On the use of the autonomous Birkhoff equations in Lie series perturbation theory

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• 作者：T. S. Boronenko
• 关键词：Perturbation theory ; Averaging method ; Lie transformations ; Birkhoff’s equations ; Restricted three ; body problem ; Satellites dynamics ; Pfaffian
• 刊名：Celestial Mechanics and Dynamical Astronomy
• 出版年：2017
• 出版时间：February 2017
• 年：2017
• 卷：127
• 期：2
• 页码：139-161
• 全文大小：
• 刊物类别：Physics and Astronomy
• 刊物主题：Astrophysics and Astroparticles; Dynamical Systems and Ergodic Theory; Aerospace Technology and Astronautics; Geophysics/Geodesy; Classical Mechanics;
• 出版者：Springer Netherlands
• ISSN：1572-9478
• 卷排序：127

文摘

In this article, we present the Lie transformation algorithm for autonomous Birkhoff systems. Here, we are referring to Hamiltonian systems that obey a symplectic structure of the general form. The Birkhoff equations are derived from the linear first-order Pfaff–Birkhoff variational principle, which is more general than the Hamilton principle. The use of 1-form in formulating the equations of motion in dynamics makes the Birkhoff method more universal and flexible. Birkhoff’s equations have a tensorial character, so their form is independent of the coordinate system used. Two examples of normalization in the restricted three-body problem are given to illustrate the application of the algorithm in perturbation theory. The efficiency of this algorithm for problems of asymptotic integration in dynamics is discussed for the case where there is a need to use non-canonical variables in phase space.