doi: 10.1007/BF01474603.  Google Scholar [11] S.-S. Chern, The geometry of the differential equation $y'''=F(x,y,y',y'')$,, Sci. Rep. Nat. Tsing Hua Univ., 4 (1940), 97.   Google Scholar [12] S.-S. Chern, Sur la géométrie d'un systéme d'équations différentialles du second ordre,, Bull. Sci. Math. 63 (1939), 63 (1939), 206.   Google Scholar [13] S.-S. Chern, The geometry of higher path-spaces,, Journal of the Chinese Mathematical Society, 2 (1940), 247.   Google Scholar [14] M. Crampin, G. Prince and G. Thompson, A geometrical version of the Helmholtz conditions in time-dependent Lagrangian dynamics,, J. Phys. A: Math. Gen., 17 (1984).   Google Scholar [15] M. Crampin, E. Martinez and W. Sarlet, Linear connections for systems of second-order ordinary differential equations,, Ann. Inst. Henri Poincare, 65 (1996), 223.   Google Scholar [16] M. Crampin and D. Saunders, On the geometry of higher-order ordinary differential equations and the Wuenschmann invariant,, Groups, 29 (2006), 79.   Google Scholar [17] B. Doubrov, B. Komrakov and T. Morimoto, Equivalence of holonomic differential equations,, Lobachevskii Journal of Math., 3 (1999), 39.   Google Scholar [18] M. Dunajski and P. Tod, Paraconformal geometry of n-th order ODEs, and exotic holonomy in dimension four,, J. Geom. Phys., 56 (2006), 1790.  doi: 10.1016/j.geomphys.2005.10.007.  Google Scholar [19] M. E. Fels, The equivalence problem for systems of second order ordinary differential equations,, Proc. London Math. Soc., 71 (1995), 221.  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] I. M. Gelfand and I. Zakharevich, Webs, Veronese curves, and bi-Hamiltonian systems,, Journal of Functional Analysis, 99 (1991), 150.  doi: 10.1016/0022-1236(91)90057-C.  Google Scholar [22] M. Godliński and P. Nurowski, $GL(2,R)$ geometry of ODEs,, J. Geom. Phys., 60 (2010), 991.  doi: 10.1016/j.geomphys.2010.03.003.  Google Scholar [23] J. Grifone, Structure presque-tangente et connexions I,, Ann. Inst. Fourier, 22 (1972), 287.   Google Scholar [24] B. Jakubczyk, Curvatures of single-input control systems,, Control and Cybernetics, 38 (2009), 1375.   Google Scholar [25] B. Jakubczyk and W. Kryński, Relative curvatures of vector fields and their conjugate points,, in preparation., ().   Google Scholar [26] B. Jakubczyk and W. Kryński, Vector fields with distributions and invariants of ODEs., Preprint 728, (2010).   Google Scholar [27] S. Kobayashi, "Transformation Groups in Differential Geometry,", Springer-Verlag, (1972).   Google Scholar [28] D. Kosambi, System of differential equations of second order,, Quart. J. Math. Oxford Ser., 6 (1935), 1.   Google Scholar [29] D. Kosambi, Path spaces of higher order,, Quart. J. Math. Oxford Ser., 7 (1936), 97.   Google Scholar [30] W. Kryński, "Equivalence Problems for Tangent Distributions and Ordinary Differential Equations,", PhD thesis, (2008).   Google Scholar [31] W. Kryński, Paraconformal structures and differential equations,, Differential Geometry and its Applications, 28 (2010), 523.  doi: 10.1016/j.difgeo.2010.05.003.  Google Scholar [32] W. Kryński, Geometry of isotypic Kronecker webs,, Central European Journal of Mathematics, 10 (2012), 1872.  doi: 10.2478/s11533-012-0081-z.  Google Scholar [33] T. Mestdag and M. Crampin, Involutive distributions and dynamical systems of second-order type,, Diff. Geom. Appl., 29 (2011), 747.  doi: 10.1016/j.difgeo.2011.08.003.  Google Scholar [34] R. Miron, "The Geometry of Higher-Order Lagrange Spaces,", Kluwer Academic Publishers, (1997).   Google Scholar [35] W. Respondek and S. Ricardo, When is a control system mechanical?,, J. Geometric Mechanics, 2 (2010), 265.  doi: 10.3934/jgm.2010.2.265.  Google Scholar [36] Z. Shen, "Lectures on Finsler Geometry,", World Scientific, (2001).  doi: 10.1142/9789812811622.  Google Scholar [37] S. Sternberg, "Lectures on Differential Geometry,", Prentice-Hall, (1964).   Google Scholar [38] F.-J. Turiel, $C^\infty$-equivalence entre tissus de Veronese et structures bihamiltoniennes,, Comptes Rendus de l'Acad. des Sciences. Ser. I. Math., 328 (1999), 891.  doi: 10.1016/S0764-4442(99)80292-4.  Google Scholar [39] F.-J. Turiel, $C^\infty$-classification des germes de tissus de Veronese,, Comptes Rendus de l'Acad. des Sciences. Ser. I. Math., 329 (1999), 425.  doi: 10.1016/S0764-4442(00)88618-8.  Google Scholar [40] E. Wilczynski, "Projective Differential Geometry of Curves and Rules Surfaces,", Teubner, (1906).   Google Scholar [41] T. Yajima and H. Nagahama, KCC-theory and geometry of the Rikitake system,, Journal of Physics A - Mathematical and Theoretical, 40 (2007), 2755.  doi: 10.1088/1751-8113/40/11/011.  Google Scholar [42] I. Zakharevich, Kronecker webs, bihamiltonian structures, and the method of argument translation,, Transform. Groups, 6 (2001), 267.  doi: 10.1007/BF01263093.  Google Scholar
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