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Linear connections for systems of second-order ordinary differential equations. (English) Zbl 0912.58002

In this interesting and important paper the authors develop a geometrical approach to the study of systems of second-order ordinary differential equations of the form \[ \ddot x^i= f^i(t,x^j,\dot x^j). \] Such a system of equations may be represented as a certain type of vector field on a differentiable manifold of the form \(\mathbb{R}\times TM\), where \(M\) is a manifold and \(TM\) is its tangent bundle. This geometrical approach to tackling many problems is encountered in the study of systems of second-order ODE’s, for example, in problems concerning conditions for the existence of coordinates with respect to which the equations take a special form – in which the right-hand sides vanish, or are linear, or in which the equations decouple, and also in problems concerned with the qualitative behaviour of families of solutions. It is shown that a convenient linear connection is a very effective tool for the investigation of problems of the kind described above. In particular, the vanishing of the curvature of the connection is a necessary and sufficient condition for the existence of coordinates with respect to which the solution curves of the equations are straight lines.
The paper is well written.
Reviewer: A.Klíč (Praha)

MSC:

58A20 Jets in global analysis
37-XX Dynamical systems and ergodic theory
34A26 Geometric methods in ordinary differential equations
53C07 Special connections and metrics on vector bundles (Hermite-Einstein, Yang-Mills)
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References:

