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

Vector fields with distributions and invariants of ODEs

BronisŁaw Jakubczyk 1, and  Wojciech Kryński 1,

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.
Keywords: equivalence, distribution, ordinary differential equations, connection, G-structure, Vector field, semispray, invariants., control system.
Mathematics Subject Classification: Primary: 34C14, 53A55; Secondary: 58A3.
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), 26-46. doi: 10.1134/S0081543807010026.

[2]

A. Agrachev and R. Gamkrelidze, Feedback-invariant optimal control theory and differential geometry, I. Regular extremals, J. Dynamical and Control Systems, 3 (1997), 343-389. doi: 10.1007/BF02463256.

[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), 297-327. doi: 10.1007/s10883-005-6581-4.

[4]

C. Boehmer and T. Harko, Nonlinear stability analysis of the Emden-Fowler equation, J. Nonlinear Math. Phys., 17 (2010), 503-516. doi: 10.1142/S1402925110001100.

[5]

R. Bryant, Two exotic holonomies in dimension four, path geometries, and twistor theory, Proc. Sympos. Pure Math., 53 (1991), 33-88.

[6]

I. Bucataru, Linear connections for systems of higher order differential equations, Houston Journal of Mathematics, 31 (2005), 315-332.

[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), 1291-1327. doi: 10.1142/S0219887811005701.

[8]

F. Bullo and A. D. Lewis, "Geometric Control of Mechanical Systems," Springer Verlag, New York, 2004.

[9]

E. Cartan, Sur les variétés a connexion projective, Bull. Soc. Math. France, 52 (1924), 205-241.

[10]

E. Cartan, Observations sur le mémoir précédent, Math. Z., 37 (1933), 619-622. doi: 10.1007/BF01474603.

[11]

S.-S. Chern, The geometry of the differential equation $y'''=F(x,y,y',y'')$, Sci. Rep. Nat. Tsing Hua Univ., 4 (1940), 97-111.

[12]

S.-S. Chern, Sur la géométrie d'un systéme d'équations différentialles du second ordre, Bull. Sci. Math. 63 (1939), 206-212.

[13]

S.-S. Chern, The geometry of higher path-spaces, Journal of the Chinese Mathematical Society, 2 (1940), 247-276.

[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).

[15]

M. Crampin, E. Martinez and W. Sarlet, Linear connections for systems of second-order ordinary differential equations, Ann. Inst. Henri Poincare, 65 (1996), 223-249.

[16]

M. Crampin and D. Saunders, On the geometry of higher-order ordinary differential equations and the Wuenschmann invariant, Groups, geometry and physics, Monogr. Real Acad. Ci. Exact. Fs.-Qum. Nat. Zaragoza, 29 (2006), 79-92.

[17]

B. Doubrov, B. Komrakov and T. Morimoto, Equivalence of holonomic differential equations, Lobachevskii Journal of Math., 3 (1999), 39-71.

[18]

M. Dunajski and P. Tod, Paraconformal geometry of n-th order ODEs, and exotic holonomy in dimension four, J. Geom. Phys., 56 (2006), 1790-1809. doi: 10.1016/j.geomphys.2005.10.007.

[19]

M. E. Fels, The equivalence problem for systems of second order ordinary differential equations, Proc. London Math. Soc., 71 (1995), 221-240. doi: 10.1112/plms/s3-71.1.221.

[20]

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

[21]

I. M. Gelfand and I. Zakharevich, Webs, Veronese curves, and bi-Hamiltonian systems, Journal of Functional Analysis, 99 (1991), 150-178. doi: 10.1016/0022-1236(91)90057-C.

[22]

M. Godliński and P. Nurowski, $GL(2,R)$ geometry of ODEs, J. Geom. Phys., 60 (2010), 991-1027. doi: 10.1016/j.geomphys.2010.03.003.

[23]

J. Grifone, Structure presque-tangente et connexions I, Ann. Inst. Fourier, Grenoble, 22 (1972), 287-334.

[24]

B. Jakubczyk, Curvatures of single-input control systems, Control and Cybernetics, 38 (2009), 1375-1391.

[25]

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

[26]

B. Jakubczyk and W. Kryński, Vector fields with distributions and invariants of ODEs. Preprint 728, Institute of Mathematics, Polish Academy of Sciences, Warsaw (2010).

