Inverse approach in ordinary differential equations: applications to Lagrangian and Hamiltonian mechanics
type: Item
instance of: scholarly article
title: Inverse approach in ordinary differential equations: applications to Lagrangian and Hamiltonian mechanics
zbMATH Open document ID: 1307.70011
author: Jaume Llibre
author: Natalia Sadovskaia
author: R. Ramirez
published in: Journal of Dynamics and Differential Equations
publication date: 2014-12-12T00:00:00Z
full work available at URL: http://ddd.uab.cat/record/150742
review text: The authors study inverse problems of ordinary differential equations which consist in finding a differential system satisfying a set of given properties. For instance, to find what are the differential systems in \(\mathbb{R}^N\) having a given set of invariant hypersurfaces, or having a set of first integrals. The first inverse problem was stated and solved by Newton determining the potential field of a force that ensures the planetary motion in accordance to Kepler's laws. In 1877, \textit{J. Bertrand} [C. R. Acad. Sci., Paris 84, 671--674 (1877; JFM 09.0638.01)] proved that the expression of the Newton's force of attraction can be obtained directly from Kepler's first law. Bertrand stated also a more general problem of determining a positional force, under which a particle describes a conic section under any initial conditions. These ideas were developed in series of works until our days. Hence, the origin of the inverse problem comes from celestial mechanics. According to the second Newton law, the mentioned inverse problems are equivalent to determine second-order differential equations from given properties that must be satisfied. The inverse problem of finding more general differential systems of first order satisfying a set of given properties was stated by \textit{N. P. Erugin} [Prikl. Mat. Mekh. 16, 659--670 (1952; Zbl 0047.32805)] for dimension 2 and later developed by \textit{A. S. Galiullin} [Inverse problems of dynamics. (Obratnye zadachi dinamiki) (Russian). Moskva: ''Nauka'' (1981; Zbl 0516.70002)]. In the present work, the authors propose a new approach to an inverse problem which uses the Nambu bracket [\textit{Y. Nambu}, Phys. Rev. D (3) 7, 2405--2412 (1973; Zbl 1027.70503)]. The authors deduce new properties of this bracket which play a very important role in the proof of all the results of this work and in their applications. The applications of the Nambu bracket given in this paper are original and present a new direction in order to develop N…
reviewed by: Jaume Giné
Mathematics Subject Classification ID: 70F17
Mathematics Subject Classification ID: 70H03
Mathematics Subject Classification ID: 70H06
Mathematics Subject Classification ID: 34A55
zbMATH DE Number: 6380036
zbMATH Keywords: algebraic limit circle
zbMATH Keywords: completely integrable systems
zbMATH Keywords: polynomial vector fields
zbMATH Keywords: invariant algebraic circle
MaRDI profile type: Publication
OpenAlex ID: W1985569815
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DOI: 10.1007/S10884-014-9390-1
description: scientific article; zbMATH DE number 6380036