Section 24
Stability in mechanics and dynamics; a brief historical tour from Newton to modern tools, including Riemannian structures and entropy
Roland Gunesch,
Corresponding Author
Roland Gunesch
- [email protected]
- +43 5522 311 554
University of Education Vorarlberg, Liechtensteinerstr. 33-37, A-6800 Feldkirch, Austria
Roland Gunesch
University of Education Vorarlberg, Liechtensteinerstr. 33-37, A-6800 Feldkirch, Austria
Email: [email protected]
Telephone: +43 5522 311 554
Search for more papers by this authorRoland Gunesch,
Corresponding Author
Roland Gunesch
- [email protected]
- +43 5522 311 554
University of Education Vorarlberg, Liechtensteinerstr. 33-37, A-6800 Feldkirch, Austria
Roland Gunesch
University of Education Vorarlberg, Liechtensteinerstr. 33-37, A-6800 Feldkirch, Austria
Email: [email protected]
Telephone: +43 5522 311 554
Search for more papers by this authorFirst published: 31 May 2023
Abstract
This article presents some historical developments in the theory of stability in mechanics from a dynamical systems point of view, focusing on mathematical methods from the earliest developments (Newton) and explaining some classical and some recent methods used, including Riemannian structures, entropy and notions of stability in dynamics.
References
- 1 D. V. Anosov and Y. G. Sinai, Some smooth ergodic systems, Uspekhi Mat. Nauk 22(5(137)), 107–172 (1967) (in Russian). Engl. transl.: Russian Math. Surveys 22(5), 103–167 (1967).
- 2 C. Bär, Elementary differential geometry (Cambridge University Press, Cambridge, 2010)
- 3 M. I. Brin and Y. B. Pesin, Partially hyperbolic dynamical systems, Izv. Akad. Nauk SSSR Ser. Mat. 38(1), 170–212 (1974) (in Russian). Engl. transl.: Math. USSR Izv. 8 (1), 177–218 (1974).
- 4 M. I. Brin and Y. B. Pesin, D. V. Anosov and our road to partial hyperbolicity, Modern Theory of Dynamical Systems (Contemporary Mathematics, 692) (American Mathematical Society, Providence, RI, 2017), pp. 23–28.
- 5 M. I. Brin, G. Stuck, Introduction to Dynamical Systems (Cambridge University Press, New York, 2002).
- 6 M. I. Brin and A. B. Katok, On local entropy, Geometric Dynamics (Rio de Janeiro, 1981) (Lecture Notes in Mathematics, 1007) (Springer, Berlin, 1983), pp. 30–38.
- 7 K. Burns and A. B. Katok, Manifolds with nonpositive curvature, Ergod. Th. & Dynam. Sys. 5(2), 307–317 (1985).
- 8 M. P. do Carmo, Riemannian Geometry (Birkhäuser, Boston, 1992).
- 9 R. Gunesch, Precise asymptotics for periodic orbits of the geodesic flow in nonpositive curvature, PhD Thesis, The Pennsylvania State University (ProQuest LLC, Ann Arbor, MI, 2002).
- 10 R. Gunesch, Counting closed geodesics on rank one manifolds, https://arxiv.org/abs/0706.2845 (2007).
- 11 R. Gunesch, Computer simulations in lectures in mathematics and in the natural sciences: learning-theoretical aspects, PAMM, 10.1002/pamm.201900493, 19, 1, (2019).
- 12 R. Gunesch, How the mathematical understanding of the theory of dynamical systems can be improved with computer-assisted experimentation, PAMM, 10.1002/pamm.202000343, 20, 1, (2021).
- 13 R. Gunesch, Selected historical aspects of wave phenomena in the context of dynamical systems, and applications in seamless learning, PAMM, 10.1002/pamm.202100256, 21, 1, (2021).
- 14 B. Hasselblatt, Introduction to hyperbolic dynamics and ergodic theory, Ergodic Theory and Negative Curvature (Lecture Notes in Mathematics, 2164), (Springer, Cham, 2017), pp. 1–124.
- 15 B. Hasselblatt and A. B. Katok, A First Course in Dynamics: With a Panorama of Recent Developments (Cambridge University Press, New York, 2003).
- 16 B. Hasselblatt and A. B. Katok, The development of dynamics in the 20th century and the contribution of Jürgen Moser, Ergod. Th. & Dynam. Sys. 22(5), 1343–1364 (2002).
- 17 B. Hasselblatt and A. B. Katok (eds.), Handbook of Dynamical Systems, Vol. 1A (North-Holland, Amsterdam, 2002).
- 18 B. Hasselblatt and A. B. Katok (eds.), Handbook of Dynamical Systems, Vol. 1B (Elsevier B. V., Amsterdam, 2006).
- 19 A. B. Katok, Lyapunov exponents, entropy and periodic orbits for diffeomorphisms, Publ. Math. Inst. Hautes Études Sci. 51, 137–173 (1980).
