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Mathematics 2017, 5(4), 75; https://doi.org/10.3390/math5040075

Geometric Structure of the Classical Lagrange-d’Alambert Principle and Its Application to Integrable Nonlinear Dynamical Systems

1
Department of Physics, Mathematics and Computer Science at the Cracov University of Technology, 30-155 Krakow, Poland
2
The Institute for Applied Problems of Mechanics and Mathematics at the NAS, 79060 Lviv, Ukraine
3
Department of Applied Mathematics, University of Agriculture, 30059 Krakow, Poland
*
Author to whom correspondence should be addressed.
Received: 14 September 2017 / Revised: 11 November 2017 / Accepted: 25 November 2017 / Published: 5 December 2017
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Abstract

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 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. View Full-Text
Keywords: Lagrange-d’Alembert principle; M. Buhl vector field symmetry problem; Lax–Sato equations; heavenly equations; Lax integrability; Hamiltonian system; torus diffeomorphisms; loop Lie algebra; Lie-algebraic scheme; Casimir invariants; ℜ-structure; Lie-Poisson structure Lagrange-d’Alembert principle; M. Buhl vector field symmetry problem; Lax–Sato equations; heavenly equations; Lax integrability; Hamiltonian system; torus diffeomorphisms; loop Lie algebra; Lie-algebraic scheme; Casimir invariants; ℜ-structure; Lie-Poisson structure
This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. (CC BY 4.0).
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Prykarpatski, A.K.; Hentosh, O.E.; Prykarpatsky, Y.A. Geometric Structure of the Classical Lagrange-d’Alambert Principle and Its Application to Integrable Nonlinear Dynamical Systems. Mathematics 2017, 5, 75.

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