WE STUDY THE ROTO-ORBITAL DYNAMICS OF A UNIFORM SPHERE AND A BODY WITH AXIAL SYMMETRY BY MEANS OF A RADIAL INTERMEDIARY, WHICH DEFINES AN INTEGRABLE SYSTEM. NUMERICAL COMPARISONS OF THE MACCULLAGH?S TRUNCATION OF THE GRAVITY GRADIENT POTENTIAL AND INTERMEDIARY MODELS ARE PERFORMED, CONCLUDING THAT THE INTERMEDIARY PROVIDES A VALUABLE APPROXIMATION WITH SMALL DIFFERENCES WHEN COMPARED WITH THE MACCULLAGH?S ONE. OUR ANALYSIS INCLUDES THE ANALYTICAL INTEGRATION AND A STUDY OF THE SPECIAL SOLUTIONS AND RELATIVE EQUILIBRIA.

WE STUDY THE ROTO-ORBITAL DYNAMICS OF A UNIFORM SPHERE AND A TRIAXIAL BODY BYMEANS OF A MODEL WHICH DE?NES A 2-DOF HAMILTONIAN SYSTEM USING VARIABLES REFERRED TOTHE TOTAL ANGULAR MOMENTUM. THE VALIDITY AND APPLICABILITY OF OUR MODEL IS BEEN ASSESSEDNUMERICALLY. WE PRESENT A CLASSI?CATION OF SOME RELATIVE EQUILIBRIA, ?NDING CONSTANT RA-DIUS SOLUTIONS ?LLING 4-D AND LOWER DIMENSIONAL TORI. THESE FAMILIES OF RELATIVE EQUILIBRIAINCLUDE SOME OF THE CLASSICAL ONES REPORTED IN THE LITERATURE AND SOME NEW TYPES SHOW-ING THE TRIAXIALITY IN?UENCE ON BOTH. FOR A NUMBER OF SCENARIOS THE RELATION BETWEEN THETRIAXIALITY AND THE INCLINATION CONNECTED WITH RELATIVE EQUILIBRIA ARE DISCUSSED AND A FULLANALYSIS IN IN PROGRESS.

THIS WORK ESTABLISHES THE EXISTENCE OF ADDITION THEOREMS AND DOUBLE-ANGLE FORMULAS FOR CK REAL SCALAR FUNCTIONS. MOREOVER, WE DETERMINE NECESSARY AND SUFFICIENT CONDITIONS FOR A BIVARIATE FUNCTION TO BE AN ADDITION FORMULA FOR A CK REAL FUNCTION. THE DOUBLE-ANGLE FORMULAS ALLOW US TO GENERATE A DUPLICATION ALGORITHM, WHICH CAN BE USED AS AN ALTERNATIVE TO THE CLASSICAL NUMERICAL METHODS TO OBTAIN AN APPROXIMATION FOR THE SOLUTION OF AN ORDINARY DIFFERENTIAL EQUATION. WE DEMONSTRATE THAT THIS ALGORITHM CONVERGES UNIFORMLY IN ANY COMPACT DOMAIN CONTAINED IN THE MAXIMAL DOMAIN OF THAT SOLUTION. FINALLY, WE CARRY OUT SOME NUMERICAL SIMULATIONS SHOWING A GOOD PERFORMANCE OF THE DUPLICATION ALGORITHM WHEN COMPARED WITH STANDARD NUMERICAL METHODS

