WE CONSIDER THE DYNAMICS OF A PASSIVE TRACER, ADVECTED BY THE PRESENCE OF A LATITUDINAL RING OF IDENTICAL POINT VORTICES. THE CORRESPONDING INSTANTANEOUS MOTION IS MODELED BY A ONE DEGREE OF FREEDOM HAMILTONIAN SYSTEM. SUCH A DYNAMICS PRESENTS A RICH VARIETY OF BEHAVIORS WITH RESPECT TO THE NUMBER OF VORTICES, N, AND THE RING?S CO-LATITUDE, ?O?OR, EQUIVALENTLY, ITS VERTICAL POSITION QO = COS??O. WE CARRY OUT A COMPLETE DESCRIPTION OF THE GLOBAL PHASE PORTRAIT FOR THE CASES N = 2, 3, 4 BY DETERMINING EQUILIBRIUM POINTS, THEIR STABILITY, AND BIFURCATIONS WITH RESPECT TO THE PARAMETER ?O, AND BY CHARACTERIZING THE SEPARATRIX SKELETON. MOREOVER, FOR N ? 5, WE PROVE THE EXISTENCE OF A VALUE OF BIFURCATION ?ON SUCH THAT WHEN ?O = ?ON (?O = ? ? ?ON, RESPECTIVELY) THE SOUTH (NORTH, RESPECTIVELY) POLE BECOMES A N-BIFURCATION POINT, I.E., A SYMMETRIC WEB OF N CENTERS AND N SADDLES BIFURCATES FROM THE CORRESPONDING POLE.

ABSTRACT THE DYNAMICS OF A TEST PARTICLE (A PARTICLE WITH ZERO VORTICITY) ADVECTED BY THE VELOCITY FIELD OF N POINT-VORTICES WITH VORTICITIES ?J, ?J = 1, ?N, IS CONSIDERED. MAKING AN ANALOGY WITH SIMILAR STUDIES IN CELESTIAL MECHANICS, WE CALL SUCH A STUDY A ?RESTRICTED N-VORTEX PROBLEM? OR (N + 1)-VORTEX PROBLEM. IN PARTICULAR, WE STUDY AND CHARACTERIZE THE GLOBAL PLANAR DYNAMICS OF SOME RESTRICTED 3 AND 4-VORTEX PROBLEMS, AS A FUNCTION OF THE VORTICITIES ?J OF THE VORTICES.

FINITE ENERGY QCD SUM RULES INVOLVING NUCLEON CURRENT CORRELATORS ARE USED TO DETERMINE SEVERAL QCD AND HADRONIC PARAMETERS IN THE PRESENCE OF AN EXTERNAL, UNIFORM, LARGE MAGNETIC FIELD. THE CONTINUUM HADRONIC THRESHOLD S0, NUCLEON MASS MN, CURRENT-NUCLEON COUPLINGN, TRANSVERSE VELOCITY V?, THE SPIN POLARIZATION CONDENSATE ?¯Q?12Q?, AND THE MAGNETIC SUSCEPTIBILITY OF THE QUARK CONDENSATE ?Q, ARE OBTAINED FOR THE CASE OF PROTONS AND NEUTRONS. DUE TO THE MAGNETIC FIELD, AND CHARGE ASYMMETRY OF LIGHT QUARKS UP AND DOWN, ALL THE OBTAINED QUANTITIES EVOLVE DIFFERENTLY WITH THE MAGNETIC FIELD, FOR EACH NUCLEON OR QUARK FLAVOR. WITH THIS APPROACH IT IS POSSIBLE TO OBTAIN THE EVOLUTION OF THE ABOVE PARAMETERS UP TO A MAGNETIC FIELD STRENGTH EB

