Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)


SIGMA 8 (2012), 059, 17 pages      arXiv:1203.1716      https://doi.org/10.3842/SIGMA.2012.059

Formal Integrability for the Nonautonomous Case of the Inverse Problem of the Calculus of Variations

Oana Constantinescu
Faculty of Mathematics, Alexandru Ioan Cuza University, Bd. Carol no. 11, 700506, Iasi, Romania

Received March 16, 2012, in final form September 03, 2012; Published online September 06, 2012

Abstract
We address the integrability conditions of the inverse problem of the calculus of variations for time-dependent SODE using the Spencer version of the Cartan-Kähler theorem. We consider a linear partial differential operator P given by the two Helmholtz conditions expressed in terms of semi-basic 1-forms and study its formal integrability. We prove that P is involutive and there is only one obstruction for the formal integrability of this operator. The obstruction is expressed in terms of the curvature tensor R of the induced nonlinear connection. We recover some of the classes of Lagrangian semisprays: flat semisprays, isotropic semisprays and arbitrary semisprays on 2-dimensional manifolds.

Key words: formal integrability; partial differential operators; Lagrangian semisprays; Helmholtz conditions.

pdf (453 kb)   tex (94 kb)

References

  1. Aldridge J.E., Prince G.E., Sarlet W., Thompson G., An EDS approach to the inverse problem in the calculus of variations, J. Math. Phys. 47 (2006), 103508, 22 pages.
  2. Anderson I., Thompson G., The inverse problem of the calculus of variations for ordinary differential equations, Mem. Amer. Math. Soc. 98 (1992), no. 473, 110 pages.
  3. Antonelli P.L., Bucataru I., Volterra-Hamilton production models with discounting: general theory and worked examples, Nonlinear Anal. Real World Appl. 2 (2001), 337-356.
  4. Bryant R.L., Chern S.S., Gardner R.B., Goldschmidt H.L., Griffiths P.A., Exterior differential systems, Mathematical Sciences Research Institute Publications, Vol. 18, Springer-Verlag, New York, 1991.
  5. Bucataru I., Constantinescu O., Helmholtz conditions and symmetries for the time dependent case of the inverse problem of the calculus of variations, J. Geom. Phys. 60 (2010), 1710-1725, arXiv:0908.1631.
  6. Bucataru I., Dahl M.F., Semi-basic 1-forms and Helmholtz conditions for the inverse problem of the calculus of variations, J. Geom. Mech. 1 (2009), 159-180, arXiv:0903.1169.
  7. Bucataru I., Muzsnay Z., Projective metrizability and formal integrability, SIGMA 7 (2011), 114, 22 pages, arXiv:1105.2142.
  8. Cantrijn F., Cariñena J.F., Crampin M., Ibort L.A., Reduction of degenerate Lagrangian systems, J. Geom. Phys. 3 (1986), 353-400.
  9. Cantrijn F., Sarlet W., Vandecasteele A., Martínez E., Complete separability of time-dependent second-order ordinary differential equations, Acta Appl. Math. 42 (1996), 309-334.
  10. Cariñena J.F., Gràcia X., Marmo G., Martínez E., Muñoz-Lecanda M.C., Román-Roy N., Geometric Hamilton-Jacobi theory, Int. J. Geom. Methods Mod. Phys. 3 (2006), 1417-1458, math-ph/0604063.
  11. Cariñena J.F., Martínez E., Symmetry theory and Lagrangian inverse problem for time-dependent second-order differential equations, J. Phys. A: Math. Gen. 22 (1989), 2659-2665.
  12. Cartan É., Les systèmes différentiels extérieurs et leurs applications géométriques, Actualités Sci. Ind., no. 994, Hermann et Cie., Paris, 1945.
  13. Crampin M., On the differential geometry of the Euler-Lagrange equations, and the inverse problem of Lagrangian dynamics, J. Phys. A: Math. Gen. 14 (1981), 2567-2575.
  14. Crampin M., Prince G.E., Sarlet W., Thompson G., The inverse problem of the calculus of variations: separable systems, Acta Appl. Math. 57 (1999), 239-254.
  15. Crampin M., Prince G.E., Thompson G., A geometrical version of the Helmholtz conditions in time-dependent Lagrangian dynamics, J. Phys. A: Math. Gen. 17 (1984), 1437-1447.
