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  • 1
    Electronic Resource
    Electronic Resource
    On a first integral of the Kepler problem (1997)
    College Park, Md. : American Institute of Physics (AIP)
    Journal of Mathematical Physics 38 (1997), S. 4561-4569 
    Details
    ISSN: 1089-7658
    Source: AIP Digital Archive
    Topics: Mathematics , Physics
    Notes: A quadratic first integral of the Kepler problem, obtained by Benenti ("Orthogonal separable dynamical systems," 5th International Conference on Differential Geometry and its Applications, 24–28 August 1992, Silesian University at Opava) through separation in elliptic coordinates, is shown to be intimately connected with the pre-quantization of the Kepler manifold, thus acquiring a physical interpretation. © 1997 American Institute of Physics.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1063/1.532143
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  • 2
    Electronic Resource
    Electronic Resource
    The KS-transformation revisited (1994)
    Springer
    Meccanica 29 (1994), S. 15-26 
    Details
    ISSN: 1572-9648
    Keywords: Kepler problem ; KS-regularization ; theoretical mechanics
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mechanical Engineering, Materials Science, Production Engineering, Mining and Metallurgy, Traffic Engineering, Precision Mechanics , Physics
    Description / Table of Contents: Sommario Si mostra come la trasformazione di Kustaanheimo e Stiefel (KS) per regolarizzare il moto kepleriano perturbato sia cosí legata agli elementi orbitali da condurre, tramite una particolare roto-omotetia nello spazio fisico euclideo, al vettore posizione di una particella sull'orbita osculatrice. Avvalendosi del consueto calcolo vettoriale, si introducono solo all'occorrenza quaternioni e spinori. Fondamentali risultano (i): l'introduzione per via odografica degli integrali primi caratteristici del moto kepleriano (momento della quantità di moto, vettore di Runge-Lenz); (ii) l'interpretazione polarizzata della terna di assi che dà la posizione dell'orbita nello spazio; (iii) le semplici espressioni cinematiche degli elementi orbitali. L'energia meccanica e la relazione bilineare, basilari nella teoria KS, acquistano interessanti interpretazioni cinematiche.
    Notes: Abstract We show that the Kustaanheimo-Stiefel (KS) regularizing transformation for the perturbed Kepler motion is so deeply rooted to the Keplerian orbital elements as to yield the position vector of a particle on the osculating orbit as the effect of a peculiar roto-dilatation in the physical Euclidean space. Adopting the conventional vector formulation, quaternions and spinors are also involved. A key role is played by (i): a simple hodographical approach to the integrals of the Kepler motion (angular momentum vector, Runge-Lenz vector); (ii) a polarized outlook on the attitude frame of the Kepler orbit; (iii) a simple kinematical expression for the orbital elements. The mechanical energy, the bilinear relation, the gauge transformation — fundamental in the KS-theory — are naturally arrived at, acquiring interesting kinematical interpretations.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00989522
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  • 3
    Electronic Resource
    Electronic Resource
    The Sum Vector S and the Fictitious Time s in the Kepler Problem (2000)
    Springer
    Meccanica 35 (2000), S. 55-67 
    Details
    ISSN: 1572-9648
    Keywords: Kepler problem ; Regularization ; Orbital mechanics ; General mechanics
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mechanical Engineering, Materials Science, Production Engineering, Mining and Metallurgy, Traffic Engineering, Precision Mechanics , Physics
    Notes: Abstract The well-known regularizing fictitious time variable s for the Kepler motion is shown to fit naturally in the unitary configuration description of the classical Kepler problem we presented in our previous works and which is based on the introduction of the sum vector S.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1023/A:1004840415066
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  • 4
    Electronic Resource
    Electronic Resource
    On the connection among three classical mechanical problems via the hypercomplex KS-transformation (1990)
    Springer
    Celestial mechanics and dynamical astronomy 50 (1990), S. 109-124 
    Details
    ISSN: 1572-9478
    Keywords: KS-transformation ; regularization ; two body problem ; Euler-Poinsot problem
    Source: Springer Online Journal Archives 1860-2000
    Topics: Physics
    Notes: Abstract With reference to our previous notes [23–30] on the hypercomplex regularizations of the classical Kepler problem and on the hypercomplex descriptions of the classical rotational mechanics, we arrive at an intimate mathematical relationship among the three classical mechanical problems: the elliptic Kepler problem, the spherical Euler-Poinsot problem and the four-dimensional isotropic harmonic oscillator. A key role is played: (i) by the Kustaanheimo-Stiefel (KS) regularizing transformation viewed, in the light of Souriau's quantization procedure, as the projection map in the Hopf fibering of the contact 3-sphere; (ii) by a novel Lagrangian description of rotational kinematics expressed in terms of the Euler-Rodrigues parameters; (iii) by the unit vector a (characterizing both the attitude frame of the rotator and the direction of the major axis of the Kepler orbit) and shown to link ‘à la Cartan’ the KS-theory with the rotation theory. The Kepler-Poinsot-Oscillator connection is examined also in a quantistic Schroedinger approach and is related to the results of Ikeda and Miyachi.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00051045
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  • 5
    Electronic Resource
    Electronic Resource
    Geometrical and physical outlook on the cross product of two quaternions (1987)
    Springer
    Celestial mechanics and dynamical astronomy 41 (1987), S. 359-370 
    Details
    ISSN: 1572-9478
    Source: Springer Online Journal Archives 1860-2000
    Topics: Physics
    Notes: Abstract As an outcome of our previous notes [13, 14] on the quaternion regularization of the classical Kepler problem and pre-quantization of the Kepler manifold we show, first, that both the cross product of two quaternions and the cross product of their anti-involutes are susceptible of a simple geometrical representation in the ordinary 3-dimensional euclidean spaceR 3 and, secondly, that they satisfy anSO(4)-invariant relation that implies projection of curves from the quaternion space onto the spaceR 3. ThisSO(4)-invariance allows—in the particular case of orthogonal quaternions of equal norm—a straight derivation: (i) of the correspondence between the free motion on the surface of a sphereS 3 and the physical elliptical Kepler motion (collisions included) on a plane denoted by П w ; (ii) of the celebrated Kepler equation and (iii) of the Levi-Civita regularizing time transformation. With (i) and (ii) we recover some of Györgyi's [3] results. The aforesaid orbital plane П w and the orbital plane П*, arrived at independently by exploiting the Kustaanheimo-Stiefel regularizing transformation, are shown to be inclined exactly at an angle characterizing the ratio of the semi-axes of the elliptical orbits and intimately related to the cross product representation. Thus the eventual superimposition of the two planes confirms the intimate connection between the various regularization procedures—transforming the classical Kepler problem into the geodesic flow onS 3—and the Fock's procedure for the quantum theoretical Kepler problem of the hydrogen atom (‘accidental degeneracy’).
