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Geometric dynamics on the automorphism group of principal bundles: Geodesic flows, dual pairs and chromomorphism groups
Vector fields with distributions and invariants of ODEs
1. | Institute of Mathematics, Polish Academy of Sciences, ul. Śniadeckich 8, 00-956 Warszawa, Poland, Poland |
References:
[1] |
A. Agrachev, The curvature and hyperbolicity of Hamiltonian systems,, Proceed. Steklov Math. Inst., 256 (2007), 26.
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.
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.
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.
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.
|
[6] |
I. Bucataru, Linear connections for systems of higher order differential equations,, Houston Journal of Mathematics, 31 (2005), 315.
|
[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. |
[8] |
F. Bullo and A. D. Lewis, "Geometric Control of Mechanical Systems,", Springer Verlag, (2004).
|
[9] |
E. Cartan, Sur les variétés a connexion projective,, Bull. Soc. Math. France, 52 (1924), 205.
|
[10] |
E. Cartan, Observations sur le mémoir précédent,, Math. Z., 37 (1933), 619.
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.
|
[12] |
S.-S. Chern, Sur la géométrie d'un systéme d'équations différentialles du second ordre,, Bull. Sci. Math. 63 (1939), 63 (1939), 206.
|
[13] |
S.-S. Chern, The geometry of higher path-spaces,, Journal of the Chinese Mathematical Society, 2 (1940), 247.
|
[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.
|
[16] |
M. Crampin and D. Saunders, On the geometry of higher-order ordinary differential equations and the Wuenschmann invariant,, Groups, 29 (2006), 79.
|
[17] |
B. Doubrov, B. Komrakov and T. Morimoto, Equivalence of holonomic differential equations,, Lobachevskii Journal of Math., 3 (1999), 39.
|
[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. |
[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. |
[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.
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.
doi: 10.1016/j.geomphys.2010.03.003. |
[23] |
J. Grifone, Structure presque-tangente et connexions I,, Ann. Inst. Fourier, 22 (1972), 287.
|
[24] |
B. Jakubczyk, Curvatures of single-input control systems,, Control and Cybernetics, 38 (2009), 1375.
|
[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).
|
[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. |
[32] |
W. Kryński, Geometry of isotypic Kronecker webs,, Central European Journal of Mathematics, 10 (2012), 1872.
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.
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.
doi: 10.3934/jgm.2010.2.265. |
[36] |
Z. Shen, "Lectures on Finsler Geometry,", World Scientific, (2001).
doi: 10.1142/9789812811622. |
[37] |
S. Sternberg, "Lectures on Differential Geometry,", Prentice-Hall, (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.
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.
doi: 10.1016/S0764-4442(00)88618-8. |
[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. |
[42] |
I. Zakharevich, Kronecker webs, bihamiltonian structures, and the method of argument translation,, Transform. Groups, 6 (2001), 267.
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.
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.
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.
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.
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.
|
[6] |
I. Bucataru, Linear connections for systems of higher order differential equations,, Houston Journal of Mathematics, 31 (2005), 315.
|
[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. |
[8] |
F. Bullo and A. D. Lewis, "Geometric Control of Mechanical Systems,", Springer Verlag, (2004).
|
[9] |
E. Cartan, Sur les variétés a connexion projective,, Bull. Soc. Math. France, 52 (1924), 205.
|
[10] |
E. Cartan, Observations sur le mémoir précédent,, Math. Z., 37 (1933), 619.
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.
|
[12] |
S.-S. Chern, Sur la géométrie d'un systéme d'équations différentialles du second ordre,, Bull. Sci. Math. 63 (1939), 63 (1939), 206.
|
[13] |
S.-S. Chern, The geometry of higher path-spaces,, Journal of the Chinese Mathematical Society, 2 (1940), 247.
|
[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.
|
[16] |
M. Crampin and D. Saunders, On the geometry of higher-order ordinary differential equations and the Wuenschmann invariant,, Groups, 29 (2006), 79.
|
[17] |
B. Doubrov, B. Komrakov and T. Morimoto, Equivalence of holonomic differential equations,, Lobachevskii Journal of Math., 3 (1999), 39.
|
[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. |
[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. |
[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.
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.
doi: 10.1016/j.geomphys.2010.03.003. |
[23] |
J. Grifone, Structure presque-tangente et connexions I,, Ann. Inst. Fourier, 22 (1972), 287.
|
[24] |
B. Jakubczyk, Curvatures of single-input control systems,, Control and Cybernetics, 38 (2009), 1375.
|
[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).
|
[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. |
[32] |
W. Kryński, Geometry of isotypic Kronecker webs,, Central European Journal of Mathematics, 10 (2012), 1872.
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.
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.
doi: 10.3934/jgm.2010.2.265. |
[36] |
Z. Shen, "Lectures on Finsler Geometry,", World Scientific, (2001).
doi: 10.1142/9789812811622. |
[37] |
S. Sternberg, "Lectures on Differential Geometry,", Prentice-Hall, (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.
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.
doi: 10.1016/S0764-4442(00)88618-8. |
[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. |
[42] |
I. Zakharevich, Kronecker webs, bihamiltonian structures, and the method of argument translation,, Transform. Groups, 6 (2001), 267.
doi: 10.1007/BF01263093. |
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