[27]

S. Kobayashi, "Transformation Groups in Differential Geometry," Springer-Verlag, 1972.

[28]

D. Kosambi, System of differential equations of second order, Quart. J. Math. Oxford Ser., 6 (1935), 1-12.

[29]

D. Kosambi, Path spaces of higher order, Quart. J. Math. Oxford Ser., 7 (1936), 97-104.

[30]

W. Kryński, "Equivalence Problems for Tangent Distributions and Ordinary Differential Equations," PhD thesis, Institute of Mathematics, Polish Academy of Sciences, 2008 (in Polish).

[31]

W. Kryński, Paraconformal structures and differential equations, Differential Geometry and its Applications, 28 (2010), 523-531. doi: 10.1016/j.difgeo.2010.05.003.

[32]

W. Kryński, Geometry of isotypic Kronecker webs, Central European Journal of Mathematics, 10 (2012), 1872-1888. doi: 10.2478/s11533-012-0081-z.

[33]

T. Mestdag and M. Crampin, Involutive distributions and dynamical systems of second-order type, Diff. Geom. Appl., 29 (2011), 747-757. doi: 10.1016/j.difgeo.2011.08.003.

[34]

R. Miron, "The Geometry of Higher-Order Lagrange Spaces," Kluwer Academic Publishers, 1997.

[35]

W. Respondek and S. Ricardo, When is a control system mechanical?, J. Geometric Mechanics, 2 (2010), 265-302. doi: 10.3934/jgm.2010.2.265.

[36]

Z. Shen, "Lectures on Finsler Geometry," World Scientific, Singapore, 2001. doi: 10.1142/9789812811622.

[37]

S. Sternberg, "Lectures on Differential Geometry," Prentice-Hall, Englewood Cliffs, N. J., 1964.

[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), 891-894. doi: 10.1016/S0764-4442(99)80292-4.

[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), 425-428. doi: 10.1016/S0764-4442(00)88618-8.

[40]

E. Wilczynski, "Projective Differential Geometry of Curves and Rules Surfaces," Teubner, 1906.

[41]

T. Yajima and H. Nagahama, KCC-theory and geometry of the Rikitake system, Journal of Physics A - Mathematical and Theoretical, 40 (2007), 2755-2772. doi: 10.1088/1751-8113/40/11/011.

[42]

I. Zakharevich, Kronecker webs, bihamiltonian structures, and the method of argument translation, Transform. Groups, 6 (2001), 267-300. doi: 10.1007/BF01263093.

show all references

References:
[1]

A. Agrachev, The curvature and hyperbolicity of Hamiltonian systems, Proceed. Steklov Math. Inst., 256 (2007), 26-46. doi: 10.1134/S0081543807010026.

[2]

A. Agrachev and R. Gamkrelidze, Feedback-invariant optimal control theory and differential geometry, I. Regular extremals, J. Dynamical and Control Systems, 3 (1997), 343-389. doi: 10.1007/BF02463256.

[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), 297-327. doi: 10.1007/s10883-005-6581-4.

[4]

C. Boehmer and T. Harko, Nonlinear stability analysis of the Emden-Fowler equation, J. Nonlinear Math. Phys., 17 (2010), 503-516. doi: 10.1142/S1402925110001100.

[5]

R. Bryant, Two exotic holonomies in dimension four, path geometries, and twistor theory, Proc. Sympos. Pure Math., 53 (1991), 33-88.

[6]

I. Bucataru, Linear connections for systems of higher order differential equations, Houston Journal of Mathematics, 31 (2005), 315-332.

[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), 1291-1327. doi: 10.1142/S0219887811005701.

[8]

F. Bullo and A. D. Lewis, "Geometric Control of Mechanical Systems," Springer Verlag, New York, 2004.

[9]

E. Cartan, Sur les variétés a connexion projective, Bull. Soc. Math. France, 52 (1924), 205-241.

[10]

E. Cartan, Observations sur le mémoir précédent, Math. Z., 37 (1933), 619-622. doi: 10.1007/BF01474603.

[11]

S.-S. Chern, The geometry of the differential equation $y'''=F(x,y,y',y'')$, Sci. Rep. Nat. Tsing Hua Univ., 4 (1940), 97-111.

[12]

S.-S. Chern, Sur la géométrie d'un systéme d'équations différentialles du second ordre, Bull. Sci. Math. 63 (1939), 206-212.

[13]

S.-S. Chern, The geometry of higher path-spaces, Journal of the Chinese Mathematical Society, 2 (1940), 247-276.

[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).

