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.
Highlights
The classical Lagrange-d’Alembert principle had a decisive influence on formation of modern analytical mechanics which culminated in modern Hamilton and Poisson mechanics
It is well known that modern analytical mechanics was founded mainly by such giants as Newton, Lagrange, d’Alembert, Posson, Hamilton, Maupertui and Jacobi, whose oevres strongly influenced the whole modern mechanical and physical sciences
It is worthwhile to stress here that the above constructive algorithm for generating completely integrable nonlinear multidimensional dynamical systems still does not make it possible to directly show they are Hamiltonian and construct other related mathematical structures. This important problem is solved by employing other mathematical theories; for example, the analytical properties of the related loop diffeomorphisms groups generated by the hierarchy of vector fields (28)
Summary
It is well known that modern analytical mechanics was founded mainly by such giants as Newton, Lagrange, d’Alembert, Posson, Hamilton, Maupertui and Jacobi, whose oevres strongly influenced the whole modern mechanical and physical sciences. Inasmuch the coefficients of vector fields (35) are suitably smooth functions on the manifold RZ+ × Tn , the compatibility conditions (36) yield the corresponding sets of differential-algebraic relationships on their coefficients, which have the common infinite set of invariants, thereby comprising an infinite hierarchy of completely integrable so called heavenly nonlinear dynamical systems on the corresponding multidimensional functional manifolds. It is worthwhile to stress here that the above constructive algorithm for generating completely integrable nonlinear multidimensional dynamical systems still does not make it possible to directly show they are Hamiltonian and construct other related mathematical structures This important problem is solved by employing other mathematical theories; for example, the analytical properties of the related loop diffeomorphisms groups generated by the hierarchy of vector fields (28). These and other interesting related aspects of the integrable heavenly dynamical systems shall be investigated separately elsewhere
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