# KAM Stability and Celestial Mechanics

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*Alessandra Celletti; Luigi Chierchia*

KAM theory is a powerful tool apt to prove perpetual
stability in Hamiltonian systems, which are a perturbation of integrable ones.
The smallness requirements for its applicability are well known to be
extremely stringent. A long standing problem, in this context, is the
application of KAM theory to “physical systems” for
“observable” values of the perturbation parameters.

The authors consider the Restricted, Circular, Planar, Three-Body Problem
(RCP3BP), i.e., the problem of studying the planar motions of a small body
subject to the gravitational attraction of two primary bodies revolving on
circular Keplerian orbits (which are assumed not to be influenced by the small
body). When the mass ratio of the two primary bodies is small, the RCP3BP is
described by a nearly-integrable Hamiltonian system with two degrees of
freedom; in a region of phase space corresponding to nearly elliptical motions
with non-small eccentricities, the system is well described by Delaunay
variables. The Sun-Jupiter observed motion is nearly circular and an asteroid
of the Asteroidal belt may be assumed not to influence the Sun-Jupiter motion.
The Jupiter-Sun mass ratio is slightly less than 1/1000.

The authors consider the motion of the asteroid 12 Victoria taking into
account only the Sun-Jupiter gravitational attraction regarding such a system
as a prototype of a RCP3BP. For values of mass ratios up to 1/1000, they prove
the existence of two-dimensional KAM tori on a fixed three-dimensional energy
level corresponding to the observed energy of the Sun-Jupiter-Victoria system.
Such tori trap the evolution of phase points “close” to the
observed physical data of the Sun-Jupiter-Victoria system. As a consequence, in
the RCP3BP description, the motion of Victoria is proven to be forever close
to an elliptical motion.

The proof is based on: 1) a new iso-energetic KAM theory; 2) an
algorithm for computing iso-energetic, approximate Lindstedt series; 3) a
computer-aided application of 1)+2) to the Sun-Jupiter-Victoria system.

The paper is self-contained but does not include the (\(\sim\)
12000 lines) computer programs, which may be obtained by sending an e-mail to
one of the authors.

#### Table of Contents

# Table of Contents

## KAM Stability and Celestial Mechanics

- Contents v6 free
- Chapter 1. Introduction 110 free
- 1.1. Quasi-periodic solutions for the n-body problem 110
- 1.2. A stability theorem for the Sun-Jupiter-Victoria system viewed as a restricted, circular, planar three-body problem 312
- 1.3. About the proof of the Sun-Jupiter-Victoria stability theorem 615
- 1.4. A short history of KAM stability estimates 817
- 1.5. A section-by-section summary 1019

- Chapter 2. Iso-energetic KAM Theory 1322
- Chapter 3. The Restricted, Circular, Planar Three-body Problem 7584
- Chapter 4. KAM Stability of the Sun-Jupiter-Victoria Problem 95104
- 4.1. Iso-energetic Lindstedt series for the Sun-Jupiter-Asteroid problem and choice of the initial approximate tori (u[sup((0)±)], v[sup((0)±)], w[sup((0)±)]) 96105
- 4.2. Evaluation of the input parameters of the KAM norm map associated to the approximate tori (u[sup((0)±)], v[sup((0)±)], w[sup((0)±)]) 100109
- 4.3. Iterations of the KAM map 108117
- 4.4. Application of the iso-energetic KAM theorem and perpetual stability of the Sun-Jupiter-Victoria problem 112121

- Appendix A. The Ellipse 117126
- Appendix B. Diophantine Estimates 121130
- Appendix C. Interval Arithmetic 125134
- Appendix D. A Guide to the Computer Programs 127136
- Bibliography 129138