[15]

M. Crampin, E. Martinez and W. Sarlet, Linear connections for systems of second-order ordinary differential equations, Ann. Inst. Henri Poincare, 65 (1996), 223-249.

[16]

M. Crampin and D. Saunders, On the geometry of higher-order ordinary differential equations and the Wuenschmann invariant, Groups, geometry and physics, Monogr. Real Acad. Ci. Exact. Fs.-Qum. Nat. Zaragoza, 29 (2006), 79-92.

[17]

B. Doubrov, B. Komrakov and T. Morimoto, Equivalence of holonomic differential equations, Lobachevskii Journal of Math., 3 (1999), 39-71.

[18]

M. Dunajski and P. Tod, Paraconformal geometry of n-th order ODEs, and exotic holonomy in dimension four, J. Geom. Phys., 56 (2006), 1790-1809. doi: 10.1016/j.geomphys.2005.10.007.

[19]

M. E. Fels, The equivalence problem for systems of second order ordinary differential equations, Proc. London Math. Soc., 71 (1995), 221-240. doi: 10.1112/plms/s3-71.1.221.

[20]

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

[21]

I. M. Gelfand and I. Zakharevich, Webs, Veronese curves, and bi-Hamiltonian systems, Journal of Functional Analysis, 99 (1991), 150-178. doi: 10.1016/0022-1236(91)90057-C.

[22]

M. Godliński and P. Nurowski, $GL(2,R)$ geometry of ODEs, J. Geom. Phys., 60 (2010), 991-1027. doi: 10.1016/j.geomphys.2010.03.003.

[23]

J. Grifone, Structure presque-tangente et connexions I, Ann. Inst. Fourier, Grenoble, 22 (1972), 287-334.

[24]

B. Jakubczyk, Curvatures of single-input control systems, Control and Cybernetics, 38 (2009), 1375-1391.

[25]

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

[26]

B. Jakubczyk and W. Kryński, Vector fields with distributions and invariants of ODEs. Preprint 728, Institute of Mathematics, Polish Academy of Sciences, Warsaw (2010).

[27]

S. Kobayashi, "Transformation Groups in Differential Geometry," Springer-Verlag, 1972.

[28]

D. Kosambi, System of differential equations of second order, Quart. J. Math. Oxford Ser., 6 (1935), 1-12.

[29]

D. Kosambi, Path spaces of higher order, Quart. J. Math. Oxford Ser., 7 (1936), 97-104.

[30]

W. Kryński, "Equivalence Problems for Tangent Distributions and Ordinary Differential Equations," PhD thesis, Institute of Mathematics, Polish Academy of Sciences, 2008 (in Polish).

[31]

W. Kryński, Paraconformal structures and differential equations, Differential Geometry and its Applications, 28 (2010), 523-531. doi: 10.1016/j.difgeo.2010.05.003.

[32]

W. Kryński, Geometry of isotypic Kronecker webs, Central European Journal of Mathematics, 10 (2012), 1872-1888. doi: 10.2478/s11533-012-0081-z.

[33]

T. Mestdag and M. Crampin, Involutive distributions and dynamical systems of second-order type, Diff. Geom. Appl., 29 (2011), 747-757. doi: 10.1016/j.difgeo.2011.08.003.

[34]

R. Miron, "The Geometry of Higher-Order Lagrange Spaces," Kluwer Academic Publishers, 1997.

[35]

W. Respondek and S. Ricardo, When is a control system mechanical?, J. Geometric Mechanics, 2 (2010), 265-302. doi: 10.3934/jgm.2010.2.265.

[36]

Z. Shen, "Lectures on Finsler Geometry," World Scientific, Singapore, 2001. doi: 10.1142/9789812811622.

[37]

S. Sternberg, "Lectures on Differential Geometry," Prentice-Hall, Englewood Cliffs, N. J., 1964.

[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), 891-894. doi: 10.1016/S0764-4442(99)80292-4.

[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), 425-428. doi: 10.1016/S0764-4442(00)88618-8.

[40]

E. Wilczynski, "Projective Differential Geometry of Curves and Rules Surfaces," Teubner, 1906.

[41]

T. Yajima and H. Nagahama, KCC-theory and geometry of the Rikitake system, Journal of Physics A - Mathematical and Theoretical, 40 (2007), 2755-2772. doi: 10.1088/1751-8113/40/11/011.

[42]

I. Zakharevich, Kronecker webs, bihamiltonian structures, and the method of argument translation, Transform. Groups, 6 (2001), 267-300. doi: 10.1007/BF01263093.

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