In this article we develop an analogue of Aubry-Mather theory for a class of dissipative systems, namely conformally symplectic systems, and prove the existence of interesting invariant sets, which, in analogy to the conservative case, will be called the Aubry and the Mather sets. Besides describing their structure and their dynamical significance, we shall analyze their attracting/repelling properties, as well as their noteworthy role in driving the asymptotic dynamics of the system.
We consider the model describing the vertical motion of a ball falling with constant acceleration on a wall and elastically reflected. The wall is supposed to move in the vertical direction according to a given periodic function f . We apply the Aubry-Mather theory to the generating function in order to prove the existence of bounded motions with prescribed mean time between the bounces. As the existence of unbounded motions is known, it is possible to find a class of functions f that allow both bounded and unbounded motions.
We present the results of our investigation on the use of the twobody integrals to compute preliminary orbits by linking too short arcs of observations of celestial bodies. This work introduces a significant improvement with respect to the previous papers on the same subject [3], [4]. Here we find a univariate polynomial equation of degree 9 in the radial distance ρ of the orbit at the mean epoch of one of the two arcs. This is obtained by a combination of the algebraic integrals of the two-body problem. Moreover, the elimination step, which in [3], [4] was done by resultant theory coupled with the discrete Fourier transform, is here obtained by elementary calculations. We also show some numerical tests to illustrate the performance of the new algorithm.
h i g h l i g h t s• We consider the model of a bouncing ball on a moving plate.• If the velocity of the plate is small in norm C 5 then the ball cannot speed-up.• We construct movements of the racket with arbitrary small C 0 -norm that allow to speed-up the ball. • These constitute a mechanical example of the necessity of regularity in KAM theory.
a b s t r a c tWe give a mechanical example concerning the fact that some regularity is necessary in KAM theory. We consider the model given by the vertical bouncing motion of a ball on a periodically moving plate. Denoting with f the motion of the plate, some variants of Moser invariant curve theorem apply ifḟ is small in norm C 5 and every motion has bounded velocity. This is not possible if the function f is only C 1 .Indeed we construct a function f ∈ C 1 with arbitrary small derivative in norm C 0 for which a motion with unbounded velocity exists.
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