THE KS MAP IS REVISITED IN TERMS OF AN S-1-ACTION IN (TH0)-H-* WITH THE BILINEAR FUNCTION AS THE ASSOCIATED MOMENTUM MAP. INDEED, THE KS TRANSFORMATION MAPS THE S-1-FIBERS RELATED TO THE MENTIONED ACTION TO SINGLE POINTS. BY MEANS OF THIS PERSPECTIVE A SECOND TWIN-BILINEAR FUNCTION IS OBTAINED WITH AN ANALOGOUS S-1-ACTION. WE ALSO SHOW THAT THE CONNECTION BETWEEN THE 4-D ISOTROPIC HARMONIC OSCILLATOR AND THE SPATIAL KEPLER SYSTEMS CAN BE DONE IN A STRAIGHTFORWARD WAY AFTER REGULARIZATION AND THROUGH THE EXTENSION TO 4 DEGREES OF FREEDOM OF THE EULER ANGLES, WHEN THE BILINEAR RELATION IS IMPOSED. THIS CONNECTION INCORPORATES BOTH BILINEAR FUNCTIONS AMONG THE VARIABLES. WE WILL SHOW THAT AN ALTERNATIVE REGULARIZATION SEPARATES THE OSCILLATOR EXPRESSED IN PROJECTIVE EULER VARIABLES. THIS SETTING TAKES ADVANTAGE OF THE TWO BILINEAR FUNCTIONS AND ANOTHER INTEGRAL OF THE SYSTEM INCLUDING THEM AMONG A NEW SET OF VARIABLES THAT ALLOWS TO CONNECT THE 4-D ISOTROPIC HARMONIC OSCILLATOR AND THE PLANAR KEPLER SYSTEM. IN ADDITION, OUR APPROACH MAKES TRANSPARENT THAT ONLY WHEN WE REFER TO RECTILINEAR SOLUTIONS, BOTH BILINEAR RELATIONS DEFINING THE KS TRANSFORMATIONS ARE NEEDED.

THIS WORK DEALS WITH THE FULL REDUCTION OF THE SPATIAL KEPLER SYSTEM FOR BOUNDED AND UNBOUNDED MOTIONS. PRECISELY, WE CONSIDER THE FOUR-DIMENSIONAL OSCILLATOR ASSOCIATED TO THE KEPLER SYSTEM AND CARRY OUT OUR PROGRAM IN THREE STAGES: AXIAL-AXIAL-ENERGY RATHER THAN ENERGY-AXIAL-AXIAL AS IS CUSTOMARY. THIS APPROACH REVEALS THE TRUE ROLE OF THE KS MAP AND THE BILINEAR RELATION. OUR DEVELOPMENT ALLOWS FOR A GLOBAL ANALYSIS AT EACH REDUCTION STAGE PROVIDING A COMPLETE DESCRIPTION OF EACH REDUCED SPACE. THE FIRST REDUCED SPACE IS DESCRIBED BY MEANS OF ONLY SIX INVARIANTS LEADING TO A NONSINGULAR SIX-DIMENSIONAL POISSON MANIFOLD. THEN, THE CLASSICAL BILINEAR RELATION IS REPLACED BY THE MOMENTUM MAP OF A GEOMETRIC REDUCTION. FURTHERMORE, THE SECOND REDUCED SPACE IS GIVEN AS THE PRODUCT OF TWO HYPERBOLOIDS AND HAS CONES AS SINGULAR STRATA. FOR THE LAST STAGE, WE DISTINGUISH AMONG THE POSSIBLE SIGN OF THE ENERGY. THE POSITIVE AND ZERO CASES BRING NEW NONCOMPACT REDUCED SPACES WHICH ARE DESCRIBED IN DETAIL.

