Abstract
We study the motion of a particle in a plane subject to an attractive central force with inverse-square law on one side of a wall at which it is reflected elastically. This model is a special case of a class of systems considered by Boltzmann which was recently shown by Gallavotti and Jauslin to admit a second integral of motion additionally to the energy. By recording the subsequent positions and momenta of the particle as it hits the wall, we obtain a three-dimensional discrete-time dynamical system. We show that this system has the Poncelet property: If for given generic values of the integrals one orbit is periodic, then all orbits for these values are periodic and have the same period. The reason for this is the same as in the case of the Poncelet theorem: The generic level set of the integrals of motion is an elliptic curve, and the Poincaré map is the composition of two involutions with fixed points and is thus the translation by a fixed element. Another consequence of our result is the proof of a conjecture of Gallavotti and Jauslin on the quasi-periodicity of the integrable Boltzmann system, implying the applicability of KAM perturbation theory to the Boltzmann system with weak centrifugal force.
Highlights
In a 1868 paper with the unpretentious1 title “Solution of a mechanical problem” [1], L. In his search of candidate dynamical systems obeying his Ergodic Hypothesis, introduced and studied a simple mechanical system. It describes of a particle moving in the region of a plane on one side of a straight line and subject to a central force whose centre is not on the wall
The force considered by Boltzmann is the sum of an attractive one with inverse-square law and a centrifugal force with inverse-cube law
The map sending a point to the point at the collision is called the Poincaré map, and the orbits are obtained by iterating the Poincaré map. We focus on this integrable case of the Boltzmann system, with zero centrifugal force, and we show that it has the Poncelet property: For given values of the two integrals of motion, either there are no periodic orbits or all orbits are periodic
Summary
In a 1868 paper with the unpretentious title “Solution of a mechanical problem” [1], L. Our observation is that the integrable Boltzmann system behaves very much in the same way: we consider for fixed generic values of the two integrals of motion the pairs (P, K ) consisting of a Kepler conic K and a intersection point P ∈ K with the wall We show that this space (after complexification and throwing in a couple of points at infinity) is a smooth curve of genus one carrying two involutions. The orbits of the Boltzmann system are obtained by iterating the composition t = j ◦i of these involution, implying the Poncelet property Another consequence of this observation is the proof of a conjecture of Gallavotti and Jauslin [7] that the motion is quasi-periodic for generic values of the integrals, namely that on generic level sets of the integrals there is an angle variable, a map to the circle, whose value increases by a fixed amount α at each iteration of t.
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