Perturbation theory and canonical coordinates in celestial mechanics (Q6584158)

This paper comprises the notes of two courses given by the author, respectively, at the 18th School of Interaction Between Dynamical Systems and Partial Differential Equations (Barcelona, June 27--July 1, 2022) and at the XLVII Summer School on Mathematical Physics (Ravello, 12--24 September, 2022). Its purpose is to introduce classical perturbation theory and canonical coordinates, as well as new ideas and methods developed in recent years, to celestial mechanics.\N\NA key ingredient in this work is the Fundamental Theorem of Arnold that refers to a result related to the stability of motion in nearly integrable Hamiltonian systems, which is encapsulated in the Kolmogorov-Arnold-Moser (KAM) Theorem. Arnold extended the work started by Kolmogorov, providing rigorous conditions under which quasi-periodic orbits in integrable systems persist under small perturbations. This theorem became particularly relevant to the planetary problem once it became possible to use a coordinate system that effectively incorporated rotational invariance and the system's proximity to integrability. Since then, significant progress has been made in the symplectic analysis of the problem, and this work reviews these developments.\N\NThe manuscript under review is richly wrought, combining clear exposition, coherence, sharp mathematical insight, and concise communication of ideas. As such, this outstanding notes can be used not only as an introductory course at the graduate level in mathematics, but also as course material for engineering graduate students.\N\NSection 1 provides background on various sets of canonical coordinates used in many-body problems, specifically focusing on the following topics: \((1 + n)\)-body problem, Delaunay-Poincaré coordinates and the fundamental theorem of Arnold, Birkhoff theory for rotational invariant Hamiltonian systems, the rotational degeneracy, Jacobi reduction of the nodes, Deprit coordinates. In the last part, the maps \({\mathcal K}\) and \({\mathcal P}\) are introduced and their behavior under reflections is studied.\N\NIn Section 2, after the reader has developed a solid and intuitive understanding of the canonical coordinates used in many-body problems, the author introduces applications. Subsection 2.1 revisits the key ideas behind the proof of the Theorem 1.1 formulated by \textit{V. I. Arnol'd} in Chapter III, p. 125, of [Russ. Math. Surv. 18, No. 6, 85--191 (1963; Zbl 0135.42701); translation from Usp. Mat. Nauk 18, No. 6(114), 91--192 (1963)].\N\N\textbf{Theorem 1.1} (Theorem of stability of planetary motions). For the majority of initial conditions under which the instantaneous orbits of the planets are close to circles lying in a single plane, perturbation of the planets on one another produces, in the course of an infinite interval of time, little change on these orbits provided the masses of the planets are sufficiently small. [\dots] In particular [\dots] in the \(n\)-body problem there exists a set of initial conditions having a positive Lebesgue measure and such that, if the initial positions and velocities of the bodies belong to this set, the distances of the bodies from each other will remain perpetually bounded.\N\NAfter the symplectic reduction of the linear momentum, the \((1 + n)\)-body problem with masses \(m_0\), \(m_1,\dots , m_n\) is governed by a \(3n\)-degrees of freedom Hamiltonian. In this context, the Birkhoff normal form theory is applied under certain generic assumptions, indicating that a nearly integrable Hamiltonian system is close to an integrable one, and then KAM theory is applied. The quasi-periodic motions described in Theorem 1.1 lead to almost circular and nearly planar orbits. The final part is dedicated to showing how the use two different sets of coordinates leading to prove the co-existence of stable and whiskered tori. Additionally, the author notes that further detailed information on this topic is available in her articles e.g. [\textit{G. Pinzari}, J. Math. Phys. 59, No. 5, 052701, 37 p. (2018; Zbl 1391.70024); \textit{G. Pinzari} and \textit{X. Liu}, J. Nonlinear Sci. 33, No. 5, Paper No. 90, 45 p. (2023; Zbl 1525.37068)].