Linear connections for systems of second-order ordinary differential equations

M. Crampin; E. Martínez; W. Sarlet

Annales de l'I.H.P. Physique théorique (1996)

  • Volume: 65, Issue: 2, page 223-249
  • ISSN: 0246-0211

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Crampin, M., Martínez, E., and Sarlet, W.. "Linear connections for systems of second-order ordinary differential equations." Annales de l'I.H.P. Physique théorique 65.2 (1996): 223-249. <http://eudml.org/doc/76741>.

@article{Crampin1996,
author = {Crampin, M., Martínez, E., Sarlet, W.},
journal = {Annales de l'I.H.P. Physique théorique},
keywords = {tangent bundle; jet bundle; horizontal distribution; systems of second-order ordinary differential equations; connections},
language = {eng},
number = {2},
pages = {223-249},
publisher = {Gauthier-Villars},
title = {Linear connections for systems of second-order ordinary differential equations},
url = {http://eudml.org/doc/76741},
volume = {65},
year = {1996},
}

TY - JOUR
AU - Crampin, M.
AU - Martínez, E.
AU - Sarlet, W.
TI - Linear connections for systems of second-order ordinary differential equations
JO - Annales de l'I.H.P. Physique théorique
PY - 1996
PB - Gauthier-Villars
VL - 65
IS - 2
SP - 223
EP - 249
LA - eng
KW - tangent bundle; jet bundle; horizontal distribution; systems of second-order ordinary differential equations; connections
UR - http://eudml.org/doc/76741
ER -

References

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  1. [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. Zbl0760.49021MR1115829
  2. [2] V.I. Arnold, Geometrical Methods in the Theory of Ordinary Differential Equations, Springer Verlag, 1983. Zbl0507.34003MR695786
  3. [3] L. Auslander, On curvature in Finsler geometry, Trans. Amer. Math. Soc., Vol. 79, 1954, pp. 378-388. Zbl0066.16202MR71833
  4. [4] D. Bao and S.S. Chern, On a notable connection in Finsler geometry, Houston J. Math., Vol. 19, 1993, pp. 135-180. Zbl0787.53018MR1218087
  5. [5] A. Bejancu, Finsler Geometry and Applications, Ellis Horwood, 1990. Zbl0702.53001MR1071171
  6. [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. Zbl0851.53008MR1306452
  7. [7] F. Cantrijn, W. Sarlet, A. Vandecasteele and E. Martínez, Complete separability of time-dependent second-order equations, Acta Appl. Math. Zbl0842.34009
  8. [8] M. Crampin, Generalized Bianchi identities for horizontal distributions, Math. Proc. Camb. Phil. Soc., Vol. 94, 1983, pp. 125-132. Zbl0521.53023MR704806
  9. [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. Zbl0545.58020MR748776
  10. [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. Zbl0826.58015MR1269007
  11. [11] J. Douglas, Solution of the inverse problem of the calculus of variations, Trans. Amer. Math. Soc., Vol. 50, 1941, pp. 71-128. Zbl0025.18102MR4740JFM67.1038.01
  12. [12] P. Foulon, Géometrie des équations différentielles du second ordre, Ann. Inst. H. Poincaré, Phys. Théor., Vol. 45, 1986, pp. 1-28. Zbl0624.58011MR856446
  13. [13] P. Foulon, Réductibilité de systèmes dynamiques variationnels, Ann. Inst. H. Poincaré, Phys. Théor., Vol. 45, 1986, pp. 359-388. Zbl0614.70017MR880743
  14. [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. Zbl0806.58029MR1184886
  15. [15] J. Grifone, Structure presque tangente et connections II, Ann. Inst. Fourier (Grenoble), Vol. 22, 1972, pp. 291-338. Zbl0236.53027MR341361
  16. [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. Zbl0671.34012MR980540
  17. [17] S. Kobayashi and K. Nomizu, Foundations of Differential Geometry, Interscience, 1963. Zbl0119.37502MR152974
  18. [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. [19] E. Martínez and J.F. Cariñena, Geometric characterization of linearizable second-order differential equations, Math. Proc. Camb. Phil. Soc, in press. Zbl0851.53009
  20. [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. Zbl0748.58002MR1244454
  21. [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. Zbl0770.53018MR1245556
  22. [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. Zbl0803.34010MR1188830
  23. [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. Zbl0813.70004MR1303184
  24. [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. Zbl1211.58008MR1050526
  25. [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. Zbl0831.58003MR1334841
  26. [26] D.J. Saunders, The Geometry of Jet Bundles, London Mathematical Society Lecture Note Series, Vol. 142, Cambridge University Press, 1989. Zbl0665.58002MR989588
  27. [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. Zbl0957.34009
  28. [28] A. Vondra, Sprays and homogeneous connections on R × TM, Archiv. Math. (Brno), Vol. 28, 1992, pp. 163-173. Zbl0790.53028MR1222283

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