- 20 A. B. Katok, Entropy and closed geodesics, Ergod. Th. & Dynam. Sys. 2(3–4) (1982), 339–365 (1983).
- 21 A. B. Katok (ed.), Nonuniform hyperbolicity and structure of smooth dynamical systems, Proc. Int. Congress of Mathematicians (Warsaw, 1983), vols. 1 and 2 (PWN, Warsaw, 1984), pp. 1245–1253.
- 22 A. B. Katok (ed.), Fifty years of entropy in dynamics: 1958–2007, J. Mod. Dyn. 1(4), 545–596 (2007).
- 23 A. B. Katok and V. Climenhaga, Lectures on Surfaces: (Almost) Everything You Wanted to Know About Them (Student Mathematical Library, 46), American Mathematical Society, Providence, RI; Mathematics Advanced Study Semesters, University Park, PA, 2008.
- 24 A. B. Katok and B. Hasselblatt, Introduction to the Modern Theory of Dynamical Systems, with a supplement by Anatole Katok and Leonardo Mendoza (Encyclopedia of Mathematics and Its Applications, 54) (Cambridge University Press, Cambridge, 1995).
- 25 A. B. Katok, G. Knieper, M. Pollicott and H. Weiss, Differentiability and analyticity of topological entropy for Anosov and geodesic flows, Invent. Math. 98(3), 581–597 (1989).
- 26 A. B. Katok, G. Knieper, M. Pollicott and H. Weiss, Differentiability of entropy for Anosov and geodesic flows, Bull. Amer. Math. Soc. (N.S.) 22(2), 285–293 (1990).
- 27 A. B. Katok, G. Knieper and H. Weiss, Formulas for the derivative and critical points of topological entropy for Anosov and geodesic flows, Comm. Math. Phys. 138(1), 19–31 (1991).
- 28 A. B. Katok, Y. B. Pesin and F. R. Hertz (eds.), A tribute to Dmitry Victorovich Anosov, Modern Theory of Dynamical Systems (Contemporary Mathematics, 692), American Mathematical Society, Providence, RI, (2017).
- 29 A. N. Kolmogorov, Théorie générale des systèmes dynamiques et mécanique classique, Proc. Int. Congress of Mathematicians (Amsterdam, 1954), Vol. 1 (Erven P. Noordhoff N.V., Groningen; North-Holland, Amsterdam, 1957), pp. 315–333 (in French), Engl. transl. in: R. Abraham and J. E. Marsden, Foundations of Mechanics, 2nd edn, Benjamin/Cummings, Advanced Book Program, (Reading, MA, 1978), pp. 741–757.
- 30 A. N. Kolmogorov, On conservation of conditionally periodic motions for a small change in Hamilton's function, Dokl. Akad. Nauk SSSR (N.S.) 98, 527–530 (1954)(in Russian). Engl. transl.: G. Casati and J. Ford (eds.), Stochastic Behavior in Classical and Quantum Hamiltonian Systems (Lecture Notes in Physics, 93), (Springer, Berlin), pp. 51–56.
- 31 A. N. Kolmogorov, A new metric invariant of transient dynamical systems and automorphisms in Lebesgue spaces, Dokl. Akad. Nauk SSSR (N.S.) 119, 861–864 (1958) (in Russian).
- 32 W. Kühnel, Differential geometry: curves - surfaces - manifolds, translated by Bruce Hunt (American Mathematical Society, Providence, RI, 2015).
- 33 I. Newton, Philosophiae Naturalis Principia Mathematica (Mathematical Principles of Natural Philosophy), London (1687).
- 34 Y. B. Pesin, Families of invariant manifolds corresponding to nonzero characteristic exponents, Math. USSR, Isvestia, 10 no. 6, 1261–1305 (1976).
- 35 Y. B. Pesin, Characteristic exponents and smooth ergodic theory, Russ. Math. Surveys, 32 no. 4, 55–114 (1977).
- 36 Y. B. Pesin, Formulas for the entropy of the geodesic flow on a compact Riemannian manifold without conjugate points, Mat. Zametki 24(4) (1978), 553–570, 591 (in Russian). Engl. transl.: Math. Notes 24(3–4) (1978), 796–805 (1979).
- 37 Y. B. Pesin, M. I. Brin and B. Hasselblatt, Anatole Katok, Modern Dynamical Systems and Applications, Dedicated to Anatole Katok on His 60th Birthday, ed. M. Brin, B. Hasselblatt and Y. Pesin (Cambridge University Press, Cambridge, 2004), pp. xi–xiv.
- 38 Y. G. Sinai, On the notion of entropy of a dynamical system, Dokl. Akad. Nauk SSSR 124 (1959), 768–771 (in Russian). Engl. transl.: Y. G. Sinai, Selecta I, Ergodic Theory and Dynamical Systems (Springer, Berlin, 2010, reprinted 2019), pp. 3–10, available at https://nikolaivivanov.files.wordpress.com/2015/05/definitionentropy2014-20151.pdf .
