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  • 1
    Electronic Resource
    Electronic Resource
    The global flow of the Manev problem (1996)
    College Park, Md. : American Institute of Physics (AIP)
    Journal of Mathematical Physics 37 (1996), S. 2748-2761 
    Details
    ISSN: 1089-7658
    Source: AIP Digital Archive
    Topics: Mathematics , Physics
    Notes: The Manev problem (a two-body problem given by a potential of the form A/r+B/r2, where r is the distance between particles and A,B are positive constants) comprises several important physical models, having its roots in research done by Isaac Newton. We provide its analytic solution, then completely describe its global flow using McGehee coordinates and topological methods, and offer the physical interpretation of all solutions. We prove that if the energy constant is negative, the orbits are, generically, precessional ellipses, except for a zero-measure set of initial data, for which they are ellipses. For zero energy, the orbits are precessional parabolas, and for positive energy they are precessional hyperbolas. In all these cases, the set of initial data leading to collisions has positive measure. © 1996 American Institute of Physics.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1063/1.531539
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  • 2
    Electronic Resource
    Electronic Resource
    Topology of Jacobi levels with no zero velocity curves in the restricted three-body problem (1987)
    Springer
    Zeitschrift für angewandte Mathematik und Physik 38 (1987), S. 10-20 
    Details
    ISSN: 1420-9039
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Physics
    Description / Table of Contents: Zusammenfassung Wir betrachten die Topologie der Jacobi-Flächen im ebenen eingeschränkten 3-Körperproblem im Fall, daß das Massenverhältnisμ klein und die Jacobi-Konstante Null oder fast Null ist. Für diese Werte gibt es keine „zero velocity curves“, so daß es möglich ist, daß der dritte Körper aus dem Unendlichen kommt und mit den beiden andern kollidiert. Nach Regularisierung der Kollisionen und Hinzufügung eines 2-Torus im Unendlichen durch „blow-up“, kann gezeigt werden, daß die Jacobi-Fläche topologisch äquivalent zum orientierbaren [0,1]-Bündel über der „Klein bottle“ ist. Es werden Koordinaten definiert, so daß die Topologie diejenige des Würfels mit geeigneter Identifikation von Seiten ist. Andere Koordinaten, die der Physik des Problems besser angepaßt sind, erlauben eine detailliertere Beschreibung „im Unendlichen“, wo hyperbolische oder parabolische Bahnen die ins Unendliche entweichen oder daher kommen asymptotisch zu periodischen Lösungen auf dem Torus im Unendlichen sind.
    Notes: Abstract We consider the topological description of the Jacobi levels in the circular planar restricted 3-body problem, when the mass parameterμ is close to zero and the Jacobi constant is zero or close to zero. For these values of the constant there are no zero velocity curves, so that it is possible that the infinitesimal body comes from infinity and has collisions with the other 2 bodies. After regularization of such collisions and addition of a 2-torus at infinity through blow up, we show that the Jacobi level is topologically equivalent to the unique orientable [0, 1]-bundle over the Klein bottle. Then we find coordinates making explicit this topology as a cube where some of the faces are identified. More physical coordinates at infinity give a better description, where hyperbolic or parabolic orbits escaping to (or coming from) infinity are asymptotic to periodic orbits on the torus at infinity.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00944918
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  • 3
    Electronic Resource
    Electronic Resource
    Invariant sets and polhodes in the rigid body problem (1980)
    Springer
    Celestial mechanics and dynamical astronomy 21 (1980), S. 45-53 
    Details
    ISSN: 1572-9478
    Source: Springer Online Journal Archives 1860-2000
    Topics: Physics
    Notes: Abstract In this work we will describe the sets in the rigid body phase space where the energy and angular momentum are constant, and it will turn out that in nontrivial cases they will simply take the form of cartesian products of the polhodes byS 1. These sets are important for the global study of said geodesic mechanical system for being invariant under Euler's equations (energy and momentum are constant along their solutions). To motivate from something more familiar in celestial mechanics, we will begin to relate the problem to Smale's study of the planarn-body problem (Smale, 1970) and Easton's study of the planar 3-body problem (Easton, 1971), exemplifying in particular with the central force problem. In the last Sections 4 and 5, we extent our methods to give results for generalized solids on Lie groups, mentioning the further extensions to transitive mechanical systems.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01230246
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  • 4
    Electronic Resource
    Electronic Resource
    A compact model for the planar rhomboidal 4-body problem (1992)
    Springer
    Celestial mechanics and dynamical astronomy 54 (1992), S. 343-355 
    Details
    ISSN: 1572-9478
    Keywords: Singularity collisions ; blow up ; projective transformations ; total collision manifold ; global flow
    Source: Springer Online Journal Archives 1860-2000
    Topics: Physics
    Notes: Abstract We obtain a compact model for the global study of the planar rhomboidal 4-body problem in a level of constant negative energy. This model is a variation of the non compact model obtained through a McGehee blow up transformation. but compactness permits to obtain results which are not clear in the other case.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00049146
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  • 5
    Electronic Resource
    Electronic Resource
    Time scaling as an infinitesimal canonical transformation (1987)
    Springer
    Celestial mechanics and dynamical astronomy 42 (1987), S. 201-213 
    Details
    ISSN: 1572-9478
    Source: Springer Online Journal Archives 1860-2000
    Topics: Physics
    Notes: Abstract We show that time scaling transformations for Hamiltonian systems are infinitesimal canonical transformations in a suitable extended phase space constructed from geometrical considerations. We compute its infinitesimal generating function in some examples: regularization and blow up in celestial mechanics, classical mechanical systems with homogeneous potentials and Scheifele theory of satellite motion.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01232957
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  • 6
    Electronic Resource
    Electronic Resource
    Motions close to escapes in the rhomboidal four body problem (1993)