WE ADDRESS THE ATTITUDE DYNAMICS OF A TRIAXIAL RIGID BODY IN A CIRCULAR ORBIT. THIS TASK IS DONE BY MEANS OF PARTIAL AVERAGING OF THE FAST ANGLES LEADING TO A 1-DOF SYSTEM. THIS TASK IS DONE BY MEANS OF AN INTERMEDIARY MODEL, WHICH IS OBTAINED BY SPLITTING THE HAMILTONIAN IN THE FORM H = H0 +H1, WHERE H0 DEFINES A NON-DEGENERATE INTEGRABLE 1-DOF HAMILTONIAN SYSTEM. THIS MODEL DEFINES A POISSON FLOW BASED ON THE USE OF THE INVARIANTS DEFINING A S2M × S2M REDUCED SPACE. WE ANALYZE THE COUPLING BETWEEN THE ORBITAL MEAN MOTION AND ROTATIONAL VARIABLES. THE KEY ROLE PLAYED BY THE MOMENTS OF INERTIA AND THE VALUE OF THE ANGULAR MOMENTUM IN THE DIFFERENT TYPES OF RELATIVE EQUILIBRIA AND BIFURCATIONS IS SHOWN IN DETAIL. THE ANALYSIS OF THE AVERAGED SYSTEM H? SHOWS THAT UNDER SLOW ROTATION THE CLASSIC DYNAMICS OF THE FREE RIGID BODY IS NO LONGER MAINTAINED: SEVERAL BIFURCATIONS WITH CHANGES OF STABILITY ARE DISPLAY BY THE AVERAGE SYSTEM. MOREOVER, THE PRECESSION OF THE ANGULAR MOMENTUM PLANE ALWAYS CIRCULATES, ACCORDING TO THE DYNAMICS OF THE ANGLE ?, WHICH IS GIVEN BY A TIME DEPENDENT HARMONIC OSCILLATOR. A NUMERICAL STUDY IS PRESENTED COMPARING THE DYNAMICS OF THE AVERAGING WITH THE FULL SYSTEM, SHOWING THAT IT IS A GOOD APPROXIMATION FOR MANY INITIAL CONDITIONS..

THE EVOLUTION OF MASTER AND MUTANT CANCER CELL POPULATIONS IS STUDIED BY MEANS OF THE SO-CALLED QUASISPECIES MODEL OF SWENTINA-SCHUSTER. THE GENERAL FLOW AND BIFURCATIONS OF THIS SYSTEM ARE ANALYZED. OUR STUDY RELIES IN THREE PARAMETERS: THE PROBABILITY OF REPLICATION WITHOUT ERRORS IN THE MASTER SEQUENCE, THE PROBABILITY OF ERROR IN THE MUTANT SEQUENCE LEADING TO A CELL OF THE MASTER ONE AND THE RATIO BETWEEN THE MASTER AND MUTANT SEQUENCES GROWTH RATES. THESE PARAMETERS LEAD TO A VARIETY OF DIFFERENT SCENARIOS WITH THREE EQUILIBRIA. OUR APPROACH DISTINGUISH BETWEEN THE MATHEMATICAL ANALYSIS AND THE APPLICATIONS, PROVIDED THAT NOT EVERY STATE HAS A BIOLOGICAL INTERPRETATION. HOWEVER, A COMPREHENSIVE MATHEMATICAL STUDY LEADS TO A BETTER UNDERSTANDING OF THE BIOLOGICAL BEHAVIOR. IN THIS REGARD, A COMPLETE EXPLANATION OF THE 2-DIMENSIONAL MODEL IS PROVIDED, COMPLETING AND ENLARGING PREVIOUS WORKS IN THE LITERATURE. PRECISELY, CONDITIONS ON THE PARAMETERS ARE GIVEN LEADING TO ENSURE THE EXISTENCE OF TWO EQUILIBRIA STATES WITH BIOLOGICAL MEANING. ONE OF THEM CORRESPONDS TO THE EXTINCTION OF THE MASTER POPULATION AND IS UNSTABLE. THE SECOND EQUILIBRIA IS STABLE AND IT BELONGS TO THE REGION WITH BIOLOGICAL INTERPRETATION FOR ANY VALUE OF THE PARAMETERS. THIS EQUILIBRIA GENERICALLY RESEMBLES THE COEXISTENCE BETWEEN POPULATIONS AND MAY RANGE BETWEEN THE EXTREME CASES OF THE EXTINCTION OF THE MASTER OR MUTANT CELLS.

