Acceleration of bouncing balls in external fields

We introduce two models, the Fermi-Ulam model in an external field and a one dimensional system of bouncing balls in an external field above a periodically oscillating plate. For both models we investigate the possibility of unbounded motion. In a special case the two models are equivalent

Coherent Acceleration of Material Wavepackets

We study the quantum dynamics of a material wavepacket bouncing off a modulated atomic mirror in the presence of a gravitational field. We find the occurrence of coherent accelerated dynamics for atoms. The acceleration takes place for certain initial phase space data and within specific windows of modulation strengths. The realization of the proposed acceleration scheme is within the range of present day experimental possibilities.Comment: 6 pages, 3 figures, NASA "Quantum-to-Cosmos" conference proceedings to be published in IJMP

Bouncing trimer: a random self-propelled particle, chaos and periodical motions

A trimer is an object composed of three centimetrical stainless steel beads equally distant and is predestined to show richer behaviours than the bouncing ball or the bouncing dimer. The rigid trimer has been placed on a plate of a electromagnetic shaker and has been vertically vibrated according to a sinusoidal signal. The horizontal translational and rotational motions of the trimer have been recorded for a range of frequencies between 25 and 100 Hz while the amplitude of the forcing vibration was tuned for obtaining maximal acceleration of the plate up to 10 times the gravity. Several modes have been detected like e.g. rotational and pure translational motions. These modes are found at determined accelerations of the plate and do not depend on the frequency. By recording the time delays between two successive contacts when the frequency and the amplitude are fixed, a mapping of the bouncing regime has been constructed and compared to that of the dimer and the bouncing ball. Period-2 and period-3 orbits have been experimentally observed. In these modes, according to observations, the contact between the trimer and the plate is persistent between two successive jumps. This persistence erases the memory of the jump preceding the contact. A model is proposed and allows to explain the values of the particular accelerations for which period-2 and period-3 modes are observed. Finally, numerical simulations allow to reproduce the experimental results. That allows to conclude that the friction between the beads and the plate is the major dissipative process.Comment: 22 pages, 10 figure

On the connection of the Riemann problem with properties of a dynamical system

Abstract: We give the construction of an operator acting in a Hilbert space such that the Riemann hypothesis on zeros of the zeta-function is equivalent to the problem of the existence of an eigenvector for this operator with eigenvalue -1. We give also the construction of a dynamical system which turns out to be related to the Riemann hypothesis in the following way: for each complex zero of the zeta-function not lying on the critical line, there is a periodical trajectory of order two having a special form.Note: Research direction:Mathematical problems and theory of numerical method

Generalized Continued Fractions Connected with the Gauss Transformation.

Abstract: A new theory of generalized continued fractions connected with the Gauss transformation is constructed. The theory is near to classical theory of continued fractions and allows to characterize arbitrary algebraic and transcendental numbers with the help of generalized continued fractions. The results of this theory generalize classical results of the theory of continued fractions.Note: Research direction:Mathematical problems and theory of numerical method

Discrete rotations and generalized continued fractions

Abstract: The qualitative and experimental investigation of a dynamical system representing a discretization of a rotation is established. For different values of the angle of rotation and different initial data we found the first periodic point, the period, the maximal and minimal values of the coordinates of the trajectory points. The main result we found was the discovery of such a value of the angle of rotation for which all the trajectories with different initial positions (points) go very far from their initial positions and from the origin. The first periodic point occurs after more than 1500000 non-periodic points of the trajectory appear and the maximal value of its coordinates is greater than 1022. The value of the period is 2049. The angle for which such a phenomenon takes place differs fromp/2 only at the 14th decimal place in the computer representation ofp/2. The algebraic and algorithmic structure of the discrete rotation map is covered by the concept of the new theory of generalized continued fractions.Note: Research direction:Mathematical problems and theory of numerical method

Nonequilibrium gas, entropy and generalized billiards

Abstract: Generalized billiards describe nonequilibrium gas, consisting of finitely many particles, that move in a container, whose walls heat up or cool down. Generalized billiards can be considered both in the framework of the Newtonian mechanics and of the relativity theory. In the Newtonian case, a generalized billiard may possess an invariant measure; the Gibbs entropy with respect to this measure is constant. On the contrary, generalized relativistic billiards are always dissipative, and the Gibbs entropy both with respect to the same measure and the phase volume grows under some natural conditions. In this article, we find the necessary and sufficient conditions for a generalized Newtonian billiard to possess a smooth invariant measure, which is independent of the boundary action: the corresponding classical billiard should have an additional first integral of special type. In particular, the generalized Sinai billiards do not possess a smooth invariant measure. We then consider generalized billiards inside a ball, which is one of the main examples of the Newtonian generalized billiards which does have an invariant measure. We construct explicitly this invariant measure, and find the conditions for the Gibbs entropy growth for the corresponding relativistic billiard both for monotone and periodic action of the boundary, both with respect to the measure above and withrespect to the phase volume.Note: Research direction:Mathematical modelling in actual problems of science and technic