Applied Math Seminar | Manuele Santoprete, Canonoid and Poissonoid Transformations, Symmetries and bi-Hamiltonian structures

Manuele Santoprete
​Department of Mathematics
Wilfrid Laurier Univerisity

Title

Canonoid and Poissonoid Transformations, Symmetries and bi-Hamiltonian structures

Abstract

Symplectic transformations between symplectic manifold are maps that preserve the symplectic structure. Such maps transform any Hamiltonian system on the first symplectic manifold to a Hamiltonian system on the second symplectic manifold. Canonoid transformations are a generalization of symplectic transformation. They transform a given Hamiltonian system defined on a symplectic manifold to another Hamiltonian system. The term "canonoid" was introduced by Saletan and Cromer in 1971. A modern coordinate-free definition was given by Marmo in 1976. In this talk we will give an introduction to canonoid transformations. Then we will generalize such transformations to the Poisson case, introducing "Poissonoid transformations". We will give a brief introduction to bi-Hamiltonian systems, an important type of integrable systems. We will show that Poissonoid transformations may be useful to obtain bi-Hamiltonian structures in certain examples (e.g. the rigid body, the multidimensional rigid body, and Kirchhoff'e equations for a rigid body in a fluid). If time permits we will introduce infinitesimal Poissonoid transformations and their connection to symmetries.

Bio

Manuele Santoprete is a tenured associate professor of mathematics at the Wilfrid Laurier University where his research is supported by grants from the Natural Sciences and Engineering Research Council of Canada. He is author of over 30 articles in peer reviewed journals, and of a book.

Prior to joining Laurier, he was a Visiting Assistant Professor of mathematics at University of California, Irvine. He received his B.S. in Physics from University of Pisa in 1999, and his PhD in Applied Mathematics under the direction of Florin Diacu in 2003, at the University of Victoria.

Research Interest

Celestial Mechanics, Dynamical Systems, Geometric Mechanics, Mathematical Sociology and Biology.