Euler and mathematical methods in mechanics. (English. Russian original) Zbl 1307.01009
Proc. Steklov Inst. Math. 272, Suppl. 2, S191-S207 (2011); translation from Sovrem. Probl. Mat. 11, 39-70 (2008).
While this paper contains brief descriptions concerning Euler and his work, the core of the paper is the presentation of aspects of his vast heritage. In particular, several topics associated with Euler are linked to modern work published in the last 30 years. For example, the section on divergent series and the Lagrange-Dirichlet theorem connects Euler’s summation work to a result of A. N. Kuznetsov in 1972. The paper reminds us that Euler developed the theory of motion of rotating bodies and the least-action principle. In hydrodynamics, Euler’s equations of motion for an ideal fluid lead to a description of stationary flow and the work of V. I. Arnold and the author on the hydrodynamics of Hamiltonian systems. A paper of the author from 1990 relates to vortex theory and Euler’s top. Mentioning Euler’s investigations of elasticity he introduces a synopsis of modern developments in stability theory of elastic systems. Finally, it is shown how Euler’s topic of two fixed centers of gravity has been generalized to any space of constant curvature. Altogether, the paper is a useful, even though brief, exposition of a large body of work.
Reviewer: Albert C. Lewis (Austin)
MSC:
| 01A50 | History of mathematics in the 18th century |
| 01A65 | Development of contemporary mathematics |
| 01A70 | Biographies, obituaries, personalia, bibliographies |
| 70-03 | History of mechanics of particles and systems |
| 76-03 | History of fluid mechanics |
Biographic References:
Euler, LeonhardReferences:
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| [2] | G. H. Hardy, Divergent Series (Clarendon, Oxford, 1949). |
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| [16] | V. V. Kozlov, Dynamical Systems. X. General Theory of Vortices, in Encyclopaedia Math. Sci. (Springer-Verlag, Berlin, 2003) Vol. 67. · Zbl 1104.37050 |
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| [22] | V. I. Arnold, ”Sur un principe variationnel pour lesécoulements stationnaires des liquides parfaits et ses applications aux problèmes de stabilité non linéaires,” J.Mécanique 5(1), 29–43 (1966). |
| [23] | M. V. Deryabin and Yu. N. Fedorov, ”On Reductions for a Group of Geodesic Flows with (Left-) Right-Invariant Metric, and Their Symmetry Fields,” Dokl. Akad. Nauk 391(4), 439–442 (2003) [Dokl. Math. 68 (1), 75–78 (2003)]. |
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| [25] | V. V. Kozlov, ”On the Degree of Instability,” Prikl. Mat. Mekh. 57(5), 14–19 (1993) [J. Appl. Math. Mech. 57 (5), 771–776 (1993)]. |
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| [28] | V. V. Kozlov and A. A. Karapetyan, ”On the Degree of Stability,” Differ. Uravn. 41(2), 186–192 (2005) [Differ. Equations 41 (2), 195–201 (2005)]. · Zbl 1090.34564 |
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| [30] | P. Lancaster and M. Tismenetsky, ”Inertia Characteristics of Selfajoint Matrix Polynomials,” Linear Algebra Appl. 52–53, 479–496 (1983). · Zbl 0516.15018 · doi:10.1016/0024-3795(83)80030-5 |
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| [32] | Classical Dynamics in non-Euclidean Spaces, Ed. by A. V. Borisov and I. S. Mamaev (Inst. Komp’yuter. Issled., Moscow, 2004) [in Russian]. |
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