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

WE PROPOSE A MODIFIED MATHEMATICAL MODEL OF THE QUASISPECIES TYPE TO ANALYZE AN UNSTABLE TUMOR PROGRESSION EVOLUTION. IN OUR STUDY, WE CONSIDER A HETEROGENEOUS POPULATION WITH DIFFERENT INDIVIDUALS, GENERATED BY THE ACCUMULATION OF SUCCESSIVE MUTATIONS. OUR MODEL?S MAIN FEATURE IS THAT IT ALLOWS FOR VARIABLE GROWTH RATES FOR EACH SUBPOPULATION AND TAKES INTO ACCOUNT MUTATIONS FROM NONCONSECUTIVE TYPES OF MUTANTS. BIFURCATIONS AND LINEAR STABILITY OF THE STEADY STATES ARE ANALYZED. WE FOCUS ON TWO EQUILIBRIA; ONE OF THEM IMPLIES THE COEXISTENCE OF ANOMALOUS GROWTH AND GENETICALLY UNSTABLE CELLS. THE OTHER ONE YIELDS THE DOMINANCE OF THE ANOMALOUS GROWTH POPULATION AND THE EXTINCTION OF THE MALIGNANT CELLS. HOWEVER, LINEAR STABILITY ANALYSIS OF THE SECOND EQUILIBRIUM IS INCONCLUSIVE AND SUGGESTS A SUITABLE ENVIRONMENT FOR THE STUDY OF PERIODIC THERAPY. THIS IS CARRIED OUT BY INTRODUCING A SMALL PERTURBATION MODELING THE EFFECT OF A PERIODIC MEDICAL TREATMENT. AS A RESULT, A ZERO-HOPF PERIODIC ORBIT IS IDENTIFIED, SHOWING A CYCLIC BEHAVIOR AMONG THE POPULATIONS, WITH A STRONG DOMINANCE OF THE PARENTAL ANOMALOUS GROWTH CELL POPULATION.

IN THIS WORK, WE CONSIDER A GENERIC 2-DOF AUTONOMOUS HAMILTONIAN SYSTEM, WHICH LINEAR PART HAS ALL EIGENVALUES EQUAL TO ZERO. FOR THESE TYPE OF HAMILTONIAN, WE PROVIDE, IN AN EXPLICIT AND CONSTRUCTIVE WAY, THE NORMAL FORM UP TO THE TERM OF THE FOURTH ORDER. ADDITIONALLY, WE ANALYZE THE STABILITY OF THE NORMALIZED HAMILTONIAN IN SEVERAL CASES. PRECISELY, WE OBTAIN INSTABILITY OF THE TRUNCATED HAMILTONIAN BY EXPLICITLY BUILDING AN INVARIANT RAY SOLUTION.

WE PRESENT A GENERALIZATION OF MOSER?S THEOREM ON THE REGULARIZATION OF KEPLERIAN SYSTEMS THAT INCLUDE POSITIVE AND NEGATIVE ENERGIES. OUR APPROACH DOES NOT CONSIDER THE GEODESICS OF THE HYPERBOLOID EMBEDDED IN A LORENTZ SPACE FOR THE UNBOUNDED ORBITS, AS IT IS PREVIOUSLY DONE IN THE LITERATURE. INSTEAD, WE CONNECT THE KEPLERIAN POSITIVE AND NEGATIVE ENERGY ORBITS WITH THE HARMONIC OSCILLATOR WITH NEGATIVE AND POSITIVE FREQUENCIES. THE CONNECTION IS ESTABLISHED THROUGH THE CANONICAL EXTENSION OF THE STEREOGRAPHIC PROJECTION, AS IT IS DONE IN MOSER?S ORIGINAL PAPER. HOW WE BASE OUR STUDY REVEALS THAT KUSTAANHEIMO?STIEFEL MAP KS AND MOSER REGULARIZATIONS ARE ALTERNATIVE WAYS OF SHOWING THE SPATIAL KEPLER SYSTEM AS A SUBDYNAMICS OF THE 4D HARMONIC OSCILLATOR.

