| Title:
|
Geometry of second-order connections and ordinary differential equations (English) |
| Author:
|
Vondra, Alexandr |
| Language:
|
English |
| Journal:
|
Mathematica Bohemica |
| ISSN:
|
0862-7959 (print) |
| ISSN:
|
2464-7136 (online) |
| Volume:
|
120 |
| Issue:
|
2 |
| Year:
|
1995 |
| Pages:
|
145-167 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
The geometry of second-order systems of ordinary differential equations represented by $2$-connections on the trivial bundle $\operatorname{pr_1} \Bbb R\times M\to\Bbb R$ is studied. The formalism used, being completely utilizable within the framework of more general situations (partial equations), turns out to be of interest in confrontation with a traditional approach (semisprays), moreover, it amounts to certain new ideas and results. The paper is aimed at discussion on the interrelations between all types of connections having to do with integral sections (geodesics), integrals and symmetries of the equations studied. (English) |
| Keyword:
|
geometry of second-order systems of ordinary differential equations |
| Keyword:
|
$2$- connections |
| Keyword:
|
connection |
| Keyword:
|
semispray |
| Keyword:
|
differential equation |
| Keyword:
|
integral |
| Keyword:
|
symmetry |
| MSC:
|
34A26 |
| MSC:
|
53C05 |
| MSC:
|
58A20 |
| MSC:
|
70H03 |
| MSC:
|
70H35 |
| idZBL:
|
Zbl 0836.34007 |
| idMR:
|
MR1357599 |
| DOI:
|
10.21136/MB.1995.126226 |
| . |
| Date available:
|
2009-09-24T21:10:01Z |
| Last updated:
|
2020-07-29 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/126226 |
| . |
| Reference:
|
[1] M. Anastasiei, H. Kawaguchi: A geometrical theory of the time dependent Lagrangians I. Non-linear connections.Tensor, N.S. 48 (1989), 273-282; II. M-connections. Tensor, N.S. 48 (1989), 283-293; III. Applications. Tensor, N.S. 49 (1990), 296-304. |
| Reference:
|
[2] M. Crampin: On the differential geometry of the Euler-Lagrange equations and the inveгse pгoblem of Lagrangian dynamics.J. Phys. A: Math. Gen. 14 (1981), 2567-2575. MR 0629315, 10.1088/0305-4470/14/10/012 |
| Reference:
|
[3] M. Crampin: Alternative Lagrangians in particle dynamics.Pгoc. Conf. Diff. Geom. and Its Appl., Bгno, 1986. J. E. Purkyně University, Brno, 1986, pp. 1-12. MR 0923340 |
| Reference:
|
[4] A. Dekrét: Mechanical structures and connections.Proc. Conf. Diff. Geom. and Appl., Dubrovnik, 1988. University of Novi Sad, Novi Sad, 1989, pp. 121-131. MR 1040061 |
| Reference:
|
[5] A. Dekrét: Vector fields and connections on TM.Časopis Pěst. Mat. 115 (1990), 360-367. MR 1090860 |
| Reference:
|
[6] M. de León, P. R. Rodrigues: Methods of Differential Geometry in Analytical Mechanics.North-Holland Mathematics Studies 158, North-Holland, Amsterdam, 1989. MR 1021489 |
| Reference:
|
[7] M. Doupovec, I. Kolář: Natural affinors on time-dependent Weil bundles.Arch. Math. (Brno) 27 (1991), 205-209. MR 1189217 |
| Reference:
|
[8] M. Doupovec, A. Vondra: On certain natural transformations between connections.Proc. Conf. Diff. Geom. and Its Appl., Opava, 1992. Silesian University, Opava, 1993, pp. 273-279. MR 1255548 |
| Reference:
|
[9] M. Doupovec, A. Vondra: Some natural operations between connections on fibered manifolds.Preprint, Brno, 1993. MR 1396603 |
| Reference:
|
[10] J. Grifone: Structure presque-tangente et connexions, I.Ann. Inst. Fourier, Grenoble 22, 3 (1972), 287-334; Structure presque-tangente et connexions, II. Ann. Inst. Fourier, Grenoble 22, 4 (1972), 291-338. Zbl 0219.53032, MR 0336636, 10.5802/aif.431 |
| Reference:
|
[11] D. Chinea M. de León, J. C. Marrero: The constraint algoritm for the time-dependent Lagrangians.Preprint, 1992. MR 1230594 |
