The application of monomial Lie groups to the problem of asymptotically integrating equations of mechanics (Q1093387)

The basis of the algorithm of the asymptotic integration of equations of mechanics discussed below is the representation of the initial system as a monomial Lie group of transformations of the phase space into itself. Transformations of the system which reduce it to a simpler form are also sought in a class of systems possessing group properties. Matching the instrument of the analysis to the objective of the analysis enables us to limit the operations used in the algorithm to those from the corresponding operator algebra. Hori's paper, in which Lie series were used to construct an additional first integral in an autonomous Hamiltonian system, was followed by a number of papers which extended this approach to autonomous systems of general form. Note that all these papers are essentially only different forms of deriving Hausdorff's formula, which is well-known from the theory of Lie groups, complicated somewhat by the concept of parameter identification and order separation. Now results can only be obtained by refusing to consider systems of general form and by proceeding to analyse the more special types of systems that are characteristic for those or other areas of asymptotic theory, with the aim of improving existing procedures. One should proceed from Hausdorff's formula, without repeating its conclusion. That course is taken here, where we consider systems in so-called single-frequency standard form. This form of system is basic for the well-known Krylov-Bogolyubov method, which also achieves substantial simplification using group-theoretical principles. The objective features of that simplification are: 1) there is no need to solve the derivatives transformed at each step of the system, or to invert the equation of change; 2) the algorithm does not use power series of the small parameter, and all discussion of the procedure can be carried out in terms of the required asymptotic forms; 3) the expression for the arbitrary approximation can be obtained in the form of an explicit recurrence formula which is convenient when using computers that perform symbol calculations. Following the author, ibid. 47, 559-565 (1983; Zbl 0562.70022), the proposed algorithm can be generalized to multifrequency systems, essentially non-linear systems and resonance cases can be handled. Examples of the application of the method are considered.