Article
Full entry |
PDF
(0.3 MB)
Feedback
PDF
(0.3 MB)
Feedback
Keywords:
Euler-Lagrange equation; metrizability; projective metrizability; geodesics; spray; formal integrability
Euler-Lagrange equation; metrizability; projective metrizability; geodesics; spray; formal integrability
Summary:
A. Rapcsák obtained necessary and sufficient conditions for the projective Finsler metrizability in terms of a second order partial differential system. In this paper we investigate the integrability of the Rapcsák system and the extended Rapcsák system, by using the Spencer version of the Cartan-Kähler theorem. We also consider the extended Rapcsák system completed with the curvature condition. We prove that in the non-isotropic case there is a nontrivial Spencer cohomology group in the sequences determining the \hbox {2-acyclicity} of the symbol of the corresponding differential operator. Therefore the system is not integrable and higher order obstruction exists.
References:
[1] Bácsó, S., Szilasi, Z.: On the projective theory of sprays. Acta Math. Acad. Paedagog. Nyházi. (N.S.) (electronic only) 26 (2010), 171-207. MR 2754415 | Zbl 1240.53047
[2] Bryant, R. L., Chern, S. S., Gardner, R. B., Goldschmidt, H. L., Griffiths, P. A.: Exterior Differential Systems. Mathematical Sciences Research Institute Publications 18, Springer, New York (1991). DOI 10.1007/978-1-4613-9714-4 | MR 1083148 | Zbl 0726.58002
[3] Bucataru, I., Milkovszki, T., Muzsnay, Z.: Invariant metrizability and projective metrizability on Lie groups and homogeneous spaces. Mediterr. J. Math. (2016), 4567-4580. DOI 10.1007/s00009-016-0762-0 | MR 3564521 | Zbl 1356.53021
[4] Bucataru, I., Muzsnay, Z.: Projective metrizability and formal integrability. SIGMA, Symmetry Integrability Geom. Methods Appl. (electronic only) 7 (2011), Paper 114, 22 pages. DOI 10.3842/SIGMA.2011.114 | MR 2861227 | Zbl 1244.49072
[5] Bucataru, I., Muzsnay, Z.: Projective and Finsler metrizability: parameterization-rigidity of the geodesics. Int. J. Math. 23 (2012), 1250099, 15 pages. DOI 10.1142/S0129167X12500991 | MR 2959445 | Zbl 1263.53070
[6] Crampin, M.: Isotropic and R-flat sprays. Houston J. Math. 33 (2007), 451-459. MR 2308989 | Zbl 1125.53012
[7] Crampin, M.: On the inverse problem for sprays. Publ. Math. 70 (2007), 319-335. MR 2310654 | Zbl 1127.53015
[8] Crampin, M.: Some remarks on the Finslerian version of Hilbert's fourth problem. Houston J. Math. 37 (2011), 369-391. MR 2794554 | Zbl 1228.53085
[9] Crampin, M., Mestdag, T., Saunders, D. J.: The multiplier approach to the projective Finsler metrizability problem. Differ. Geom. Appl. 30 (2012), 604-621. DOI 10.1016/j.difgeo.2012.07.004 | MR 2996856 | Zbl 1257.53105
[10] Crampin, M., Mestdag, T., Saunders, D. J.: Hilbert forms for a Finsler metrizable projective class of sprays. Differ. Geom. Appl. 31 (2013), 63-79. DOI 10.1016/j.difgeo.2012.10.012 | MR 3010078 | Zbl 1262.53064
[11] Do, T., Prince, G.: New progress in the inverse problem in the calculus of variations. Differ. Geom. Appl. 45 (2016), 148-179. DOI 10.1016/j.difgeo.2016.01.005 | MR 3457392 | Zbl 1333.37070
[12] Frölicher, A., Nijenhuis, A.: Theory of vector-valued differential forms. I: Derivations in the graded ring of differential forms. Nederl. Akad. Wet., Proc., Ser. A. 59 (1956), 338-359. MR 0082554 | Zbl 0079.37502
[13] Grifone, J., Muzsnay, Z.: Variational Principles for Second-Order Differential Equations. Application of the Spencer Theory to Characterize Variational Sprays. World Scientific Publishing, Singapore (2000). DOI 10.1142/9789812813596 | MR 1769337 | Zbl 1023.49027
[14] Klein, J., Voutier, A.: Formes extérieures génératrices de sprays. Ann. Inst. Fourier 18 French (1968), 241-260. DOI 10.5802/aif.282 | MR 0247599 | Zbl 0181.49902
[15] Matveev, V. S.: On projective equivalence and pointwise projective relation of Randers metrics. Int. J. Math. 23 (2012), 1250093, 14 pages. DOI 10.1142/S0129167X12500930 | MR 2959439 | Zbl 1253.53018
[16] Mestdag, T.: Finsler geodesics of Lagrangian systems through Routh reduction. Mediterr. J. Math. 13 (2016), 825-839. DOI 10.1007/s00009-014-0505-z | MR 3483865 | Zbl 1338.53103
[17] Muzsnay, Z.: The Euler-Lagrange PDE and Finsler metrizability. Houston J. Math. 32 (2006), 79-98. MR 2202354 | Zbl 1113.53049
[18] Rapcsák, A.: Über die bahntreuen Abbildungen metrischer Räume. Publ. Math. German 8 (1961), 285-290. MR 0138079 | Zbl 0101.39901
[19] 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. Am. Math. Soc. 354 (2002), 2897-2919. DOI 10.1090/S0002-9947-02-02994-X | MR 1895208 | Zbl 1038.37044
[20] Shen, Z.: Differential Geometry of Spray and Finsler Spaces. Kluwer Academic Publishers, Dordrecht (2001). DOI 10.1007/978-94-015-9727-2 | MR 1967666 | Zbl 1009.53004
[21] Szilasi, J., Lovas, R. L., Kertész, D. C.: Connections, Sprays and Finsler Structures. World Scientific Publishing, Hackensack (2014). MR 3156183 | Zbl 06171673
[22] Szilasi, J., Vattamány, S.: On the Finsler-metrizabilities of spray manifolds. Period. Math. Hung. 44 (2002), 81-100. DOI 10.1023/A:1014928103275 | MR 1892276 | Zbl 0997.53056
Search
Partner of