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01238771
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  • 6
    Electronic Resource
    Electronic Resource
    The Kepler problem: A unifying view (1994)
    Springer
    Celestial mechanics and dynamical astronomy 60 (1994), S. 291-305 
    Details
    ISSN: 1572-9478
    Keywords: Kepler problem ; constants ; regularization ; KS-transformation
    Source: Springer Online Journal Archives 1860-2000
    Topics: Physics
    Notes: Abstract This paper is an attempt to bring unity in the study of the classical Kepler problem by combining, through simple vectorial and quaternionic techniques, its two peculiar aspects: the determination of the constants of the motion and the regularization at the origin.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00691898
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  • 7
    Electronic Resource
    Electronic Resource
    The KS-transformation in hypercomplex form (1983)
    Springer
    Celestial mechanics and dynamical astronomy 29 (1983), S. 45-50 
    Details
    ISSN: 1572-9478
    Source: Springer Online Journal Archives 1860-2000
    Topics: Physics
    Notes: Abstract In this note the KS-transformation introduced by Kustaanheimo and Stiefel into Celestial Mechanics is formulated in terms of hypercomplex numbers as the product of a quaternion and its antiinvolute. Therefore it represents a particular morphism of the real algebra of quaternions-having for image a three-dimensional real linear subspace-and also anatural generalization of the Levi-Civita transformation. The quaternion matrix of the product leads to the KS-matrix; the bilinear relation and the two identities which play a central role in the KS-theory are easily derived. A suitable quaternion gauge-transformation is given which leads to the well-known fibration of the four-dimensional space. In addition several geometrical interpretations are brought out.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01358597
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  • 8
    Electronic Resource
    Electronic Resource
    A Configuration Counterpart of the Kepler Problem Hodograph (1997)
    Springer
    Celestial mechanics and dynamical astronomy 68 (1997), S. 305-311 
    Details
    ISSN: 1572-9478
    Keywords: Kepler problem ; hodograph ; Minkowskian parameterizatio
    Source: Springer Online Journal Archives 1860-2000
    Topics: Physics
    Notes: Abstract This paper presents two peculiar geometrical structures in the configuration space of the classical Kepler problem: the circularU-graph and the circularS-graph. TheS-graph shows up a configuration space counterpart of the well-known velocity space hodograph. Several interesting results are brought out, such as a peculiar description of the mechanical energy. An extension to the three-dimensional space, through theU-sphere and theS-sphere, characterizes the regular Kepler orbits by means of the north pole of the associatedS-sphere. The Minkowskian parameterization introduced in [1] is easily recovered and is shown to fit naturally in theS-sphere description of the Kepler problem.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1023/A:1008223102613
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  • 9
    Electronic Resource
    Electronic Resource
    Development of spinor descriptions of rotational mechanics from Euler's rigid body displacement theorem (1984)
    Springer
    Celestial mechanics and dynamical astronomy 32 (1984), S. 193-207 
    Details
    ISSN: 1572-9478
    Source: Springer Online Journal Archives 1860-2000
    Topics: Physics
    Notes: Abstract A new approach to quaternions and to one-index spinors in the Euclidean three-space is offered,directly derived from the vector formulation of the Euler's theorem on the general displacement of a rigid body with a fixed point. A linear and injective correspondence between quaternions and one-index spinors is introduced, leading to a spinor formulation of rotations as a natural extension of the usual quaternion automorphism and providing other kinematic relations in the spin space as well as the recovery of the usual Pauli spinor formalism.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01236599
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  • 10
    Electronic Resource
    Electronic Resource
    The Levi-Civita Time Tranformation and a Killing Vector for the Kepler Problem (1997)
    Springer
    Meccanica 32 (1997), S. 135-142 
    Details
    ISSN: 1572-9648
    Keywords: Kepler problem ; Regularization ; Killing vector ; Orbital mechanics.
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mechanical Engineering, Materials Science, Production Engineering, Mining and Metallurgy, Traffic Engineering, Precision Mechanics , Physics
    Notes: Abstract As an outcome of our hypercomplex unitary description [9] of some peculiaraspects of the Kepler problem (regularization, prequantization of theKepler manifold), we show how both the differential time transformationadopted by Levi-Civita for the regularization of the plane problem, and thequaternionic transformation which prequantizes the Kepler manifold, derivefrom the same quaternionic definition of a Keplerian orbit. Moreover, weshow how the stereographic projection of an hodographical sphere leads to aKilling vector, the counterpart of the position vector of the particle.Interesting results are brought out.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1023/A:1004292916946
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