| Title:
|
On the projective Finsler metrizability and the integrability of Rapcsák equation (English) |
| Author:
|
Milkovszki, Tamás |
| Author:
|
Muzsnay, Zoltán |
| Language:
|
English |
| Journal:
|
Czechoslovak Mathematical Journal |
| ISSN:
|
0011-4642 (print) |
| ISSN:
|
1572-9141 (online) |
| Volume:
|
67 |
| Issue:
|
2 |
| Year:
|
2017 |
| Pages:
|
469-495 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| 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. (English) |
| Keyword:
|
Euler-Lagrange equation |
| Keyword:
|
metrizability |
| Keyword:
|
projective metrizability |
| Keyword:
|
geodesics |
| Keyword:
|
spray |
| Keyword:
|
formal integrability |
| MSC:
|
49N45 |
| MSC:
|
53C22 |
| MSC:
|
53C60 |
| MSC:
|
58E30 |
| idZBL:
|
Zbl 06738532 |
| idMR:
|
MR3661054 |
| DOI:
|
10.21136/CMJ.2017.0010-16 |
| . |
| Date available:
|
2017-06-01T14:30:36Z |
| Last updated:
|
2020-07-03 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/146769 |
| . |
| Reference:
|
[1] Bácsó, S., Szilasi, Z.: On the projective theory of sprays.Acta Math. Acad. Paedagog. Nyházi. (N.S.) (electronic only) 26 (2010), 171-207. Zbl 1240.53047, MR 2754415 |
| Reference:
|
[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). Zbl 0726.58002, MR 1083148, 10.1007/978-1-4613-9714-4 |
| Reference:
|
[3] Bucataru, I., Milkovszki, T., Muzsnay, Z.: Invariant metrizability and projective metrizability on Lie groups and homogeneous spaces.Mediterr. J. Math. (2016), 4567-4580. Zbl 1356.53021, MR 3564521, 10.1007/s00009-016-0762-0 |
| Reference:
|
[4] Bucataru, I., Muzsnay, Z.: Projective metrizability and formal integrability.SIGMA, Symmetry Integrability Geom. Methods Appl. (electronic only) 7 (2011), Paper 114, 22 pages. Zbl 1244.49072, MR 2861227, 10.3842/SIGMA.2011.114 |
| Reference:
|
[5] Bucataru, I., Muzsnay, Z.: Projective and Finsler metrizability: parameterization-rigidity of the geodesics.Int. J. Math. 23 (2012), 1250099, 15 pages. Zbl 1263.53070, MR 2959445, 10.1142/S0129167X12500991 |
| Reference:
|
[6] Crampin, M.: Isotropic and R-flat sprays.Houston J. Math. 33 (2007), 451-459. Zbl 1125.53012, MR 2308989 |
| Reference:
|
[7] Crampin, M.: On the inverse problem for sprays.Publ. Math. 70 (2007), 319-335. Zbl 1127.53015, MR 2310654 |
| Reference:
|
[8] Crampin, M.: Some remarks on the Finslerian version of Hilbert's fourth problem.Houston J. Math. 37 (2011), 369-391. Zbl 1228.53085, MR 2794554 |
| Reference:
|
[9] Crampin, M., Mestdag, T., Saunders, D. J.: The multiplier approach to the projective Finsler metrizability problem.Differ. Geom. Appl. 30 (2012), 604-621. Zbl 1257.53105, MR 2996856, 10.1016/j.difgeo.2012.07.004 |
| Reference:
|
[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. Zbl 1262.53064, MR 3010078, 10.1016/j.difgeo.2012.10.012 |
| Reference:
|
[11] Do, T., Prince, G.: New progress in the inverse problem in the calculus of variations.Differ. Geom. Appl. 45 (2016), 148-179. Zbl 1333.37070, MR 3457392, 10.1016/j.difgeo.2016.01.005 |
| Reference:
|
[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. Zbl 0079.37502, MR 0082554 |
| Reference:
|
[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). Zbl 1023.49027, MR 1769337, 10.1142/9789812813596 |
| Reference:
|
[14] Klein, J., Voutier, A.: Formes extérieures génératrices de sprays.Ann. Inst. Fourier 18 French (1968), 241-260. Zbl 0181.49902, MR 0247599, 10.5802/aif.282 |
| Reference:
|
[15] Matveev, V. S.: On projective equivalence and pointwise projective relation of Randers metrics.Int. J. Math. 23 (2012), 1250093, 14 pages. Zbl 1253.53018, MR 2959439, 10.1142/S0129167X12500930 |
| Reference:
|
[16] Mestdag, T.: Finsler geodesics of Lagrangian systems through Routh reduction.Mediterr. J. Math. 13 (2016), 825-839. Zbl 1338.53103, MR 3483865, 10.1007/s00009-014-0505-z |
| Reference:
|
[17] Muzsnay, Z.: The Euler-Lagrange PDE and Finsler metrizability.Houston J. Math. 32 (2006), 79-98. Zbl 1113.53049, MR 2202354 |
| Reference:
|
[18] Rapcsák, A.: Über die bahntreuen Abbildungen metrischer Räume.Publ. Math. German 8 (1961), 285-290. Zbl 0101.39901, MR 0138079 |
| Reference:
|
[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. Zbl 1038.37044, MR 1895208, 10.1090/S0002-9947-02-02994-X |
| Reference:
|
[20] Shen, Z.: Differential Geometry of Spray and Finsler Spaces.Kluwer Academic Publishers, Dordrecht (2001). Zbl 1009.53004, MR 1967666, 10.1007/978-94-015-9727-2 |
| Reference:
|
[21] Szilasi, J., Lovas, R. L., Kertész, D. C.: Connections, Sprays and Finsler Structures.World Scientific Publishing, Hackensack (2014). Zbl 06171673, MR 3156183 |
| Reference:
|
[22] Szilasi, J., Vattamány, S.: On the Finsler-metrizabilities of spray manifolds.Period. Math. Hung. 44 (2002), 81-100. Zbl 0997.53056, MR 1892276, 10.1023/A:1014928103275 |
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