THIS PAPER DEALS WITH THE FULL GRAVITATIONAL N-BODY PROBLEM. WE EXPRESS THE NEWTON?EULER EQUA- TIONS OF MOTION INTO THE HAMILTONIAN FORMALISM EMPLOYING A NONCANONICAL POISSON STRUCTURE. FOR THIS SYSTEM, WE IDENTIFY THE FULL-SYMMETRY GROUP GIVEN BY THE GALILEAN GROUP. WE CARRY OUT A PARTIAL REDUCTION IN TWO STAGES: TRANSLATIONAL AND ROTATIONAL SYMMETRIES. MOREOVER, WE IDENTIFY THE POISSON STRUCTURE AT EACH STAGE OF THE REDUCTION PROCESS. A CHARACTERIZATION OF RELATIVE EQUILIBRIA IS PROVIDED. IT ALLOWS FOR CLASSIFICATION AND SHOWS THAT THE CENTERS OF MASS OF ALL BODIES MOVE IN PARALLEL PLANES. THE CASE OF N > 2 REVEALS A RATHER DIFFERENT SCENARIO FROM N = 2. PRECISELY, THE RELATIVE EQUILIBRIA ARE CLASSIFIED AS FOLLOWS: LAGRANGIAN EQUILIBRIA, IN WHICH ALL THE BODIES ARE MOVING IN THE SAME PLANE; NONLAGRANGIAN EQUILIBRIA, IN WHICH EACH BODY IS IN A DIFFERENT PLANE; AND SEMI-LAGRANGIAN EQUILIBRIA, IN WHICH SOME OF THE BODIES SHARE A COMMON PLANE BUT NOT ALL OF THEM ARE IN THE SAME ONE. THE MAIN NOVELTY IN THE EQUILIBRIA CLASSIFICATION IS THAT THE PLANE OF MOTION OF THE CENTER OF MASS DOES NOT NEED TO BE PARALLEL TO THE PLANE DETERMINED BY THE TOTAL ANGULAR MOMENTUM. IN OUR ANALYSIS, WE SPECIFY SUFFICIENT CONDITIONS ENSURING THE PLANES OF MOTION ARE PARALLEL TO THE TOTAL ANGULAR MOMENTUM PLANE.

WE PRESENT A GEOMETRICAL DESCRIPTION OF THE SYMMETRIES AND REDUCTION OF THE FULL GRAVITATIONAL 2-BODY PROBLEM AFTER COMPLETE AVERAGING OVER FAST ANGLES. OUR VARIABLES ALLOW FOR A WELL-SUITED FORMULATION IN ACTION-ANGLE TYPE COORDINATES ASSOCIATED WITH THE AVERAGED ANGLES, WHICH PROVIDE GEOMETRIC INSIGHT INTO THE PROBLEM. AFTER INTRODUCING EXTRA FICTITIOUS VARIABLES AND THROUGH A SYMPLECTIC TRANSFORMATION, WE MOVE TO A SINGULARITY-FREE QUATERNIONIC TRIPLE-CHART. THIS CHOICE ALLOWS FOR A GLOBAL CHART TO AVOID THE CLASSICAL SINGULARITIES ASSOCIATED WITH ANGLES AND RENDERS ALL THE INVARIANTS AS HOMOGENEOUS QUADRATIC POLYNOMIALS. ADDITIONALLY, IT PERMITS ONE TO QUICKLY WRITE THE HAMILTONIAN OF THE SYSTEM IN TERMS OF THE INVARIANTS AND THE POISSON STRUCTURE AT EACH STAGE OF THE REDUCTION PROCESS. IN CONTRAST WITH EXISTING LITERATURE, THE GEOMETRICAL APPROACH OF THIS RESEARCH COMPLETELY DESCRIBES ALL THE DYNAMICAL ASPECTS OF THE FULL REDUCED SPACE SINCE IT INVOLVES THE RELATIVE POSITION OF THE ROTATIONAL AND ORBITAL ANGULAR MOMENTA AND THEIR ORIENTATION, WHICH HAS YET TO BE CONSIDERED IN PREVIOUS STUDIES. OUR PROGRAM INCLUDES A PRELIMINARY PARAMETRIC ANALYSIS OF RELATIVE EQUILIBRIA AND A COMPLETE DESCRIPTION OF THE FIBERS IN THE RECONSTRUCTION OF THE REDUCED SYSTEM.

