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On Some Geometric Methods in Mathematics and Mechanics

§ Keldysh Institute of Applied Mathematics of RAS

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On Some Geometric Methods in Mathematics and Mechanics

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Abstract

We give a survey of geometric methods used in papers and books of V.I. Arnold and V.V. Kozlov. They are methods of different normal forms, of some polyhedra, of small denominators and asymptotic expansions.

References

34 Cites in Article
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  2. 2. V Arnold (1968). LETTER TO THE EDITOR. Jewish Education, 38(4), 57-64.
  3. 3. V Arnold (1978). Mathematical Methods in Classical Mechanics. Mathematical Methods in Classical Mechanics
  4. 4. V Arnold (1998). Geometrical Methods in the Theory of Ordinary Differential Equations. Geometrical Methods in the Theory of Ordinary Differential Equations
  5. 5. A Bruno (1988). The normal form of a Hamiltonian system. Russian Mathematical Surveys, 43(1), 25-66.
  6. 6. A Bruno (1989). Local Methods in Nonlinear Differential Equations. Local Methods in Nonlinear Differential Equations
  7. 7. Alexander Bruno (1994). The Restricted 3-Body Problem: Plane Periodic Orbits. The Restricted 3-Body Problem: Plane Periodic Orbits
  8. 8. A Bruno,V Parusnikov (1994). Klein polyhedrals for two cubic Davenport forms. Math. Notes, 56, 3-4.
  9. 9. A Bruno (2000). Power Geometry in Algebraic and Differential Equations. Power Geometry in Algebraic and Differential Equations
  10. 10. A Bruno,A Parusnikova (2004). Local expansions of solutions to the fifth Painlevé equation. Doklady Mathematics, 83(3), 348-352.
  11. 11. A Bruno,I Goruchkina (2004). Expansions of solutions to the sixth Painlevé equation. Doklady Mathematics, 69(2), 268-272.
  12. 12. A Bruno (2005). The structure of multidimensional diophantine approximations. Doklady Mathematics, 82(1), 587-589.
  13. 13. A Bruno (2005). Generalizied continued fraction algorithm. Doklady Mathematics, 71(3), 446-450.
  14. 14. A Bruno,A Petrov (2006). On computation of the Hamiltonian normal form. Doklady Physics, 51(10), 555-559.
  15. 15. A Bruno (2007). Analysis of the Euler–Poisson equations by methods of power geometry and normal form. Journal of Applied Mathematics and Mechanics, 71(2), 168-199.
  16. 16. A Bruno,V Parusnikov (2009). Two-way generalization of the continued fraction. Doklady Mathematics, 80(3), 887-890.
  17. 17. A Bruno (2010). The structure of multidimensional diophantine approximations. Doklady Mathematics, 82(1), 587-589.
  18. 18. Alexander Bruno (2010). New generalization of continued fraction, I. Functiones et Approximatio Commentarii Mathematici, 43(1), 55-104.
  19. 19. A Bruno,I Goryuchkina (2010). Asymptotic expansions of solutions of the sixth Painlevé equation. Transactions of the Moscow Mathematical Society, 71, 1-1.
  20. 20. A Bruno,A Parusnikova (2011). Local expansions of solutions to the fifth Painlevé equation. Doklady Mathematics, 83(3), 348-352.
  21. 21. A Bruno (2014). On an integrable Hamiltonian system. Doklady Mathematics, 90(1), 499-502.
  22. 22. A Bruno (2015). Asymptotic Solution of Nonlinear Algebraic and Differential Equations. International Mathematical Forum, 10, 11-535.
  23. 23. A Bruno (2015). Power geometry and elliptic expansions of solutions to the Painlevé equations. International Journal of Differential Equations
  24. 24. A Bruno (2015). Universal generalization of the continued fraction algorithm. Chebyshevskii Sbornik, 16(2), 35-65.
  25. 25. H Dulac (1912). Solutions d'un système d'équations différentielles dans le voisinage de valeurs singulières. Bulletin de la Société mathématique de France, 2, 324-383.
  26. 26. D Galin (1982). Versal deformations of linear Hamiltonian systems. Sixteen Papers on Differential Equations, 118, 1-12.
  27. 27. B Khesin,S Tabachnikov (2012). Tribute to Vladimir Arnold. Notices Amer. Math. Soc, 59(3), 378-399.
  28. 28. Valerij Kozlov (1976). Polynomial Integrals of Hamiltonian Systems. Symmetries, Topology and Resonances in Hamiltonian Mechanics, 1, 337-366.
  29. 29. Valery Kozlov,Stanislav Furta (2013). Asymptotic Solutions of Strongly Nonlinear Systems of Differential Equations. Asymptotic Solutions of Strongly Nonlinear Systems of Differential Equations, 262.
  30. 30. V Kozlov (1996). Symmetries, Topology and Resonances in Hamiltonian Mechanics. Symmetries, Topology and Resonances in Hamiltonian Mechanics
  31. 31. G Lauchand (1993). Polyèdre d'Arnol'd et voile d'in cone simplicial: analogues du théoreme de Lagrange. C. R. Acad. Sci. Ser. 1, 317, 711-716.
  32. 32. Jurgen Moser (1968). Lectures on Hamiltonian Systems *. Hamiltonian Dynamical Systems, 81, 77-136.
  33. 33. H Poincaré (1879). Sur les propriétés des fonctions définies par les équations aux différences partielles. Sur les propriétés des fonctions définies par les équations aux différences partielles
  34. 34. C Siegel,J Moser (1971). Lectures on Celestial Mechanics. Lectures on Celestial Mechanics

Funding

No external funding was declared for this work.

Conflict of Interest

The authors declare no conflict of interest.

Ethical Approval

No ethics committee approval was required for this article type.

Data Availability

Not applicable for this article.

How to Cite This Article

Alexander Bruno. 2017. "On Some Geometric Methods in Mathematics and Mechanics". Global Journal of Science Frontier Research - F: Mathematics & Decision GJSFR-F Volume 17 (GJSFR Volume 17 Issue F8).

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Journal Specifications

Crossref Journal DOI 10.17406/GJSFR

Print ISSN 0975-5896

e-ISSN 2249-4626

Keywords
Classification
GJSFR-F Classification MSC 2010: 05B35
Version of record
v1.2

Issue date
December 19, 2017

Language
English
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On Some Geometric Methods in Mathematics and Mechanics

Alexander Bruno
Alexander Bruno Keldysh Institute of Applied Mathematics of RAS