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  • 1
    Electronic Resource
    Electronic Resource
    Nonlinearly coupled oscillators in quantum mechanics: A normal form approach (1988)
    College Park, Md. : American Institute of Physics (AIP)
    Journal of Mathematical Physics 29 (1988), S. 1405-1416 
    Details
    ISSN: 1089-7658
    Source: AIP Digital Archive
    Topics: Mathematics , Physics
    Notes: Quantum mechanical Hamiltonians of the form of n harmonic oscillators coupled via an interaction of the form ε times a polynomial in the position and momenta variables are studied in a rigorous Hilbert space setting. In particular, normal form theory is used to define the mth approximation to the associated Schrödinger initial value problem and it is shown that it deviates in norm from the exact solution by a term of order εm+1 ||t|| (t=time) provided only that the initial vector is confined to an appropriate dense subspace. The main concentration is on the case in which there exist no resonances of order ≤m between the frequencies of the n oscillators, but the case of two oscillators in 1:1 resonance is also taken up.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1063/1.527933
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  • 2
    Electronic Resource
    Electronic Resource
    On the regularization of the Kepler problem (1990)
    Springer
    Communications in mathematical physics 131 (1990), S. 221-221 
    Details
    ISSN: 1432-0916
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Physics
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02097686
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  • 3
    Electronic Resource
    Electronic Resource
    On the regularization of the Kepler problem (1982)
    Springer
    Communications in mathematical physics 84 (1982), S. 133-152 
    Details
    ISSN: 1432-0916
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Physics
    Notes: Abstract An intimate relationship between Moser's regularization [1] and the KS-regularization [3] of the 3-dimensional Kepler problem is established. Explicit formulae linking Moser's and the KS-transformation are obtained in the case of negative as well as in the case of positive energies. As a side result it is shown that the KS-transformation owes its existence to the local isomorphism of SO(2,4) and SU(2,2).
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01208375
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  • 4
    Electronic Resource
    Electronic Resource
    On resonant non linearly coupled oscillators with two equal frequencies (1976)
    Springer
    Communications in mathematical physics 48 (1976), S. 53-79 
    Details
    ISSN: 1432-0916
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Physics
    Notes: Abstract This paper contains a detailed study of the flow that the classical Hamiltonian $$H = \tfrac{1}{2}(x\begin{array}{*{20}c} 2 \\ 1 \\ \end{array} + y\begin{array}{*{20}c} 2 \\ 1 \\ \end{array} ) + \tfrac{1}{2}(x\begin{array}{*{20}c} 2 \\ 2 \\ \end{array} + y\begin{array}{*{20}c} 2 \\ 2 \\ \end{array} ) + \mathcal{O}_3 $$ induces inR 4, $$\mathcal{O}_3 $$ representing a convergent power series that begins with a third order term. In particular the existence and stability of periodic orbits is investigated.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01609411
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  • 5
    Electronic Resource
    Electronic Resource
    On resonant classical Hamiltonians with two equal frequencies (1978)
    Springer
    Communications in mathematical physics 58 (1978), S. 85-112 
    Details
    ISSN: 1432-0916
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Physics
    Notes: Abstract This paper contains a detailed study of the flow that the classical Hamiltonian $$H = \tfrac{1}{2}(x_1^2 + y_1^2 ) - \tfrac{1}{2}(x_2^2 + y_2^2 ) + \mathcal{O}_3 $$ induces near the origin of its phase spaceR 4. Here the perturbation term $$\mathcal{O}_3 $$ represents a convergent power series. In particular, criteria for the existence and stability of periodic orbits are developed and expressed in terms of canonical invariants that are extracted from the perturbation term.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01624789
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  • 6
    Electronic Resource
    Electronic Resource
    Transcendentally small transversality in the rapidly forced pendulum (1993)
    Springer
    Journal of dynamics and differential equations 5 (1993), S. 241-277 
    Details
    ISSN: 1572-9222
    Keywords: Rapidly forced pendulum ; transversality ; separatrix splitting ; asymptotics beyond all orders ; exponentially small ; stable manifold theorem ; Hamiltonian
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract The rapidly forced pendulum equation with forcing δ sin((t/ε), where δ=〈δ0εp,p = 5, forδ 0,ε sufficiently small, is considered. We prove that stable and unstable manifolds split and that the splitting distanced(t) in the ( $$\dot x$$ ,t) plane satisfiesd(t) = πδ sin(t/ε) sech(π/2ε) +O(δ 0 δ exp(−π/2ε)) (2.3a) and the angle of transversal intersection,ψ, in thet = 0 section satisfiesψ ∼ 2 tanψ/2 = 2S s = (πδ/2ε) sech(π/2ε) +O((δ 0 δ/ε) exp(−π/2ε)) (2.3b) It follows that the Melnikov term correctly predicts the exponentially small splitting and angle of transversality. Our method improves a previous result of Holmes, Marsden, and Scheuerle. Our proof is elementary and self-contained, includes a stable manifold theorem, and emphasizes the phase space geometry.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01053162
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  • 7
    Electronic Resource
    Electronic Resource
    The annular twist mapping of the restricted three-body problem (1997)
    Springer
    Journal of dynamics and differential equations 9 (1997), S. 249-268 
    Details
    ISSN: 1572-9222
    Keywords: Twist map ; restricted three-body problem ; rotating Kepler problem ; symplectic reduction
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract This paper presents a new derivation of the twist mapping in the planar restricted problem. It differs from other treatments in the use of a novel canonical transformation which allows for the utilization of symplectic reduction techniques.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02219222
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  • 8
    Electronic Resource
    Electronic Resource
    Some applications of computable one-one numberings (1990)
    Springer
    Archive for mathematical logic 30 (1990), S. 219-230 
    Details
    ISSN: 1432-0665
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract We present a simple proof of a Theorem of Khutoretskij on the number of incomparable one-one numberings of an r.e. family of r.e. sets. The proof directly generalizes to effective domains. In the second part, applying a Theorem of Goncharov, we show that for anyk≧ there exist total recursive functions having exactlyk recursive isomorphism classes. Using a Theorem of Selivanov, it is shown that a certain notion of computability via gödelization is different from Lacombe's notion ofV-recursiveness. Finally, we discuss the complexity (w.r.t.T-degrees) of translating a Gödelnumbering into a direct sum of Friedbergnumberings.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01792984
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  • 9
    Electronic Resource
    Electronic Resource
    A group theoretical approach to a certain class of perturbations of the Kepler problem (1985)
    Springer
    Archive for rational mechanics and analysis 91 (1985), S. 55-82 
    Details
    ISSN: 1432-0673
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Physics
    Notes: Abstract We study a particular class of perturbations of the classical Kepler Hamiltonian, first in two, then in three and finally in n dimensions. At every stage of our investigation the group theoretical nature of our constructions is fully exposed. In particular we present a new regularization of the n-dimensional Kepler problem which is based on previous constructions of Guillemin & Sternberg (see [8]). This regularization is similarily related to Moser's (see [9]) as is Kustaanheimo-Stiefel's (see [4]) in three dimensions.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00280223
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  • 10
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    Class of Perturbation Theories of Ordinary Differential Equations (1971)
    American Institute of Physics
    Details
    Publication Date: 1971-01-01
    Print ISSN: 0022-2488
    Electronic ISSN: 1089-7658
    Topics: Mathematics , Physics
    Published by American Institute of Physics
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