THE FORMULATION OF THE DYNAMICS OF N-BODIES ON THE SURFACE OF AN INFINITE CYLINDER IS CONSIDERED. WE HAVE CHOSEN SUCH A SURFACE TO BE ABLE TO STUDY THE IMPACT OF THE SURFACE?S TOPOLOGY IN THE PARTICLE?S DYNAMICS. FOR THIS PURPOSE WE NEED TO MAKE A CHOICE OF HOW TO GENERALIZE THE NOTION OF GRAVITATIONAL POTENTIAL ON A GENERAL MANIFOLD. FOLLOWING BOATTO, DRITSCHEL AND SCHAEFER [5], WE DEFINE A GRAVITATIONAL POTENTIAL AS AN ATTRACTIVE CENTRAL FORCE WHICH OBEYS MAXWELL?S LIKE FORMULAS. AS A RESULT OF OUR THEORETICAL DIFFERENTIAL GALOIS THEORY AND NUMERICAL STUDY ? POINCARÉ SECTIONS, WE PROVE THAT THE TWO-BODY DYNAMICS IS NOT INTEGRABLE. MOREOVER, FOR VERY LOW ENERGIES, WHEN THE BODIES ARE RESTRICTED TO A SMALL REGION, THE TOPOLOGICAL SIGNATURE OF THE CYLINDER IS STILL PRESENT IN THE DYNAMICS. A PERTURBATIVE EXPANSION IS DERIVED FOR THE FORCE BETWEEN THE TWO BODIES. SUCH A FORCE CAN BE VIEWED AS THE PLANAR LIMIT PLUS THE TOPOLOGICAL PERTURBATION. FINALLY, A POLYGONAL CONFIGURATION OF IDENTICAL MASSES (IDENTICAL CHARGES OR IDENTICAL VORTICES) IS PROVED TO BE AN UNSTABLE RELATIVE EQUILIBRIUM FOR ALL N > 2.

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.

IN THIS PAPER, WE STUDY THE STABILITY OF THE RING SOLUTION OF THE N-BODY PROBLEM IN THE ENTIRE SPHERE S2 BY USING THE LOGARITHMIC POTENTIAL PROPOSED IN BOATTO ET AL. (2016, PROCEEDINGS OF THE ROYAL SOCIETY OF LONDON. SERIES A. MATHEMATICAL, PHYSICAL AND ENGINEERING SCIENCES 472, 20160020) AND DRITSCHEL (2019, PHILOSOPHICAL TRANSACTIONS OF THE ROYAL SOCIETY OF LONDON. SERIES A. MATHEMATICAL, PHYSICAL AND ENGINEERING SCIENCES 377, 20180349), DERIVED THROUGH A DEFINITION OF CENTRAL FORCE AND HODGE DECOMPOSITION THEOREM FOR 1-FORMS IN MANIFOLDS. FIRST, WE CHARACTERIZE THE RING SOLUTION AND STUDY ITS SPECTRAL STABILITY, OBTAINING REGIONS (SPHERICAL CAPS) WHERE THE RING SOLUTION IS SPECTRALLY STABLE FOR 2?N?6 , WHILE, FOR N?7 , THE RING IS SPECTRALLY UNSTABLE. THE NONLINEAR STABILITY IS STUDIED BY REDUCING THE SYSTEM TO THE HOMOGRAPHIC REGULAR POLYGONAL SOLUTIONS, OBTAINING A 2-D.O.F. HAMILTONIAN SYSTEM, AND THEREFORE SOME CLASSIC RESULTS ON STABILITY FOR 2-D.O.F. HAMILTONIAN SYSTEMS ARE APPLIED TO PROVE THAT THE RING SOLUTION IS UNSTABLE AT ANY PARALLEL WHERE IT IS PLACED. ADDITIONALLY, THIS SYSTEM CAN BE REDUCED TO 1-D.O.F. BY USING THE ANGULAR MOMENTUM INTEGRAL, WHICH ENABLES US TO DESCRIBE THE PHASE PORTRAITS AND USE THEM TO FIND PERIODIC RING SOLUTIONS TO THE FULL SYSTEM. SOME OF THOSE SOLUTIONS ARE NUMERICALLY APPROXIMATED.

