Electronic ISBN:  9781470404826 
Product Code:  MEMO/187/878.E 
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Book DetailsMemoirs of the American Mathematical SocietyVolume: 187; 2007; 134 ppMSC: Primary 70; Secondary 37;
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, ThreeBody 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 nearlyintegrable Hamiltonian system with two degrees of freedom; in a region of phase space corresponding to nearly elliptical motions with nonsmall eccentricities, the system is well described by Delaunay variables. The SunJupiter observed motion is nearly circular and an asteroid of the Asteroidal belt may be assumed not to influence the SunJupiter motion. The JupiterSun mass ratio is slightly less than 1/1000.
The authors consider the motion of the asteroid 12 Victoria taking into account only the SunJupiter 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 twodimensional KAM tori on a fixed threedimensional energy level corresponding to the observed energy of the SunJupiterVictoria system. Such tori trap the evolution of phase points “close” to the observed physical data of the SunJupiterVictoria 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 isoenergetic KAM theory; 2) an algorithm for computing isoenergetic, approximate Lindstedt series; 3) a computeraided application of 1)+2) to the SunJupiterVictoria system.
The paper is selfcontained but does not include the (\(\sim\) 12000 lines) computer programs, which may be obtained by sending an email to one of the authors. 
Table of Contents

Chapters

1. Introduction

2. Isoenergetic KAM theory

3. The restricted, circular, planar threebody problem

4. KAM stability of the SunJupiterVictoria problem


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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, ThreeBody 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 nearlyintegrable Hamiltonian system with two degrees of freedom; in a region of phase space corresponding to nearly elliptical motions with nonsmall eccentricities, the system is well described by Delaunay variables. The SunJupiter observed motion is nearly circular and an asteroid of the Asteroidal belt may be assumed not to influence the SunJupiter motion. The JupiterSun mass ratio is slightly less than 1/1000.
The authors consider the motion of the asteroid 12 Victoria taking into account only the SunJupiter 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 twodimensional KAM tori on a fixed threedimensional energy level corresponding to the observed energy of the SunJupiterVictoria system. Such tori trap the evolution of phase points “close” to the observed physical data of the SunJupiterVictoria 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 isoenergetic KAM theory; 2) an algorithm for computing isoenergetic, approximate Lindstedt series; 3) a computeraided application of 1)+2) to the SunJupiterVictoria system.
The paper is selfcontained but does not include the (\(\sim\) 12000 lines) computer programs, which may be obtained by sending an email to one of the authors.

Chapters

1. Introduction

2. Isoenergetic KAM theory

3. The restricted, circular, planar threebody problem

4. KAM stability of the SunJupiterVictoria problem