Quantitative predictions with detuned normal forms

Pucacco, Giuseppe; Boccaletti, Dino; Belmonte, Cinzia

doi:10.1007/s10569-008-9141-x

Cited by 33 publications

(50 citation statements)

References 24 publications

(41 reference statements)

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“…which can be considered as 'realistic' for elliptical galaxies, the thresholds (65) give estimates correct within a 10% if compared to numerical computations [1,2].…”

Section: The Stability Of Axial Orbits In Systems With Elliptical Equmentioning

confidence: 76%

Relevance of the 1:1 resonance in galactic dynamics

Marchesiello

Pucacco

2011

Eur. Phys. J. Plus

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Abstract. This paper aims to illustrate the applications of resonant Hamiltonian normal forms to some problems of galactic dynamics. We detail the construction of the 1:1 resonant normal form corresponding to a wide class of potentials with self-similar elliptical equi-potentials and apply it to investigate relevant features of the orbit structure of the system. We show how to compute the bifurcation of the main periodic orbits in the symmetry planes of a triaxial ellipsoid and in the meridional plane of an axi-symmetric spheroid and briefly discuss how to refine these results with higher-order approaches. DescriptionNonlinear dynamics has often profited, since the pioneering works of Jeans, Lindblad, etc., of the progress in galactic dynamics. The interplay between the two disciplines has gradually established a common body of methods and results, as much as in the case of the interplay with celestial mechanics and is well documented in the literature [5,6,7]. Nowadays, advanced techniques of mathematical physics provide rigorous analysis of the main phenomena occurring in non-integrable dynamical systems potentially useful in galactic dynamics. However, the language of these modern methods often prevent a direct use of those results to applied fields like astrophysics.Aim of this work is to show how to apply modern perturbative methods in analytical mechanics to provide a simple and comprehensive description of a common phenomenon in the dynamics of stellar systems: the onset of instability of a given motion due to a nonlinear resonance and the bifurcation of a new solution. We limit our analysis to the case in which the two involved frequencies, that of the perturbed solution and that of the 'perturbation', happen to become equal as functions in a suitable parameter space. This is called a '1:1 resonance' and plays a distinguished role in several concrete cases. The approach we use is that of mimicking the dynamics of the original physical system, that cannot be exactly solved, with an approximating system which is exactly solvable. The original system is assumed to be a generic non-integrable Hamiltonian system: the mimicking system is an integrable resonant Hamiltonian 'normal form'. We give explicit formulae to construct the normal form of a fairly generic class of systems and show how to use it to gather a general view of the phase space structure around the resonance. We prove that, in addition to a complete qualitative understanding, the theory also provides quite accurate quantitative predictions, even at the simplest level of the procedure. We finally discuss how to refine the predictions by going to higher-orders and how to generalize the method in a wider context.The plane of the paper is as follows: in section 2 we introduce the procedure to construct the approximating integrable system by recalling the method of the Lie transform [11,12]; in sections 3 and 4 we apply this approach to investigate some aspects of the dynamics in a symmetry plane of a triaxial ellipsoid [1] and in the meridional pla...

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“…which can be considered as 'realistic' for elliptical galaxies, the thresholds (65) give estimates correct within a 10% if compared to numerical computations [1,2].…”

Section: The Stability Of Axial Orbits In Systems With Elliptical Equmentioning

confidence: 76%

Relevance of the 1:1 resonance in galactic dynamics

Marchesiello

Pucacco

2011

Eur. Phys. J. Plus

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“…The potential √ 1 + x 2 + y 2 q 2 has an absolute minimum and a reflection symmetry with respect the two axis x and y. The motivation for the choice of these symmetries comes from the interest of this potential in galactic dynamics, see for instance [2,3,4,9,10,11]. The parameter q gives the ellipticity of the potential, which ranges in the interval 0.6 ≤ q ≤ 1.…”

Section: Introduction and Statement Of The Main Resultsmentioning

confidence: 99%

On the analytic integrability of the cored galactic Hamiltonian

Llibre

Valls

2014

Applied Mathematics Letters

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“…We consider two degrees of freedom natural Hamiltonian systems of the form with V a smooth potential with an absolute minimum and reflection symmetry with respect to both axes, see for more details [17]. The reason for the choice of these symmetric systems is due to their interest in problems of galactic dynamics.…”

Section: Introduction and Statements Of Main Resultsmentioning

confidence: 99%