Related papers: The characteristic exponents of the falling ball m…
The system of falling balls is an autonomous Hamiltonian system with a smooth invariant measure and non-zero Lyapunov exponents almost everywhere. For almost three decades new, the question of its ergodicity remains open. We contribute to…
We consider the system of N (\ge 2) elastically colliding hard balls with masses m_1,..., m_N, radius r, moving uniformly in the flat torus T_L^{\nu}= R^\nu/L \cdot Z^\nu, \nu \ge 2. It is proved here that the relevant Lyapunov exponents of…
Wojtkowski's system of $N$, $N \geq 2$, falling balls is a nonuniformly hyperbolic smooth dynamical system with singularities. It is still an open question whether this system is ergodic. We contribute towards an affirmative answer, by…
We study the class of nonholonomic mechanical systems formed by a heavy symmetric ball that rolls without sliding on a surface of revolution, which is either at rest or rotates about its (vertical) figure axis with uniform angular velocity.…
The system of falling balls is an autonomous Hamiltonian system with a smooth invariant measure and non-zero Lyapunov exponents almost everywhere. Since almost three decades, the question of ergodicity is still open. The subject of this…
In this paper we present an unconditional proof of Wojtkowski's Ergodicity Conjecture for almost every system of 1D perfectly elastic balls falling down in a half line under constant gravitational acceleration. Namely, by introducing a new…
In this paper we present a conditional proof of Wojtkowski's Ergodicity Conjecture for the system of 1D perfectly elastic balls falling down in a half line under constant gravitational acceleration. Namely, we prove that almost every such…
Mean-field systems provide a natural framework in which collective effects persist as the number of degrees of freedom N increases, raising fundamental questions about the emergence of integrability and the nature of chaos in large but…
We investigate the dynamical stability of a fully-coupled system of $N$ inertial rotators, the so-called Hamiltonian Mean Field model. In the limit $N \to \infty$, and after proper scaling of the interactions, the $\mu$-space dynamics is…
Building upon previous works by Young, Chernov-Zhang and Bruin-Melbourne-Terhesiu, we present a general scheme to improve bounds on the statistical properties (in particular, decay of correlations, and rates in the almost sure invariant…
From the mesoscopic point of view, a new concept of soft matching for mass points is proposed. Then a soft Lasso's approach to learn the soft dynamical equation for the physical mechanical relationship is proposed, too. Furthermore, a…
We develop Hamiltonian mechanics on Aristotelian manifolds, which lack local boost symmetry and admit absolute time and space structures. We construct invariant phase space dynamics, define free Hamiltonians, and establish a generalized…
A new class of random quantum--dynamical systems in continuous space is introduced and studied in some detail. Each member of the class is characterized by a Hamiltonian which is the sum of two parts. While one part is deterministic,…
The spinless Falicov-Kimball model is studied by the use of a recently developed projector-based renormalization method (PRM) for many-particle Hamiltonians. The method is used to evaluate static and dynamic quantities of the…
This work is to investigate the (top) Lyapunov exponent for a class of Hamiltonian systems under small non-Gaussian L\'evy noise. In a suitable moving frame, the linearisation of such a system can be regarded as a small perturbation of a…
We study analytically the dynamics of a ball bouncing inelastically on a randomly vibrating platform, as a simple toy model of inelastic collapse. Of principal interest are the distributions of the number of flights n_f till the collapse…
We construct an example of a Hamiltonian flow $f^t$ on a $4$-dimensional smooth manifold $\mathcal{M}$ which after being restricted to an energy surface $\mathcal{M}_e$ demonstrates essential coexistence of regular and chaotic dynamics that…
We consider a set of macroscopic (classical) degrees of freedom coupled to an arbitrary many-particle Hamiltonian system, quantum or classical. These degrees of freedom can represent positions of objects in space, their angles, shape…
A framework for statistical-mechanical analysis of quantum Hamiltonians is introduced. The approach is based upon a gradient flow equation in the space of Hamiltonians such that the eigenvectors of the initial Hamiltonian evolve toward…
For a free falling particle moving in a media which has quadratic velocity force effect on the particle, two equivalent constants of motion, with units of energy, two Lagrangians, and two Hamiltonians are deduced. These quantities describe…