| Title:
|
Sprays and homogeneous connections on ${\bf R}\times {\it TM}$ (English) |
| Author:
|
Vondra, Alexandr |
| Language:
|
English |
| Journal:
|
Archivum Mathematicum |
| ISSN:
|
0044-8753 (print) |
| ISSN:
|
1212-5059 (online) |
| Volume:
|
28 |
| Issue:
|
2 |
| Year:
|
1992 |
| Pages:
|
163-173 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
The homogeneity properties of two different families of geometric objects playing a crutial role in the non-autonomous first-order dynamics - semisprays and dynamical connections on $R\times TM$ - are studied. A natural correspondence between sprays and a special class of homogeneous connections is presented. (English) |
| Keyword:
|
semispray |
| Keyword:
|
connection |
| Keyword:
|
path |
| Keyword:
|
torsion |
| Keyword:
|
tension |
| Keyword:
|
spray |
| Keyword:
|
homogeneous dynamical connection |
| MSC:
|
53C05 |
| MSC:
|
53C15 |
| MSC:
|
58F05 |
| MSC:
|
70H03 |
| MSC:
|
70H35 |
| idZBL:
|
Zbl 0790.53028 |
| idMR:
|
MR1222283 |
| . |
| Date available:
|
2008-06-06T21:22:37Z |
| Last updated:
|
2012-05-10 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/107447 |
| . |
| Reference:
|
[1] Abraham, R., Marsden, J.E.: Foundations of Mechanics.Benjamin/Cummings Publ. Comp., Reading, 1978. MR 0515141 |
| Reference:
|
[2] 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. MR 0748776 |
| Reference:
|
[3] de Andres, L.C., de León, M., Rodrigues, P.R.: Connections on tangent bundles of higher order.Demonstratio Mathematica 22 (3) (1989), 607-632. MR 1041903 |
| Reference:
|
[4] de Andres, L.C., de León, M., Rodrigues, P.R.: Connections on tangent bundles of higher order associated to regular Lagrangians.Geometriae Dedicata 39 (1991), 12-18. MR 1116206 |
| Reference:
|
[5] de León, M., Mello, M.H., Rodrigues, P.R.: A geometrical approach to the inverse problem of non-autonomous lagrangian dynamics.(preprint, 1990). |
| Reference:
|
[6] de León, M., Rodrigues, P.R.: Dynamical connections and non-autonomous lagrangian systems.Ann. Fac. Sci. IX (1988), Tolouse, 171-181. MR 1425006 |
| Reference:
|
[7] de León, M., Rodrigues, P.R.: Generalized Classical Mechanics and Field Theory.Mathematical Studies 112 (1985), North - Holland, Amsterdam. MR 0808964 |
| Reference:
|
[8] Doupovec, M., Kolář, I.: Natural affinors on time-dependent Weil bundles.Arch. Math. (Brno) 27 (1991), 205-209. MR 1189217 |
| Reference:
|
[9] Klein, J.: Almost symplectic structures in dynamics.Proc. Conf. Diff. Geom. and Its Appl. (1986), Brno, 79-91. MR 0923345 |
| Reference:
|
[10] Klein, J.: Geometry of sprays. Lagrangian case. Principle of least curvature.Proc.IUTAM-ISIMM Symposium on Modern Developments in Analytical Mechanics, Torino, Volume I-Geometrical Dynamics (1982), 177-196. Zbl 0566.58012, MR 0773487 |
| Reference:
|
[11] Kolář, I., Modugno, M.: Torsions of connections on some natural bundles.(to appear). MR 1244453 |
| Reference:
|
[12] Mangiarotti, L., Modugno, M.: Connections and differential calculus on fibred manifolds. Applications to field theory.Instituto di Matematica Applicata 11 “G. Sansone”, Firenze (to appear). |
| Reference:
|
[13] Modugno, M.: Torsion and Ricci tensor for non-linear connections.(to appear). |
| Reference:
|
[14] Saunders, D.J.: The Geometry of Jet Bundles.London Mathematical Society Lecture Note Series 142 (1989), Cambridge University Press. Zbl 0665.58002, MR 0989588 |
| Reference:
|
[15] Sternberg, S.: Lectures on Differential Geometry.Prentice-Hall, 1965. MR 0193578 |
| Reference:
|
[16] Thompson, G.: Second-order equation fields and the inverse problem of Lagrangian dynamics.J. Math. Phys. 28 (12) (1987), 2851-2857. Zbl 0638.70013, MR 0917639 |
| Reference:
|
[17] Vondra, A.: On some connections related to the geometry of regular higher-order dynamics.(preprint, 1990). |
| Reference:
|
[18] Vondra, A.: Semisprays, connections and regular equations in higher order mechanics.Proc. Conf. Diff. Geom. and Its Appl. (19891990), World Scientific, Brno Singapore, 276-287. MR 1062030 |
| . |
