Complete integrability of dynamical systems associated with the problem of nonlinear vibrations of a longitudinally compressed elastic beam. (English. Russian original) Zbl 0626.73055

Kurbatov, A. M.; Prikarpatskij, A. K.; Chelomej, S. V.

Sov. Phys., Dokl. 31, 732-734 (1986); translation from Dokl. Akad. Nauk SSSR 290, 304-308 (1986).

Methods of the theory of dynamical systems and algebraic geometry are used to investigate the mechanical vibrations of a uniform elastic beam compressed at the ends. The nonlinear differential equation describing these vibrations has been shown to be a completely integrable Hamiltonian dynamical system on a special infinite-dimensional simple function space and admits an infinite sequence of first integrals in involution. However, embedding of nonlinear equation in the scheme of symplectic analysis as a dynamical system is based on certain propositions. The author has proposed four lemmata. The paper is of interest to mathematicians working in the area of nonlinear vibrations.

Reviewer: G.Ramaiah

MSC:

74H45	Vibrations in dynamical problems in solid mechanics
70H05	Hamilton’s equations
37J35	Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests
37K10	Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.)
74K10	Rods (beams, columns, shafts, arches, rings, etc.)
70K99	Nonlinear dynamics in mechanics
58J90	Applications of PDEs on manifolds

Keywords:

uniform elastic beam; infinite-dimensional simple function space; infinite sequence of first integrals; symplectic analysis