Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · 2307.13610

Construction of stationary trajectories for a model of a system of N particles with interaction

Igor Pavlov
View original source ↗Read source document ↗Export metadata ↓

Abstract

For the classical N-body problem, an approach is proposed based on the introduction of some natural in the physical sense optimization problems of mathematical programming for finding a conditional minimum for the characteristics of the system on the set of its possible states. The solution of these problems then makes it possible to construct families of flat stationary and periodic trajectories of the system and also to find relationships and estimates for the characteristics of the system on these trajectories. It is shown that when the system moves on a plane on trajectories generated by the global minimum in these optimization problems, at any time the minimum possible size of the system is achieved at each current level of its "cohesion" (or potential energy). Similar optimization problems are considered for finding a conditional minimum for the characteristics of a system in three-dimensional space. It is shown that the solution of these problems can be achieved only on flat trajectories of the system and is achieved, in particular, on the constructed flat stationary and periodic trajectories. In addition, it is shown that the trajectory of the system in three-dimensional space, at least at one point of which the minimum possible size of the system is achieved at the current value of its cohesion (or potential energy), can only be flat. And such trajectories are, in particular, flat stationary and periodic trajectories generated by the global minimum in the considered optimization problems.

Explore related subjects

math.DSastro-ph.GAmath-phmath.MPmath.OC

Keep this discovery

BibTeXRIS

Igor Pavlov. 2023-05-24. Construction of stationary trajectories for a model of a system of N particles with interaction. https://arxiv.org/abs/2307.13610

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Admissible Fourier Lengths, KAM Reducibility, and Spectral Applications

We develop a perturbative KAM reducibility theory for one-frequency $\mathrm{SL}(2,\mathbb{R})$ cocycles based on an admissible Fourier length $\ell$. The regularity relevant to the iteration is measured by positive adapted Fourier width rather than ordinary smoothness in the Euclidean length $|n|$. The same length governs Fourier decay, truncation and resonance scales, and the arithmetic condition controlling the small divisors. This framework contains the classical analytic and Gevrey settings, while non-monotone choices of $\ell$ allow classical nowhere differentiable Weierstrass-type perturbations and continuous perturbations outside every positive H\"older class. As spectral applications, we obtain purely absolutely continuous spectrum for every phase and $1/2$-H\"older continuity of the integrated density of states for the associated quasiperiodic Schr\"odinger operators. The Aubry dual has pure point spectrum for Lebesgue almost every dual phase, with eigenfunctions exponentially localized in the metric induced by $\ell$. We also construct nowhere differentiable quasiperiodic potentials with purely absolutely continuous Cantor spectrum.

math.DS↗
arXiv2026

Dynamics inside the attracting basins of some skew products

Polynomial skew products in $\mathbb{C}^2$ are maps of the form $F(z,w)=(P(z),Q(z,w))$, where $P$ and $Q$ are polynomials. Their local dynamics have been widely investigated. In this paper, we study the global dynamics inside Fatou components of some skew products. We consider all the inverse images in a Fatou component of a given point and use the Kobayashi metric to measure the distance between points. In the cases we consider, there are always arbitrarily large Kobayashi balls in the complement of these inverse sets.

math.DS↗
arXiv2026

Ergodicity of dynamical systems without uniqueness of orbits

Recently, there has been considerable interest in the study of non-deterministic dynamical systems. To analyze the chaotic behavior of such systems from a measure-theoretic viewpoint, it is desirable to consider ergodicity. However, the classical definition of ergodicity involves invariant sets, whose definition is not unique for non-deterministic dynamical systems. Thus, we are led to the question of which invariance yields an interesting definition of ergodicity. Here, we propose a definition based on the strong backward invariance and show that analogs of classical results hold. We also consider implications of the Birkhoff ergodic theorem for systems without uniqueness of orbits.

math.DS↗