    Springer
    Celestial mechanics and dynamical astronomy 57 (1993), S. 411-437 
    Details
    ISSN: 1572-9478
    Keywords: Total collision manifold ; reversible flow ; parabolic ; escape and capture orbits ; Symbolic Dynamics
    Source: Springer Online Journal Archives 1860-2000
    Topics: Physics
    Notes: Abstract In this work we study escape and capture orbits in the planar rhomboidal 4-body problem in a level of constant negative energy. There are only two different values of the masses here. By using numerical analysis, we show certain transversal intersections of the invariant manifolds of parabolic orbits. We then introduce Symbolic Dynamics when the mass ratio is small, and when it is close to one. In the first case the escapes or captures predominate in the direction of one of the diagonals of the rhombus, while in the second case we find solutions escaping or being captured in the direction of both possible diagonals.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00695713
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  • 7
    Electronic Resource
    Electronic Resource
    Boundary manifolds for energy surfaces in celestial mechanics (1982)
    Springer
    Celestial mechanics and dynamical astronomy 28 (1982), S. 37-48 
    Details
    ISSN: 1572-9478
    Source: Springer Online Journal Archives 1860-2000
    Topics: Physics
    Notes: Abstract We complete Mc Gehee's picture of introducing a boundary (total collision) manifold to each energy surface. This is done by constructing the missing components of its boundary as other submanifolds. representing now the asymptotic behavior at infinity. It is necessary to treat each caseh=0,h〉0 orh〈0 separately. In the first case, we repeat the known result that the behavior at total escape is the same as in total collision. In particular, we explain why the situation is radically different in theh〉0 case compared with the zero energy case. In the caseh〈0 we have many infinity manifold components. and the general situation is not quite well understood. Finally, our results forh≥0 are shown to be valid for general homogeneous potentials.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01230658
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  • 8
    Electronic Resource
    Electronic Resource
    On the Dynamics and Topology of the Elliptic Rectilinear Restricted 3-Body Problem (2000)
    Springer
    Celestial mechanics and dynamical astronomy 77 (2000), S. 1-15 
    Details
    ISSN: 1572-9478
    Keywords: 3-body problem ; collinear restricted problem ; regularization
    Source: Springer Online Journal Archives 1860-2000
    Topics: Physics
    Notes: Abstract We study a symmetric collinear restricted 3-body problem, where the equal mass primaries perform elliptic collisions, while a third massless body moves in the line between the primaries, during the time between two consecutive elliptic collisions. After desingularizing binary and triple collisions, we prove the existence of a transversal heteroclinic orbit beginning and ending in triple collision. This orbit is the unique homothetic orbit that the problem possess. Finally, we describe the topology of the compact extended phase space.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1023/A:1008330921935
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  • 9
    Electronic Resource
    Electronic Resource
    Mouvements voisins de collision quadruple dans le probleme trapezoidal des 4 corps (1983)
    Springer
    Celestial mechanics and dynamical astronomy 31 (1983), S. 23-41 
    Details
    ISSN: 1572-9478
    Source: Springer Online Journal Archives 1860-2000
    Topics: Physics
    Description / Table of Contents: Abstract We continue here the study of the trapezoidal 4-body problem, started by the author in a previous work. The dimensions and interconnections of the invariant submanifolds of the equilibrium points in the total collision manifold, are studied. We conclude with the description of some important motions in the problem. This problem is roughly as hard as the zero angular momentum plane 3-body. The principal features of this problem is the nonexistence of triple collisions, and the embedding of two simpler problems with one less degree of freedom: the rectilinear trapezoidal, and the rectangular problems (the last one only if μ=m). As usual, there are some bifurcations for some values of the mass ratio. They change some of the interconnections. The set of initial conditions of orbits on a given energy surface going to quadruple collision, is a union of 4 submanifolds: two of them have dimension 2, while the others have dimension 3. Similarly for ejection orbits from quadruple collision.
    Notes: Résumé On reprend ici l'étude du problème trapezoïdal des 4 corps, qui a été commencée par l'auteur dans [2]. En particulier, on étudie les dimensions et interconnexions des sous-variétés invariantes des points d'équilibre hyperboliques dans la variété (fictive) de collision quadruple, et on conclut avec l'existence de certaines solutions remarquables du problème au voisinage des collisions quadruples. Pour certaines valeurs du paramètre rapport de masses, on a des bifurcations, qui sont expliquées par le raccordement (ou non) des orbites entre les diverses régions de repos. Même s'il s'agit d'un problème de 4 corps, certains faits simplifient l'étude. Notamment, les symétries, la non existence de collisions triples, et le plongement de deux problèmes plus simples à étudier. Le dégré de difficulté du problème etudié est le même que celui du problème général plan des 3 corps, lorsque le moment cinétique est nul.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01272558
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  • 10
    Electronic Resource
    Electronic Resource
    Origin and infinity manifolds for mechanical systems with homogeneous potentials (1988)
    Springer
    Acta applicandae mathematicae 11 (1988), S. 259-284 
    Details
    ISSN: 1572-9036
    Keywords: 58F05 ; 70H05 ; 70H15 ; 70F35 ; Blow up ; conservative mechanical system ; homogeneous function ; Hamiltonian system ; canonical transformation ; symplectic manifold ; contact manifold
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract We study manifolds describing the behavior of motions close to the origin and at infinity of configuration space, for mechanical systems with homogeneous potentials. We find an inversion between these behaviors when the sign of the degree of homogeneity is changed. In some cases, the blow up equations can be written in canonical form, by first reducing to a contact structure. A motivation for the use of blow-up techniques is given, and some examples are studied in detail.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00140121
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