3rd ENOC ProceedingsIndex



The reduction method in the theory of Lie-algebraically integrable oscillatory Hamiltonian systems

Anatoliy K. Prykarpatsky

Department of Applied Mathematics
University of Mining and Metallurgy
Krakow 30059
Poland



Abstract


As is well known symmetry analysis of nonlinear dynamical systems on a smooth manifold gives rise in many cases to exhibiting its many hidden but interesting properties, in particular such as being integrable by quadratures due to the Liouville-Arnold theorem. In case when the manifold can be represented as the cotangent space to some subgroup of a Lie group naturally acting on it, the study of the corresponding flow can be recast via the reduction method into the Hamiltonian framework due to the existence on the canonical Poisson structure. Furthermore, if the symmetry group naturally generalizes to the loop group over, then the corresponding momentum mapping provides us with a Lax type representation and related with it a complete set of commuting invariants. Such a scheme appeared to be very useful when proving the Liouville integrability of many finite-dimensional systems such as Kowalevskaya's top, Neumann type systems and other. In the report we study complete integrability of nonlinear oscillatory dynamical systems connected in particular both with the Cartan decomposition of a Lie algebra and with a Poisson action of special type on a symplectic matrix manifold.


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Contact information


Anatoliy K. Prykarpatsky

Department of Applied Mathematics
University of Mining and Metallurgy
30 Mickiewicz Al. Bl. A4 no.120
Krakow 30059
Poland

e-mail: prykanat.as@mozart.emu.edu.tr



3rd ENOC ProceedingsIndex