Serdecznie zapraszamy pracowników i studentów na posiedzenie seminarium "Geometria i układy dynamiczne", które odbędzie się w PONIEDZIAŁEK 27 czerwca o godzinie 13:15 w sali A1/6

Prelegent: Stefan Rauch-Wojciechowski, Department of Mathematics, Linköping University

Tytuł: When knowledge of one integral of motion is sufficient for integrability?

Streszczenie: 

A system of differential equations is integrable by quadratures when solutions can be expressed using integrations, algebraic operations and taking inverse functions.  A general autonomous system of n equations requires knowledge of n-1 integrals of motion and one extra integration determines time dependence of solutions. 
Usually the notion of integrability is associated with Hamiltonian integrability in 2n dimensional phase space when knowledge of only n independent and involutive integrals of motion is sufficient for Liouville integrability. This is due to the special nature of the vector-field which is determined by one function, the Hamiltonian.
But there are known systems of equations when only 2 or 1 integral of motion suffice for integrability due to special algebraic form of the equation. There is a trade off between number of integrals and algebraic features of equations. 
The purpose of this talk is to make you aware of elegant, little known classes of  n  2-nd order Newton equations for which only 1 quadratic in velocities integral of motion implies existence of further n-1 integrals. This renders equations integrable and solvable by quadratures through separation of variables. 
These equations have the triangular form: d²q_r/dt²= M_r(q₁, ... , q_r) , r=1, ... ,n where the r-th equation depends only on the preceding variables  q_j ,  j=1, ... , r .

Serdecznie zapraszamy
Aleksander Denisiuk, Adam Doliwa, Andriy Panasyuk, Vsevolod Shevchishin, Artur Siemaszko