| Reference:
|
[12] I. Kolář: Connections in 2-fibered manifolds.Arch. Math. (Brno) 17 (1981), 23-30. MR 0672485 |
| Reference:
|
[13] I. Kolář: Some natural operations with connections.J. Nat. Acad. Math. 5 (1987), 127-141. MR 0994555 |
| Reference:
|
[14] I. Kolář P. W. Michor, and J. Slovák: Natural Operations in Differential Geometry.Springer, 1993. MR 1202431 |
| Reference:
|
[15] D. Krupka, J. Janyška: Lectures on differential invariants.Folia Fac. Sci. Nat. Univ. Purk. Brun. Phys., J. E. Purkyně University, Brno, 1990. MR 1108622 |
| Reference:
|
[16] O. Krupková: Variational analysis on fibered manifolds over one-dimensional bases.PҺ.D. Thesis, Dept. of Mathematics, Silesian Univ. at Opava, Czechoslovakia, 1992. |
| Reference:
|
[17] O. Krupková, A. Vondra: On some integration methods for connections on fibered manifolds.Proc. Conf. Diff. Geom. and Its Appl., Opava, 1992. Silesian University, Opava, 1993, pp. 89-101. MR 1255531 |
| Reference:
|
[18] O. Krupková: Variational metrics on R x TM and the geometry of nonconservative mechanics.Math. Slovaca 44 (1994), 315-335. MR 1307321 |
| Reference:
|
[19] L. Mangiarotti, M. Modugno: Fibred spaces, jet spaces and connections for field theories.Proceedings of International Meeting "Geometry and Physics", Florence, 1982. Pitagora Editrice, Bologna, 1983, pp. 135-165. MR 0760841 |
| Reference:
|
[20] L. Mangiarotti, M. Modugno: Connections and differential calculus on fibred manifolds. Applications to field theory.Istituto di Matematica Applicata "G. Sansone", Firense. To appear. |
| Reference:
|
[21] E. Martínez J. F. Cariñena, W. Sarlet: Derivations of differential forms along the tangent bundle pгojection I.Diff. Geom. Appl. 2 (1992), 17-43; Derivations of differential forms along the tangent bundle projection II. Diff. Geom. Appl. 3 (1993), 1-29. |
| Reference:
|
[22] J. F. Pommaret: Systems of partial differential equations and Lie pseudogroups.Mir, Moscow, 1983. (In Russian.) Zbl 0544.58023, MR 0731697 |
| Reference:
|
[23] G. E. Prince: A complete classification of dynamical symmetries in classical mechanics.Bull. Austral. Math. Soc. 32 (1985), 299-308. Zbl 0578.58018, MR 0815371, 10.1017/S0004972700009977 |
| Reference:
|
[24] W. Sarlet E. Martínez, and A. Vandecasteele: Calculus of forms along a map adapted to the study of second-order differential equations.Proc. Conf. Diff. Geom. and Its Appl., Opava, 1992. Silesian University, Opava, 1993, pp. 123-133. MR 1255533 |
| Reference:
|
[25] D. J. Saunders: The Geometry of Jet Bundles.London Mathematical Society Lecture Note Series 142, Cambridge University Press, Cambridge, 1989. Zbl 0665.58002, MR 0989588 |
| Reference:
|
[26] A. M. Vinogradov I. S. Krasiľshchik, V. V. Lychagin: An introduction to the geometry of nonlinear differential equations.Nauka, Moscow, 1986. (In Russian.) MR 0855844 |
| Reference:
|
[27] A. Vondra: Semisprays, connections and regular equations in higher order mechanics.In Proc. Conf. Diff. Geom. and Its Appl., Brno 1989. World Scientific, Singapore, 1990, pp. 276-287. MR 1062030 |
| Reference:
|
[28] A. Vondra: On some connections related to the geometry of regular higher-order dynamics.Sborník VA, řada B 2 (1992), 7-18. |
| Reference:
|
[29] A. Vondra: Sprays and homogeneous connections on R x TM.Arch. Math. (Brno) 28 (1992), 163-173. MR 1222283 |
| Reference:
|
[30] A. Vondra: Natural dynamical connections.Czechoslovak Math. J. 41 (1991), 724-730. Zbl 0764.53019, MR 1134961 |
| Reference:
|
[31] A. Vondra: Symmetries of connections on fibered manifolds.Arch. Math. (Brno) 30 (1994), 97-115. Zbl 0813.35006, MR 1292562 |
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