[1] I. Anderson and G. Thompson , The Inverse Problem of the Calculus of Variations for Ordinary Differential Equations , Memoirs of the American Mathematical Society , Vol. 98 , 1992 . MR 1115829 | Zbl 0760.49021 · Zbl 0760.49021
[2] V.I. Arnold , Geometrical Methods in the Theory of Ordinary Differential Equations , Springer Verlag , 1983 . MR 695786 | Zbl 0507.34003 · Zbl 0507.34003
[3] L. Auslander , On curvature in Finsler geometry , Trans. Amer. Math. Soc. , Vol. 79 , 1954 , pp. 378 - 388 . MR 71833 | Zbl 0066.16202 · Zbl 0066.16202 · doi:10.2307/1993036
[4] D. Bao and S.S. Chern , On a notable connection in Finsler geometry , Houston J. Math. , Vol. 19 , 1993 , pp. 135 - 180 . MR 1218087 | Zbl 0787.53018 · Zbl 0787.53018
[5] A. Bejancu , Finsler Geometry and Applications , Ellis Horwood , 1990 . MR 1071171 | Zbl 0702.53001 · Zbl 0702.53001
[6] G. Byrnes , A complete set of Bianchi identities for tensor fields along the tangent bundle projection , J. Phys. A: Math. Gen. , Vol. 27 , 1994 , pp. 6617 - 6632 . MR 1306452 | Zbl 0851.53008 · Zbl 0851.53008 · doi:10.1088/0305-4470/27/19/030
[7] F. Cantrijn , W. Sarlet , A. Vandecasteele and E. Martínez , Complete separability of time-dependent second-order equations , Acta Appl. Math. Zbl 0842.34009 · Zbl 0842.34009 · doi:10.1007/BF01064171
[8] M. Crampin , Generalized Bianchi identities for horizontal distributions , Math. Proc. Camb. Phil. Soc. , Vol. 94 , 1983 , pp. 125 - 132 . MR 704806 | Zbl 0521.53023 · Zbl 0521.53023 · doi:10.1017/S0305004100060953
[9] M. Crampin , G.E. Prince and G. Thompson , A geometrical version of the Helmholtz conditions in time-dependent Lagrangian dynamics , Phys. A: Math. Gen. , Vol. 17 , 1984 , pp. 1437 - 1447 . MR 748776 | Zbl 0545.58020 · Zbl 0545.58020 · doi:10.1088/0305-4470/17/7/011
[10] M. Crampin , W. Sarlet , E. Martínez , G. Byrnes and G.E. Prince , Towards a geometrical understanding of Douglas’s solution of the inverse problem of the calculus of variations , Inverse Problems , Vol. 10 , 1994 , pp. 245 - 260 . MR 1269007 | Zbl 0826.58015 · Zbl 0826.58015 · doi:10.1088/0266-5611/10/2/005
[11] J. Douglas , Solution of the inverse problem of the calculus of variations , Trans. Amer. Math. Soc. , Vol. 50 , 1941 , pp. 71 - 128 . MR 4740 | Zbl 0025.18102 | JFM 67.1038.01 · Zbl 0025.18102 · doi:10.2307/1989912
[12] P. Foulon , Géometrie des équations différentielles du second ordre , Ann. Inst. H. Poincaré , Phys. Théor. , Vol. 45 , 1986 , pp. 1 - 28 . Numdam | MR 856446 | Zbl 0624.58011 · Zbl 0624.58011
[13] P. Foulon , Réductibilité de systèmes dynamiques variationnels , Ann. Inst. H. Poincaré , Phys. Théor. , Vol. 45 , 1986 , pp. 359 - 388 . Numdam | MR 880743 | Zbl 0614.70017 · Zbl 0614.70017
[14] P. Foulon , Estimation de l’entropie des systèmes Lagrangiens sans points conjugués , Ann. Inst. H. Poincaré , Phys. Théor. , Vol. 55 , 1991 , pp. 117 - 146 . Numdam | MR 1184886 | Zbl 0806.58029 · Zbl 0806.58029
[15] J. Grifone , Structure presque tangente et connections II , Ann. Inst. Fourier (Grenoble) , Vol. 22 , 1972 , pp. 291 - 338 . Numdam | MR 341361 | Zbl 0236.53027 · Zbl 0236.53027 · doi:10.5802/aif.431
[16] C. Grissom , G. Thompson and G. Wilkens , Linearization of second order ordinary differential equations via Cartan’s equivalence method , J. Diff. Equations , Vol. 77 , 1989 , pp. 1 - 15 . MR 980540 | Zbl 0671.34012 · Zbl 0671.34012 · doi:10.1016/0022-0396(89)90154-X
[17] S. Kobayashi and K. Nomizu , Foundations of Differential Geometry , Interscience , 1963 . MR 152974 | Zbl 0119.37502 · Zbl 0119.37502
[18] E. Martínez , Geometría de Ecuaciones Diferenciales Aplicada a la Mecánica , Thesis, University of Zaragoza , Spain ; Publicaciones del Seminario García Galdeano , Vol. 36 , 1991 .
[19] E. Martínez and J.F. Cariñena , Geometric characterization of linearizable second-order differential equations , Math. Proc. Camb. Phil. Soc , in press. Zbl 0851.53009 · Zbl 0851.53009 · doi:10.1017/S0305004100074235
[20] E. Martínez , J.F. Cariñena and W. Sarlet , Derivations of differential forms along the tangent bundle projection , Diff. Geom. Appl. , Vol. 2 , 1992 , pp. 17 - 43 . MR 1244454 | Zbl 0748.58002 · Zbl 0748.58002 · doi:10.1016/0926-2245(92)90007-A
[21] E. Martínez , J.F. Cariñena and W. Sarlet , Derivations of differential forms along the tangent bundle projection II , Diff. Geom. Appl. , Vol. 3 , 1993 , pp. 1 - 29 . MR 1245556 | Zbl 0770.53018 · Zbl 0770.53018 · doi:10.1016/0926-2245(93)90020-2
[22] E. Martínez , J.F. Cariñena and W. Sarlet , Geometric characterization of separable second-order equations , Math. Proc. Camb. Phil. Soc. , Vol. 113 , 1993 , pp. 205 - 224 . MR 1188830 | Zbl 0803.34010 · Zbl 0803.34010 · doi:10.1017/S0305004100075897
[23] E. Massa and E. Pagani , Jet bundle geometry, dynamical connections, and the inverse problem of Lagrangian mechanics , Ann. Inst. H. Poincaré, Phys. Théor. , Vol. 61 , 1994 , pp. 17 - 62 . Numdam | MR 1303184 | Zbl 0813.70004 · Zbl 0813.70004
[24] G. Morandi , C. Ferrario , G. , Lo Vecchio , G. Marmo and C. Rubano , The inverse problem in the calculus of variations and the geometry of the tangent bundle , Phys. Rep. , Vol. 188 , 1990 , pp. 147 - 284 . MR 1050526 · Zbl 1211.58008 · doi:10.1016/0370-1573(90)90137-Q
[25] W. Sarlet , A. Vandecasteele , F. Cantrijn and E. Martínez , Derivations of forms along a map: the framework for time-dependent second-order equations , Diff. Geom. Appl. , Vol. 5 , 1995 , pp. 171 - 203 . MR 1334841 | Zbl 0831.58003 · Zbl 0831.58003 · doi:10.1016/0926-2245(95)00013-T
[26] D.J. Saunders , The Geometry of Jet Bundles , London Mathematical Society Lecture Note Series , Vol. 142 , Cambridge University Press , 1989 . MR 989588 | Zbl 0665.58002 · Zbl 0665.58002
[27] Y.R. Romanovsky , On differential equations and Cartan’s projective connections , Geometry in Partial Differential Equations , ed A. Pràstaro and T. M. Rassias, World Scientific , 1994 , pp. 329 - 344 . Zbl 0957.34009 · Zbl 0957.34009
[28] A. Vondra , Sprays and homogeneous connections on R \times TM , Archiv. Math. (Brno) , Vol. 28 , 1992 , pp. 163 - 173 . Article | MR 1222283 | Zbl 0790.53028 · Zbl 0790.53028
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