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Regular Seminar Marta Mazzocco (Manchester)
After a short introduction to the Painleve equations, I'll study the Hamiltonian structure of the second Painleve hierarchy, an infinite sequence of nonlinear ordinary differential equations containing PII as its simplest equation. The n-th element of the hierarchy is a non linear ODE of order 2n in the independent variable z depending on n parameters denoted by t(1),...,t(n-1) and alpha(n). I'll introduce new canonical coordinates and obtain Hamiltonians for the z and t(1),...,t(n-1) evolutions. I'll give explicit formulae for these Hamiltonians showing that they are polynomials in our canonical coordinates.