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Abstract: A new class of integrable billiards has been introduced: evolutionary force billiards. They depend on a parameter and change their topology as energy (time) increases. It has been proved that they realize some important integrable systems with two degrees of freedom on the entire symplectic four-dimensional phase manifold at a time, rather than on only individual isoenergy 3-surfaces. For instance, this occurs in the Euler and Lagrange cases. It has also been proved that these two well-known systems are “billiard-equivalent”, despite the fact that the former one is square integrable, and the latter one allows a linear integral.
Bio: Anatoly Fomenko is a full member (Academician) of the Russian Academy of Sciences (1994), the International Higher Education Academy of Sciences (1993) and International Academy of Technological Sciences (2009), as well as a doctor of physics and mathematics (1972), a professor (1980), and Head of the Differential Geometry and Applications Department and the Head of Section of Mathematics of the Faculty of Mathematics and Mechanics in Moscow State University (1992). Fomenko is the author of the theory of topological invariants of an integrable Hamiltonian system. He is the author of more than 250 scientific publications, 30 monographs and textbooks on mathematics, a specialist in geometry and topology, variational calculus, symplectic topology, Hamiltonian geometry and mechanics, and computational geometry. Fomenko is also the author of a number of books on the development of new empirico-statistical methods and their application to the analysis of historical chronicles as well as the chronology of antiquity and the Middle Ages. Fomenko is also known for his original drawings inspired by topological objects and structures.
