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Order/Chaos: Genealogy of Two Concepts in the Culture of European Mathematical Physics (ORCHA)
Start date: Nov 1, 2014, End date: Oct 31, 2017 PROJECT  FINISHED 

"This project aims at understanding the generation of abstract concepts and theories in mathematical physics and their relation with theoretical practices. To achieve this, it focuses upon the emergence of the modern concepts of order and chaos in celestial and statistical mechanics. The research is articulated into three phases. The first phase discusses the problem of stability in a three-body system from the origin of celestial mechanics to Henri Poincaré’s discovery of homoclinic trajectories. Initially, the problem of stability was attacked by using analytical and algebraic techniques whose main goal was to show that approximate solutions of the equations of motion do not contain secular terms. Drawing on this vast culture of mathematical techniques and generalizing them further, Poincaré introduced a new qualitative approach, which led him to extend periodicity from periodic motion to complicate recurrence up to chaotic trajectories. The second phase focuses upon the intersections between mechanical and statistical arguments in the work of Ludwig Boltzmann. Facing the problem of dealing with mechanical systems with many degrees of freedom, Boltzmann combined the analysis of the equations of motion with statistical techniques which allowed him to describe the macroscopic behavior of the system. Thus, macroscopic order was understood as a probabilistic, long-term phenomenon deriving from a complex, microscopic disorder. Notions such as the ergodic hypothesis and molecular chaos were introduced by Boltzmann to conceptualize this disorder. The third phase concerns George D. Birkhoff’s path from the theory of dynamical systems to the ergodic theorem. Taking up Poincaré’s legacy in celestial mechanics in 1912, Birkhoff honed the analytical and topological techniques of the European mathematical culture and, in 1931, he rigorously formulated the conditions for a mechanical systems to be ergodic so unifying the two previous lines of investigation on order/chaos."
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