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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

$\frac{d{y}^{i}}{dx}={X}^{i}\left({y}^{1},\cdots ,{y}^{n},x\right),\phantom{\rule{1.em}{0ex}}i=1,\cdots ,n,$

is said to admit a superposition principle if its general solution $y$ can be expressed as $y={\Phi }\left({y}_{\left(1\right)},\cdots ,{y}_{\left(m\right)};{k}_{1},\cdots ,{k}_{n}\right)$ where $\left\{{y}_{\left(j\right)};\phantom{\rule{4pt}{0ex}}j=1,\cdots ,m\right\}$ is a set of independent particular solutions and ${k}_{1},\cdots ,{k}_{n}$ 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

$\frac{d{y}^{i}}{dx}={Z}_{1}\left(x\right){\xi }_{\left(y\right)}^{1i}+\cdots +{Z}_{r}\left(x\right){\xi }_{\left(y\right)}^{ri}$

where ${Z}_{1},\cdots ,{Z}_{r}$ being $r$ functions of only $x$, and ${\xi }^{\alpha i}$, $\alpha =1,\cdots ,r$, are functions of the variables $y=\left({y}^{1},\cdots ,{y}^{n}\right)$, such that the $r$ vector fields in ${ℝ}^{n}$, given by

${Y}^{\left(\alpha \right)}=\sum _{i=1}^{n}{\xi }^{\alpha i}\left(y\right)\frac{\partial }{\partial {y}^{i}},\phantom{\rule{1.em}{0ex}}\alpha =1,\cdots ,r,$

close on a finite-dimensional Lie algebra and $r\le mn$. 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.

##### MSC:
 34C14 Symmetries, invariants (ODE) 34A05 Methods of solution of ODE 34A26 Geometric methods in differential equations 52B15 Symmetry properties of polytopes