Theory of Orbits

Boccaletti, Dino; Pucacco, Giuseppe

doi:10.1007/978-3-662-09240-8

Cited by 78 publications

(109 citation statements)

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“…The initial conditions are obtained directly from the coordinate change (3.3) x = r cos θ, px = r sin θ, y = ρ cos(θ + α), py = ρ sin(θ + α), we proof that the two solutions (r * , α * , ρ * ) obtained in (1), (2) and (3) in Proposition 1 are initial conditions of one periodic orbit in each case; also we proof that the conditions in (4) in Proposition 1 give two initial conditions for two periodic solutions.…”

Section: Propositionmentioning

confidence: 60%

“…The Galactic dynamics is an area of the Astrophysics where recently the application of results coming from other areas as the Celestial mechanics and the Nonlinear dynamics has gradually establish common methods and results well documented (Boccaletti and Puccaco 1999) and (Contoupoulus 2002). The global dynamics of galaxies is not a simple question and represent an actual challenge for the researches.…”

Section: Introductionmentioning

confidence: 99%

“…In order to study the dynamics associated to the potentials (1. 2) it have been considered models where the potential function V (X, Y ) is expanded as a truncated series in the coordinates X and Y (1.3) H = 1 2 ω1p…”

Section: Introductionmentioning

confidence: 99%

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Periodic orbits of mechanical systems with homogeneous polynomial terms of degree five

Ortega¹

2015

Astrophys Space Sci

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Abstract. In this work the existence of periodic solutions is studied for the Hamiltonian functionswhere the first term consist of a harmonic oscillator and the second term are homogeneous polynomials of degree 5 defined by two real parameters a and b. Using the averaging method of second order we provide the sufficient conditions on the parameters to guarantee the existence of periodic solutions for positive energy and we study the stability of these periodic solutions.

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Section: Propositionmentioning

confidence: 60%

Section: Introductionmentioning

confidence: 99%

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Periodic orbits of mechanical systems with homogeneous polynomial terms of degree five

Ortega¹

2015

Astrophys Space Sci

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“…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.…”

Section: Descriptionmentioning

confidence: 99%

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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“…Normal forms for the Hamiltonian system corresponding to the potential described by Eq. (1) are constructed with standard methods (Boccaletti & Pucacco 1999;Giorgilli 2002) and were used to determine the main features of its orbit structure (Belmonte et al 2006(Belmonte et al , 2007. The starting point is the series expansion of Eq.…”

Section: Normal Forms For the Logarithmic Potentialmentioning

confidence: 99%

Periodic orbits in the logarithmic potential

Pucacco¹,

Boccaletti²,

Belmonte³

2008

A&A

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Context. We investigate periodic orbits in galactic potentials by developing analytical methods. Aims. We evaluate the quality of the approximation of periodic orbits in the logarithmic potential constructed using perturbation theory based on Hamiltonian normal forms. Methods. The solutions of the equations of motion corresponding to periodic orbits are obtained as series expansions computed by inverting the normalizing canonical transformation. To improve the convergence of the series, a resummation based on a continued fraction may be performed. This method is analogous to the Prendergast method, which searches for approximate rational solutions. Results. It is shown that with a normal form truncated at the lowest order incorporating the relevant resonance it is possible to construct accurate solutions both for normal modes and periodic orbits in general position.

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Theory of Orbits

Abstract: The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use.

Cited by 78 publications

References 0 publications

Periodic orbits of mechanical systems with homogeneous polynomial terms of degree five

Periodic orbits of mechanical systems with homogeneous polynomial terms of degree five

Relevance of the 1:1 resonance in galactic dynamics

Periodic orbits in the logarithmic potential

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