Séminaire "Equations différentielles"
In Boltzmann's billiard a particle moves in a half-plane subject to a central force and is reflected elastically when it hits the boundary. Boltzmann took the central force to be the sum of a gravitational inverse-square-law force and a centrifugal term proportional to the inverse cube of the distance to the centre. He formulated the expectation that except for special values of the parameters the system would be chaotic and would obey his Ergodic Hypothesis. Recently Gallavotti and Jauslin showed that the system is integrable if the centrifugal term is omitted: it has a second conserved quantity besides the energy. I will review this result and show that this integrable Boltzmann system has the Poncelet property: if in a level set of the conserved quantities a trajectory is periodic then all trajectories on the level set are periodic. As for the classical Poncelet theorem on inscribed-circumscribed polygons in Jacobi's interpretation, the result relies on the theory of elliptic curves. I will also present some work in progress with Michelle Wang on Boltzmann's table tennis, the three dimensional version of Boltzmann's integrable system, and the relation to QRT maps on biquadratic plane curves.
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Vasily Golyshev & Vladimir Roubtsov