Symmetry in Nonlinear Mechanics: Averaging and Normalization Procedures, New Problems and Algorithms

Abstract

The idea of introducing coordinate transformations to simplify the analytic expression of a general problem is a powerful one. Symmetry and differential equations have been close partners since the time of the founding masters, namely, Sophus Lie (1842–1899), and his disciples. To this days, symmetry has continued to play a strong role. The ideas of symmetry penetrated deep into various branches of science: mathematical physics, mechanics and so on. The role of symmetry in perturbation problems of nonlinear mechanics, which was already used by many investigators since the 70-th (G. Hori, A. Camel, U. Kirchgraber), has been developed considerably in recent time to gain further understanding and development such constructive and powerful methods as averaging and normal form methods. Normalization techniques in the context with the averaging method was considered in works by A.M. Molchanov [1], A.D. Brjuno [2], S.N. Chow, J. Mallet–Paret [3], Yu.A. Mitropolsky, A.M. Samoilenko [4], J.A. Sanders, F. Verhulst [5]. An approach, where Lie series in parameter were used as transformation, was considered in works by G. Hori [6], [7], A. Kamel [8], U. Kirchgraber [10], U. Kirchgraber, E. Stiefel [9], V.N. Bogaevsky, A.Ya. Povzner [11], V.F. Zhuravlev, D.N. Klimov [12]. Asymptotic methods of nonlinear mechanics developed by N.M. Krylov, N.N. Bogolyubov and Yu.A. Mitropolsky and known as the KBM method (see, for example, Bogolyubov N.N. and Mitropolsky Yu.A [18]) is a powerful tool for investigation of nonlinear vibrations. The present lecture deals with the development of new normalization procedures and averaging algorithms in problems of nonlinear vibrations. Namely, the development of asymptotic methods of perturbation theory is considered, making wide use of group theoretical techniques. Various assumptions about specific group properties are investigated, and are shown to lead to modifications of existing methods, such as the Bogolyubov averaging method and the Poincare–Birkhoff normal form, as well as to formulation of new ones. The development of normalization techniques on Lie groups is also treated.