WE STUDY THE REDUCTION AND REGULARIZATION PROCESSES OF PERTURBED KEPLERIAN SYSTEMS FROM AN ASTRONOMICAL POINT OF VIEW. OUR APPROACH CONNECTS AXIALLY SYMMETRIC PERTURBED 4-DOF OSCILLATORS WITH KEPLERIAN SYSTEMS, INCLUDING THE CASE OF RECTILINEAR SOLUTIONS. THIS IS DONE THROUGH A PRELIMINARY REDUCTION RECENTLY STUDIED BY THE AUTHORS. THEN, THE REDUCTION PROGRAM CONTINUES BY REMOVING THE KEPLERIAN ENERGY. FOR EACH VALUE OF THE SEMI-MAJOR AXIS, WE EXPLAIN THE ASTRONOMICAL MEANING OF THE SEXTUPLES DEFINING THE ORBIT SPACE S2×S2 AND ITS CONNECTION WITH THE ORBITAL ELEMENTS. MORE PRECISELY, WE PRESENT ALTERNATIVE SEXTUPLE COORDINATES FOR THE SET OF BOUNDED KEPLERIAN ORBITS THAT ?SEPARATE? THE NODE OF THE ORBITAL PLANE FROM THE ARGUMENT OF PERIGEE GIVING THE LAPLACE VECTOR IN THAT PLANE. STILL, THE REDUCTION OF THE AXIAL SYMMETRY DEFINED BY THE THIRD COMPONENT OF THE ANGULAR MOMENTUM IS PERFORMED. FOR THE THRICE REDUCED SPACE ?0,L,H WE PROPOSE THE CUSHMAN?DEPRIT COORDINATES, A VARIANT TO THE SET GIVEN BY CUSHMAN. THE MAIN FEATURE OF THESE VARIABLES IS THAT THEY ARE ALL WITH THE SAME DIMENSIONS, WHICH IS CONVENIENT FOR THE NORMALIZATION PROCEDURE. AS AN APPLICATION OF THE PROPOSED SCHEME, WE STUDY THE SPATIAL LUNAR PROBLEM.

WE ADDRESS THE ATTITUDE DYNAMICS OF A TRIAXIAL RIGID BODY IN A CIRCULAR ORBIT. THIS TASK IS DONE BY MEANS OF AN INTERMEDIARY MODEL, WHICH IS OBTAINED BY SPLITTING THE HAMILTONIAN IN THE FORM H = H-0 + H-1, WHERE H-0 IS REQUIRED TO BE A NON-DEGENERATE INTEGRABLE HAMILTONIAN SYSTEM. A NUMERICAL STUDY IS PRESENTED COMPARING THE DYNAMICS OF THE NEW INTERMEDIARY MODEL WITH THE FULL SYSTEM (MACCULLAGH'S TRUNCATION) AND SHOWING A COMPETITIVE PERFORMANCE FOR THE CASES SUN-ASTEROID AND EARTH-SPACECRAFT. THIS MODEL DEFINES A POISSON FLOW ENDOWED WITH INVARIANTS DEFINING A S-M(2) X S-M(2) REDUCED SPACE. WE ANALYZE THE COUPLING BETWEEN THE ORBITAL MEAN MOTION AND ROTATIONAL VARIABLES. THE KEY ROLE PLAYED BY THE MOMENTS OF INERTIA AND THE VALUE OF THE ANGULAR MOMENTUM IS SHOWN IN DETAIL. THE ANALYSIS OF THE INTERMEDIARY SHOWS THAT, UNDER SLOW ROTATION REGIME, THE CLASSIC DYNAMICS OF THE FREE RIGID BODY IS NO LONGER MAINTAINED: BIFURCATIONS WITH CHANGES OF STABILITY ARE DISPLAYED FOR SEVERAL CRITICAL INCLINATIONS OF THE ROTATIONAL ANGULAR MOMENTUM PLANE AND FOR CRITICAL ORIENTATIONS OF THE BODY FRAME. MOREOVER, THE EVOLUTION OF THE ANGULAR MOMENTUM PLANE IS GIVEN BY A TIME DEPENDENT HARMONIC OSCILLATOR. (C) 2019 COSPAR. PUBLISHED BY ELSEVIER LTD. ALL RIGHTS RESERVED.

