zbMATH — the first resource for mathematics

Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
Lie systems of differential equations and connections in fibre bundles. (English) Zbl 1091.34018
Mladenov, Ivaïlo M.(ed.) et al., Proceedings of the 6th international conference on geometry, integrability and quantization, Sts. Constantine and Elena, Bulgaria, June 3–10, 2004. Sofia: Bulgarian Academy of Sciences (ISBN 954-84952-9-5/pbk). 62-77 (2005).

A system of differential equations

dy i dx=X i (y 1 ,,y n ,x),i=1,,n,

is said to admit a superposition principle if its general solution y can be expressed as y=Φ(y (1) ,,y (m) ;k 1 ,,k n ) where {y (j) ;j=1,,m} is a set of independent particular solutions and k 1 ,,k n are n arbitrary constants. Due to Sophus Lie, it is known that this is the case if and only if the system can be written in the form

dy i dx=Z 1 (x)ξ (y) 1i ++Z r (x)ξ (y) ri

where Z 1 ,,Z r being r functions of only x, and ξ αi , α=1,,r, are functions of the variables y=(y 1 ,,y n ), such that the r vector fields in n , given by

Y (α) = i=1 n ξ αi (y) y i ,α=1,,r,

close on a finite-dimensional Lie algebra and rmn. The authors show that the study of such systems can be reduced to that of an equation on a Lie group, and that all such systems can be seen as the systems determining the horizontal curves on an appropriate connection. Some applications to the general Riccati equation and to quantum mechanics are given, too.

34C14Symmetries, invariants (ODE)
34A05Methods of solution of ODE
34A26Geometric methods in differential equations
52B15Symmetry properties of polytopes