WE CONSIDER A RESTRICTED THREE BODY PROBLEM ON SURFACES OF CONSTANT CURVATURE. AS IN THE CLASSICAL NEWTONIAN CASE THE COLLISION SINGULARITIES OCCUR WHEN THE POSITION PARTICLE WITH INFINITESIMAL MASS COINCIDES WITH THE POSITION OF ONE OF THE PRIMARIES. WE PROVE THAT THE SINGULARITIES DUE TO COLLISION CAN BE LOCALLY (EACH ONE SEPARATELY) AND GLOBALLY (BOTH AS THE SAME TIME) REGULARIZED THROUGH THE CONSTRUCTION OF LEVI-CIVITA AND BIRKHOFF TYPE TRANSFORMATIONS RESPECTIVELY. AS AN APPLICATION WE STUDY SOME GENERAL PROPERTIES OF THE HILL?S REGIONS AND WE PRESENT SOME EJECTION?COLLISION ORBITS FOR THE SYMMETRICAL PROBLEM.

IN THIS PAPER WE STUDY PART OF THE DYNAMICS OF A CIRCULAR RESTRICTED -BODY PROBLEM ON THE SPHERE AND CONSIDERING THE LOGARITHMIC POTENTIAL, WHERE PRIMARIES REMAIN IN A RING TYPE CONFIGURATION (IDENTICAL MASSES PLACED AT THE VERTICES OF A REGULAR POLYGON IN A FIXED PARALLEL AND ROTATING UNIFORMLY WITH RESPECT TO THE -AXIS) AND A -TH PRIMARY OF MASS FIXED AT THE SOUTH POLE OF . SUCH A PARTICULAR CONFIGURATION WILL BE CALLED RING-POLE CONFIGURATION (RP). AN INFINITESIMAL MASS PARTICLE HAS AN EQUILIBRIUM POSITION AT THE NORTH POLE FOR ANY VALUE OF , ANY PARALLEL WHERE THE RING HAS BEEN FIXED (WE USE AS PARAMETER , WHERE IS THE POLAR ANGLE OF THE RING) AND ANY NUMBER OF MASSES FORMING THE RING. WE STUDY THE NON-LINEAR STABILITY OF THE NORTH POLE IN TERMS OF THE PARAMETERS AND SOME BIFURCATIONS NEAR THE NORTH POLE.

IN THIS PAPER WE CONSIDER A SYMMETRIC RESTRICTED CIRCULAR THREE-BODY PROBLEM ON THE SURFACE S-2 OF CONSTANT GAUSSIAN CURVATURE KAPPA = 1. THIS PROBLEM CONSISTS IN THE DESCRIPTION OF THE DYNAMICS OF AN INFINITESIMAL MASS PARTICLE ATTRACTED BY TWO PRIMARIES WITH IDENTICAL MASSES, ROTATING WITH CONSTANT ANGULAR VELOCITY IN A FIXED PARALLEL OF RADIUS A IS AN ELEMENT OF(0, 1). IT IS VERIFIED THAT BOTH POLES OF S-2 ARE EQUILIBRIUM POINTS FOR ANY VALUE OF THE PARAMETER A. THIS PROBLEM IS MODELED THROUGH A HAMILTONIAN SYSTEM OF TWO DEGREES OF FREEDOM DEPENDING ON THE PARAMETER A. USING RESULTS CONCERNING NONLINEAR STABILITY, THE TYPE OF LYAPUNOV STABILITY (NONLINEAR) IS PROVIDED FOR THE POLAR EQUILIBRIA, ACCORDING TO THE RESONANCES. IT IS VERIFIED THAT FOR THE NORTH POLE THERE ARE TWO VALUES OF BIFURCATION (ON THE STABILITY) A = ROOT 4-ROOT 2/2 AND A = ROOT 2/3, WHILE THE SOUTH POLE HAS ONE VALUE OF BIFURCATION A = ROOT 3/2.