WE STUDY SPECIAL SOLUTIONS OF GALACTIC TIDAL MODELS AND THE EXISTENCE, STABILITY AND BIFURCATIONS OF RELATIVE EQUILIBRIA. PRECISELY, WE FOUND FOUR EQUILIBRIA AND RECTILINEAR, AS WELL AS CIRCULAR SOLUTIONS, PREVIOUS TO ANY MANIPULATION OF THE ORIGINAL SYSTEM. MOREOVER, BY AVERAGING OF THE KEPLERIAN ENERGY AND MAKING USE OF REEBS THEOREM, WE FIND FOUR PERIODIC ORBITS. THE STABILITY OF THESE RELATIVE EQUILIBRIA IS ANALYSED AND DETERMINED IN SEVERAL CASES THAT INVOLVE A RELATION BETWEEN THE KEPLERIAN ENERGY AND THE GRAVITATIONAL PARAMETER. FURTHER AVERAGING ON THE ANGLE GIVING THE ORBITAL PLANE NODE PROVIDES THE MATESE?WHITMAN MODEL. THIS HAMILTONIAN IS EXPRESSED AS A FUNCTION OF THE INVARIANTS OF THE TWICE-REDUCED SPACE, WHICH IS GIVEN BY THREE CONNECTED COMPONENTS. FOR H=0, THE THREE CONNECTED COMPONENTS BECOME ATTACHED TO EACH OTHER BY TWO SINGULAR POINTS. ALTHOUGH ONLY THE BOUNDED COMPONENT IS ENDOWED WITH PHYSICAL MEANING, WE PROVIDE A FULL ANALYSIS OF THE STABILITY AND BIFURCATIONS FOR ALL THE EQUILIBRIA. BY DOING THIS, WE GET AN INSIGHT INTO THE BIFURCATION PROCESS. PRECISELY, WE OBSERVE THAT FOR THE SINGULAR LEAF OF THE REDUCED SPACE, SOME OF THE EQUILIBRIA IN THE BOUNDED AND UNBOUNDED COMPONENTS SWITCH THEIR LOCATIONS.

WE STUDY THE ROTO-ORBITAL MOTION OF A TRIAXIAL RIGID BODY AROUND A SPHERE, WHICH IS ASSUMED TO BE MUCH MORE MASSIVE THAN THE TRIAXIAL BODY. THE ASSOCIATED DYNAMICS OF THIS SYSTEM, WHICH CONSISTS OF A NORMALIZED HAMILTONIAN WITH RESPECT TO THE FAST ANGLES (PARTIAL AVERAGING), IS INVESTIGATED MAKING USE OF VARIABLES REFERRED TO THE TOTAL ANGULAR MOMENTUM (...)

A GEOMETRICAL APPROACH TO A RADIAL INTERMEDIARY MODEL FOR AN AXISYMMETRIC RIGID BODY IN ROTO-ORBITAL MOTION IS PRESENTED. THE PRESENCE OF SYMMETRIES ENABLES A WELL-SUITED FORMULATION BY CHOOSING ACTION?ANGLE TYPE VARIABLES. SINGULARITIES ASSOCIATED WITH THE ANGLES ARE AVOIDED BY INTRODUCING EXTRA FICTITIOUS VARIABLES AND PERFORMING A SYMPLECTIC TRANSFORMATION LEADING TO A GLOBAL, QUATERNIONIC DOUBLE-CHART. THEN, MAKING USE OF THE AND SYMMETRY OF OUR MODEL, A FULL REDUCTION PROCESS BY STAGES IS CARRIED OUT, WHICH IN COMBINATION WITH THE CONSTRAINED DYNAMICS RELATED TO THE FICTITIOUS VARIABLES, LEADS TO A 1-DOF REDUCED-CONSTRAINED SYSTEM. OUR PROGRAM INCLUDES A PARAMETRIC ANALYSIS OF RELATIVE EQUILIBRIA AND A COMPLETE DESCRIPTION OF THE FIBERS IN THE RECONSTRUCTION OF THE REDUCED SYSTEM.