Geometric structure of the classical Lagrange-d'Alembert principle and its application to integrable nonlinear dynamical systems
type: Item
instance of: scholarly article
title: Geometric structure of the classical Lagrange-d'Alembert principle and its application to integrable nonlinear dynamical systems
zbMATH Open document ID: 1430.17099
author: O. E. Hentosh
author: A. K. Prykarpats'kyi
author: Yarema A. Prykarpatsky
published in: Mathematics
publication date: 2018-07-05T00:00:00Z
review text: Summary: The classical Lagrange-d'Alembert principle had a decisive influence on formation of modern analytical mechanics which culminated in modern Hamilton and Poisson mechanics. Being mainly interested in the geometric interpretation of this principle, we devoted our review to its deep relationships to modern Lie-algebraic aspects of the integrability theory of nonlinear heavenly type dynamical systems and its so called Lax-Sato counterpart. We have also analyzed old and recent investigations of the classical M. A. Buhl problem of describing compatible linear vector field equations, its general M.G. Pfeiffer and modern Lax-Sato type special solutions. Especially we analyzed the related Lie-algebraic structures and integrability properties of a very interesting class of nonlinear dynamical systems called the dispersionless heavenly type equations, which were initiated by Plebański and later analyzed in a series of articles. As effective tools the AKS-algebraic and related \(\mathcal R\)-structure schemes are used to study the orbits of the corresponding co-adjoint actions, which are intimately related to the classical Lie-Poisson structures on them. It is demonstrated that their compatibility condition coincides with the corresponding heavenly type equations under consideration. It is also shown that all these equations originate in this way and can be represented as a Lax-Sato compatibility condition for specially constructed loop vector fields on the torus. Typical examples of such heavenly type equations, demonstrating in detail their integrability via the scheme devised herein, are presented.
Mathematics Subject Classification ID: 17B80
Mathematics Subject Classification ID: 35Q53
Mathematics Subject Classification ID: 37K10
Mathematics Subject Classification ID: 58J70
zbMATH DE Number: 6898704
zbMATH Keywords: Lagrange-d'Alembert principle
zbMATH Keywords: Buhl vector field symmetry problem
zbMATH Keywords: Lax-Sato equations
zbMATH Keywords: heavenly equations
zbMATH Keywords: Lax integrability
zbMATH Keywords: Hamiltonian system
zbMATH Keywords: torus diffeomorphisms
zbMATH Keywords: loop Lie algebra
zbMATH Keywords: Lie-algebraic scheme
zbMATH Keywords: Casimir invariants
zbMATH Keywords: \(\mathcal R\)-structure
zbMATH Keywords: Lie-Poisson structure
MaRDI profile type: Publication
full work available at URL: math5040075
OpenAlex ID: W2774264867
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DOI: 10.3390/MATH5040075
description: scientific article; zbMATH DE number 6898704