Publicación: ON THE RESTRICTED (N + 1)-BODY PROBLEM ON SURFACES OF CONSTANT CURVATURE
Fecha
2023
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JOURNAL OF DIFFERENTIAL EQUATIONS
Resumen
IN THIS PAPER, WE CONSIDER A RESTRICTED -BODY PROBLEM ON SURFACES , WHERE THE CONSTANT IS THE GAUSSIAN CURVATURE, WHICH BY MEANS OF A RESCALING CAN BE REDUCED TO . THIS PROBLEM CONSISTS IN THE STUDY OF THE DYNAMICS OF AN INFINITESIMAL MASS PARTICLE ATTRACTED BY N PRIMARIES OF IDENTICAL MASSES DESCRIBING ELLIPTIC RELATIVE EQUILIBRIA OF THE N-BODY PROBLEM ON , I.E., A SOLUTION WHERE THE PRIMARIES ARE ROTATING UNIFORMLY WITH CONSTANT ANGULAR VELOCITY ? ON A FIXED PARALLEL OF OR AND PLACED AT THE VERTICES OF A REGULAR POLYGON. IN A ROTATING FRAME, THIS PROBLEM YIELDS A TWO DEGREES OF FREEDOM HAMILTONIAN SYSTEM. THE GOAL OF THIS PAPER IS TO DESCRIBE ANALYTICALLY SOME DYNAMICS FEATURES FOR . PRECISELY, WE STUDY THE RELATIVE LOCATION OF EQUILIBRIA, OBTAINING, IN PARTICULAR, THAT THE POLES OF AND VERTEX OF REPRESENT EQUILIBRIUM POINTS FOR ANY VALUE OF THE PARAMETERS. THUS, ANALYSIS OF THE NONLINEAR STABILITY OF THESE EQUILIBRIA IS CARRIED OUT, AS WELL AS TWO TYPES OF BIFURCATIONS ARE DETECTED: HAMILTONIAN-HOPF AND N-BIFURCATION. ADDITIONALLY, WE PROVE THE EXISTENCE OF A FAMILY OF HILL'S ORBITS AND A FAMILY OF PERIODIC ORBITS WHEN THE PRIMARIES ARE LOCATED NEAR THE POLES OF OR THE VERTEX OF . FINALLY WE PROVE THE EXISTENCE OF KAM 2-TORI RELATED TO THESE PERIODIC ORBITS.
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Palabras clave
Surfaces of constant curvature, Ring solution, Periodic solutions, Non-linear stability, KAM, Bifurcations
