| Title:
|
Automorphisms of spatial curves (English) |
| Author:
|
Bradáč, Ivan |
| Language:
|
English |
| Journal:
|
Archivum Mathematicum |
| ISSN:
|
0044-8753 (print) |
| ISSN:
|
1212-5059 (online) |
| Volume:
|
33 |
| Issue:
|
2 |
| Year:
|
1997 |
| Pages:
|
213-243 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
Automorphisms of curves $y= y(x)$, $z=z(x)$ in ${\bold R}^3$ are investigated; i.e. invertible transformations, where the coordinates of the transformed curve $\bar y=\bar y(\bar x)$, $\bar z= \bar z(\bar x)$ depend on the derivatives of the original one up to some finite order $m$. While in the two-dimensional space the problem is completely resolved (the only possible transformations are the well-known contact transformations), the three-dimensional case proves to be much more complicated. Therefore, results (in the form of some systems of partial differential equations for the functions, determining the automorphisms) only for the special case $\bar x =x$ and order $m\leq 2$ are obtained. Finally, the problem of infinitesimal transformations is briefly mentioned. (English) |
| Keyword:
|
automorphisms of curves |
| Keyword:
|
infinite-dimensional space |
| Keyword:
|
contact forms |
| MSC:
|
58A17 |
| MSC:
|
58A20 |
| MSC:
|
58J72 |
| idZBL:
|
Zbl 0915.58003 |
| idMR:
|
MR1478774 |
| . |
| Date available:
|
2008-06-06T21:33:25Z |
| Last updated:
|
2012-05-10 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/107612 |
| . |
| Reference:
|
[1] Lie S.: Geometrie der Berührungstransformationen.erster Band, Leipzig 1896. Zbl 0406.01015 |
| Reference:
|
[2] Anderson R., Ibragimov N.: Lie-Bäcklund transformations in applications.Philadelphia 1979. Zbl 0447.58001, MR 0520395 |
| Reference:
|
[3] Ibragimov N.: Transformation groups in mathematical physics.Moscow, Nauka, 1983 (Russian) Zbl 0529.53014, MR 0734307 |
| Reference:
|
[4] Carathèodory C.: Variationsrechnung und partielle Differentialgleichungen erster Ordnung.Band I, Theorie der partielen Differentialgleichungen erster Ordnung, Zweite Auflage, Leipzig 1956. Zbl 0070.31601, MR 0089338 |
| Reference:
|
[5] Shlomo Sternberg: Lectures on Differential Geometry.Prentice-Hall, Inc. Englewood Cliffs, New Jersey, 1965. MR 0193578 |
| Reference:
|
[6] Chrastina J.: From Elementary Algebra to Bäcklund Transformations.Czechoslovak Mathematical Journal, 40 (115) 1990, Praha. Zbl 0726.58041, MR 1046292 |
| Reference:
|
[7] Chrastina J.: Formal theory of differential equations.(to appear). Zbl 0906.35002, MR 1656843 |
| Reference:
|
[8] Chrastina J.: On the Equivalence of Variational Problems, I.Journal of Differential Equations, Vol. 98, No. 1, July 1992. Zbl 0764.49008, MR 1168972 |
| Reference:
|
[9] Stormark O.: Formal and local solvability of partial differential equations.Trita-Mat-1989-11, Mathematics, ch. 1–12, Royal Institute of Technology, Stockholm 1989. |
| Reference:
|
[10] Pressley A., Segal G.: Loop Groups.Clarendon Press, Oxford 1986, Russian translation Moscow, Mir, 1990. Zbl 0618.22011, MR 1071737 |
| Reference:
|
[11] Cartan E.: Les systèmes différentiels extérieurs et leurs applications géometriques.Gauthier-Villars, Paris 1945, Russian translation Moscow University 1962. Zbl 0063.00734, MR 0016174 |
| Reference:
|
[12] Olver P.: Applications of Lie Groups to Differential Equations.1986, Springer-Verlag, Russian translation Moscow, Mir, 1989. Zbl 0743.58003, MR 0836734 |
| Reference:
|
[13] Vinogradov A. M., Krasilščik I. S., Lygačin V. V.: Introduction into the geometry of nonlinear differential equations.Moscow 1986 (Russian). |
| . |
