Publication Frequency
Quarterly since 2025
Journal Articles
Search
Search scope
Volume Arrangement
2023
Featured Articles
If general relativity is correct, then the origin of the universe is a simple mathematical problem. The Friedmann equation in cosmology is a well-structured ordinary differential equation, and the global properties of its solutions can be qualitatively analyzed by the phase-trajectory method. In this paper we show that the total energy density of matter in the universe is positive, and the total pressure near the Big Bang is negative. By analyzing the global properties of the solutions to the Friedmann equation according to these two conditions of state functions, we find that the Big Bang is impossible, and the space must be a closed 3-dimensional sphere, the cosmological constant is likely to be zero, and the evolution of the universe should be cyclic. The analysis and the proof are simple and straight forward, therefore these conclusions should be reliable.
Fermat’s last theorem appears not as a unique property of natural numbers but as the bottom line of extended possible issues involving larger dimensions and powers when observed from a natural vector space viewpoint. The fabric of this general Fermat’s theorem structure consists of a well-defined set of vectors associated with dimensional vector spaces and the Minkowski norms one can define there. Here, a special vector set is studied and named a Fermat surface. Besides, a connection between Fermat surfaces and hypercubes is unveiled.
On types of stability in Hamiltonian systems
Article ID: 2269
Vol 1, Issue 1, 2023
DOI: https://doi.org/10.54517/mss.v1i1.2269
Vol 1, Issue 1, 2023
Received: 3 August 2023; Accepted: 21 October 2023; Available online: 3 November 2023; Issue release: 15 November 2023
Download PDF
Abstract
We consider conditions of three types of stability: Lyapunov, formal and weak of a stationary solution, and of a periodic solution in a Hamiltonian system with a finite number of degrees of freedom. The conditions contain restrictions on the order of resonances and some inequalities for initial coefficients of the normal forms of the Hamiltonian functions. We show that the number-theoretical analysis of frequencies can help in proof of stability. We also estimate the orders of solutions’ divergence from the stationary or the periodic ones under lack of formal stability.
Keywords
stationary solution; periodic solution; normal form; formal stability; weak stability
References
- Bruno AD. Analytic form of differential equations (II). Transactions of the Moscow Mathematical Society 1972; 26: 199–239.
- Bruno AD. The Restricted 3-body Problem: Plane Periodic Orbits. Walter de Gruyter; 1994.
- Dirichlet PGL. About the stability of balance (German). In: Kronecker L (editor). G. Lejeune Dirichlet’s Works (French and German). Cambridge University Press; 2012. 662p.
- Moser J. New aspects in the theory of stability of Hamiltonian systems. Communications on Pure and Applied Mathematics 1958; 11(1): 81–114. doi: 10.1002/cpa.3160110105
- Markeev AP. Libration Points in Celestial Mechanics and Cosmo Dynamics. Izdatel’stvo Nauka; 1978. 312p.
- Bruno AD, Batkhin AB. Resolution of an algebraic singularity by power geometry algorithms. Programming and Computer Software 2012; 38: 57–72. doi: 10.1134/S036176881202003X
- Bruno AD. Formal stability of Hamiltonian systems. Mathematical Notes of the Academy of Sciences of the USSR 1967; 1: 216–219. doi: 10.1007/BF01098887
- Moser JK. Lectures on Hamiltonian Systems. American Mathematical Society; 1968. 60p.
- Arnold VI. Small denominators and problems of stability of motion in classical and celestial mechanics. Russian Mathematical Surveys 1963; 18(6): 85. doi: 10.1070/RM1963v018n06ABEH001143
- Siegel CL, Moser JK. Lectures on Celestial Mechanics. Springer; 1971.
- Bruno AD. Stability in a Hamiltonian system. Mathematical Notes of the Academy of Sciences of the USSR 1986; 40: 726–730. doi: 10.1007/BF01142477
- Arnold VI. A letter to the editors. Uspekhi Matematicheskikh Nauk 1968; 23(6): 216.
- Markeev AP. Study of the Stability of Motion in Some Problems of Celestial Mechanics. Institute of Applied Mathematics; 1970. 164p.
- Batkhin AB. Segregation of stability domains of the Hamilton nonlinear system. Automation and Remote Control 2013; 74(8): 1269–1283. doi: 10.1134/S0005117913080043
- Batkhin A B. Parameterization of a set determined by the generalized discriminant of a polynomial. Programming and Computer Software 2018; 44: 75–85. doi: 10.1134/S0361768818020032
- Batkhin AB, Bruno AD, Varin VP. Stability sets of multiparameter Hamiltonian systems. Journal of Applied Mathematics and Mechanics 2012; 76(1): 56–92. doi: 10.1016/j.jappmathmech.2012.03.006
- Batkhin AB, Khaydarov ZK. Calculation of a strong resonance condition in a Hamiltonian system. Computational Mathematics and Mathematical Physics 2023; 63(5): 687–703. doi: 10.1134/S0965542523050068
- Bruno AD, Azimov AA. Computing unimodular matrices of power transformations. Programming and Computer Software 2023; 49(1): 32–41. doi: 10.1134/S0361768823010036
- Zhuravlev VF, Petrov AG, Shunderyuk MM. Selected Problems of Hamiltonian Mechanics. Lenand; 2015.
- Bruno AD. Normalization of a periodic Hamiltonian system. Programming and Computer Software 2020; 46(2): 76–83. doi: 10.1134/S0361768820020048
- Markeev AP. Stability of planar rotations of a satellite in a circular orbit. Mechanics of Solids 2006; 41(4): 46–63.
- Markeev AP. An algorithm for normalizing Hamiltonian systems in the problem of the orbital stability of periodic motions. Journal of Applied Mathematics and Mechanics 2002; 66(6): 889–896. doi: 10.1016/S0021-8928(02)00131-4figure
Refbacks
- There are currently no refbacks.
Copyright (c) 2023 Alexander Dmitrievich Bruno, Alexander Borisovich Batkhin
License URL: https://creativecommons.org/licenses/by/4.0
Editor-in-Chief
Prof. Youssri Hassan Youssri
Cairo University, Egypt
Indexing & Archiving
-
-
-
-
.jpg)
-
- J-Gate
-
-
-
.jpg)
News & Announcements
2025-05-08
Prof. Rami Ahmad El-Nabulsi is highly deserving of a Lifetime Achievement Award
Prof. Rami Ahmad El-Nabulsi, Researcher at Czech Academy of Science, Czech Republic, received the Lifetime Achievement Award!
2025-03-01
Publication Frequency of MSS changes to be quarterly!
We are pleased to announce that, effective from 2025, the publication frequency of this journal will be adjusted to a quarterly schedule, with four issues released annually in March, June, September, and December....
2024-09-20
Highly Read Article Recommendation
Since the journal launched in 2023, we have been proud to publish a plethora of insightful findings in the realms of mathematics and systems science....
2024-07-10
Meeting our Editor-in-Chief and Associate Editor
It is with great pride that we introduce Prof. Youssri Hassan Youssri and Prof. Ali Akgül, who are the Editor-in-Chief and Associate Editor of our esteemed journal.
Member Application
Journal Center
Asia Pacific Academy of Science Pte. Ltd. (APACSCI) specializes in international journal publishing. APACSCI adopts the open access publishing model and provides an important communication bridge for academic groups whose interest fields include engineering, technology, medicine, computer, mathematics, agriculture and forestry, and environment.