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| A {\it Lie system} is a system of first--order ordinary differential
equations describing the integral curves of a $t$-dependent vector field taking values in a finite-dimensional real Lie algebra of vector fields: a so-called {\it Vessiot--Guldberg Lie algebra}. We here suggest the definition of a particular class of Lie systems, the {\it $k$--symplectic Lie systems}, admitting a Vessiot--Guldberg Lie algebra of Hamiltonian vector fields with respect to the presymplectic forms of a $k$--symplectic structure. We devise new $k$-symplectic geometric methods to study their
superposition rules, time independent constants and general properties. Our results are
illustrated by examples of physical and mathematical interest. As a byproduct, we find a new interesting setting of application of $k$--symplectic geometry: systems of first-order ordinary differential equations.
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A. Awane, $k$--symplectic structures, \textit{J. Math. Phys.} \textbf{33} (1992) 4046--4052.
\bibitem{CGM07}
J.F. Cari\~nena, J. Grabowski and G. Marmo,
Superposition rules, Lie theorem and partial differential equations,
\textit{Rep. Math. Phys.} \textbf{60} (2007) 237--258.
\bibitem{Dissertations}
J.F. Cari\~nena and J. de~Lucas,
{\it Lie systems: theory, generalisations, and applications}.
Diss. Math. (Rozprawy Math.) \textbf {479}. Warsaw: Institute of Mathematics of the Polish Academy of Sciences, 2011.
\end{thebibliography} |
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