WE CONSIDER THE 2-BODY PROBLEM IN THE SPHERE S2. THIS PROBLEM IS MODELED BY A HAMILTONIAN SYSTEM WITH 4 DEGREES OF FREEDOM AND, FOLLOWING THE APPROACH GIVEN IN [4], ALLOWS US TO REDUCE THE STUDY TO A SYSTEM OF 2 DEGREES OF FREEDOM. IN THIS WORK WE WILL USE THEORETICAL TOOLS SUCH AS NORMAL FORMS AND SOME NONLINEAR STABILITY RESULTS ON HAMILTONIAN SYSTEMS FOR DEMONSTRATING A SERIES OF RESULTS THAT WILL CORRESPOND TO THE OPEN PROBLEMS PROPOSED IN [4] RELATED TO THE NONLINEAR STABILITY OF THE RELATIVE EQUILIBRIA. MOREOVER, WE STUDY THE EXISTENCE OF HAMILTONIAN PITCHFORK AND CENTER-SADDLE BIFURCATIONS.

IN THIS ARTICLE, WE DEFINE A CIRCULAR RESTRICTED (N +1)-BODY PROBLEM ON THE SURFACES M3?,WITH ? = ±1. THE MOTION OF THE PRIMARIES CORRESPONDS TO AN ELLIPTIC RELATIVE EQUILIBRIA STUDIED IN DIACU (RELATIVE EQUILIBRIA OF THE CURVED N-BODY PROBLEM. ATLANTIS STUDIES IN DYNAMICAL SYSTEMS, ATLANTIS PRESS, PARIS, 2012), WHERE N IDENTICAL MASS PARTICLES ARE ROTATING UNIFORMLY AT THE VERTICES OF A REGULAR POLYGON PLACED AT A FIXED PARALLEL OF A MAXIMAL SPHERE. BY INTRODUCING ROTATING COORDINATES, THIS PROBLEM GIVES RISE TO A 3 D.O.F. HAMILTONIAN SYSTEM. THIS PROBLEM HAS AN EQUILIBRIUM POINT PLACED AT THE POLES OF S3 AND THE VERTEX OF H3, FOR ANY VALUE OF THE PARAMETERS. WE GIVE INFORMATION ABOUT THE LINEAR AND NONLINEAR STABILITY OF THESE EQUILIBRIA. FINALLY, WE CARRY OUT A STUDY ABOUT THE EXISTENCE OF PERIODIC SOLUTIONS AND KAM TORI.

WE CONSIDER A SYMMETRIC RESTRICTED THREE-BODY PROBLEM ON SURFACES OF CONSTANT GAUSSIAN CURVATURE , WHICH CAN BE REDUCED TO THE CASES . THIS PROBLEM CONSISTS IN THE ANALYSIS OF THE DYNAMICS OF AN INFINITESIMAL MASS PARTICLE ATTRACTED BY TWO PRIMARIES OF IDENTICAL MASSES DESCRIBING ELLIPTIC RELATIVE EQUILIBRIA OF THE TWO BODY PROBLEM ON , I.E., THE PRIMARIES MOVE ON OPPOSITE SIDES OF THE SAME PARALLEL OF RADIUS A. THE HAMILTONIAN FORMULATION OF THIS PROBLEM IS POINTED OUT IN INTRINSIC COORDINATES. THE GOAL OF THIS PAPER IS TO DESCRIBE ANALYTICALLY, IMPORTANT ASPECTS OF THE GLOBAL DYNAMICS IN BOTH CASES AND DETERMINE THE MAIN DIFFERENCES WITH THE CLASSICAL NEWTONIAN CIRCULAR RESTRICTED THREE-BODY PROBLEM. IN THIS SENSE, WE DESCRIBE THE NUMBER OF EQUILIBRIA AND ITS LINEAR STABILITY DEPENDING ON ITS BIFURCATION PARAMETER CORRESPONDING TO THE RADIAL PARAMETER A. AFTER THAT, WE PROVE THE EXISTENCE OF FAMILIES OF PERIODIC ORBITS AND KAM 2-TORI RELATED TO THESE ORBITS.