  16. Crampin M., Sarlet W., Martínez E., Byrnes G.B., Prince G.E., Towards a geometrical understanding of Douglas' solution of the inverse problem of the calculus of variations, Inverse Problems 10 (1994), 245-260.
  17. Darboux G., Leçons sur la théorie générale des surfaces et les applications géométriques du calcul infinitésimal, Gauthier-Villars, Paris, 1894.
  18. Davis D.R., The inverse problem of the calculus of variations in a space of (n+1) dimensions, Bull. Amer. Math. Soc. 35 (1929), 371-380.
  19. de Leon M., Rodrigues P.R., Dynamical connections and non-autonomous Lagrangian systems, Ann. Fac. Sci. Toulouse Math. (5) 9 (1988), 171-181.
  20. Douglas J., Solution of the inverse problem of the calculus of variations, Trans. Amer. Math. Soc. 50 (1941), 71-128.
  21. Frölicher A., Nijenhuis A., Theory of vector-valued differential forms. I. Derivations of the graded ring of differential forms, Nederl. Akad. Wetensch. Proc. Ser. A 59 (1956), 338-359.
  22. Goldschmidt H., Integrability criteria for systems of nonlinear partial differential equations, J. Differential Geometry 1 (1967), 269-307.
  23. Grifone J., Muzsnay Z., Variational principles for second-order differential equations. Application of the Spencer theory to characterize variational sprays, World Scientific Publishing Co. Inc., River Edge, NJ, 2000.
  24. Klein J., Espaces variationnels et mécanique, Ann. Inst. Fourier (Grenoble) 12 (1962), 1-124.
  25. Kolár I., Michor P.W., Slovák J., Natural operations in differential geometry, Springer-Verlag, Berlin, 1993.
  26. Kosambi D., Systems of differential equations of the second order, Q. J. Math. 6 (1935), 1-12.
  27. Krupková O., The geometry of ordinary variational equations, Lecture Notes in Mathematics, Vol. 1678, Springer-Verlag, Berlin, 1997.
  28. Krupková O., Prince G.E., Second order ordinary differential equations in jet bundles and the inverse problem of the calculus of variations, in Handbook of Global Analysis, Elsevier Sci. B.V., Amsterdam, 2008, 837-904.
  29. Massa E., Pagani E., Jet bundle geometry, dynamical connections, and the inverse problem of Lagrangian mechanics, Ann. Inst. H. Poincaré Phys. Théor. 61 (1994), 17-62.
  30. Prástaro A., (Co)bordism groups in PDEs, Acta Appl. Math. 59 (1999), 111-201.
  31. Prástaro A., Geometry of PDE's. II. Variational PDE's and integral bordism groups, J. Math. Anal. Appl. 321 (2006), 930-948.
  32. Riquier C., Les systèmes d'équations aux dérivées partielles, Gauthier-Villars, Paris, 1910.
  33. Santilli R.M., Foundations of theoretical mechanics. I. The inverse problem in Newtonian mechanics, Texts and Monographs in Physics, Springer-Verlag, New York - Heidelberg, 1978.
  34. Sarlet W., Symmetries, first integrals and the inverse problem of Lagrangian mechanics, J. Phys. A: Math. Gen. 14 (1981), 2227-2238.
  35. Sarlet W., The Helmholtz conditions revisited. A new approach to the inverse problem of Lagrangian dynamics, J. Phys. A: Math. Gen. 15 (1982), 1503-1517.
  36. Sarlet W., Crampin M., Martínez E., The integrability conditions in the inverse problem of the calculus of variations for second-order ordinary differential equations, Acta Appl. Math. 54 (1998), 233-273.
  37. Sarlet W., Thompson G., Prince G.E., The inverse problem of the calculus of variations: the use of geometrical calculus in Douglas's analysis, Trans. Amer. Math. Soc. 354 (2002), 2897-2919.
  38. Sarlet W., Vandecasteele A., Cantrijn F., Martínez E., Derivations of forms along a map: the framework for time-dependent second-order equations, Differential Geom. Appl. 5 (1995), 171-203.
  39. Saunders D.J., The geometry of jet bundles, London Mathematical Society Lecture Note Series, Vol. 142, Cambridge University Press, Cambridge, 1989.
  40. Spencer D.C., Overdetermined systems of linear partial differential equations, Bull. Amer. Math. Soc. 75 (1969), 179-239.


Previous article  Next article   Contents of Volume 8 (2012)