Singular reduction of resonant Hamiltonians

Meyer, Kenneth R.; Palacián, Jesús F.; Yanguas, Patricia

doi:10.1088/1361-6544/aab591

Cited by 15 publications

(19 citation statements)

References 59 publications

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“…; (5) see, e.g., [2,13,4,15,14,18]. It is known (see [9]) that the dynamics of the full problem is well approximated by the one of the secular one as soon as no resonances between the frequencies (4) appear.…”

Section: Motivationmentioning

confidence: 99%

A first integral to the partially averaged Newtonian potential of the three-body problem

Pinzari

2019

Celest Mech Dyn Astr

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We consider the partial average i.e., the Lagrange average with respect to just one of the two mean anomalies, of the Newtonian part of the perturbing function in the three-body problem Hamiltonian. We prove that such a partial average exhibits a non-trivial first integral. We show that this integral is fully responsible of certain cancellations in the averaged Newtonian potential, including a property noticed by Harrington in the 60s. We also highlight its joint rôle (together with certain symmetries) in the appearance of the so called "Herman resonance". Finally, we discuss an application and an open problem.

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Section: Motivationmentioning

confidence: 99%

A first integral to the partially averaged Newtonian potential of the three-body problem

Pinzari

2019

Celest Mech Dyn Astr

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“…A generalisation of these variables for a fully resonant Hamiltonian with n degrees of freedom appears in [58] and in [57] for 2 degrees of freedom, deriving the construction of coordinates in full detail. Now, the coordinates (Q, P ) = (Q 1 , Q 2 , P 1 , P 2 ) will be used in the analysis of the parametric stability of O 1 C and O 2 C .…”

Section: Now Let Us Consider Periodic Solutions Whose Projection In mentioning

confidence: 99%

“…We tackle the analysis of our 3-DOF problem under the light of normal forms [18,54] and singular reduction [2,15]. In [57] we can find a geometric interpretation of singular reduction theory in the setting of resonant Hamiltonians with n degrees of freedom. A generalisation is in progress [58].…”

Section: Introductionmentioning

confidence: 99%

“…A generalisation is in progress [58]. We apply some of the results of [57] in the present problem. Our starting Hamiltonian system can be reduced both by one exact symmetry and by one approximate symmetry.…”

Section: Introductionmentioning

confidence: 99%

“…The analysis of the periodic solutions combines techniques of averaging and reduction theories [60,11], and in particular is based on Reeb's Theorem [62] for getting families of periodic solutions from the critical points of a reduced system defined on the reduced space. Additional results and some more theory on this topic have been developed in [71], see also [54,55,57], and we shall make use of them along the paper.…”

Section: Introductionmentioning

confidence: 99%

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Dynamics of Axially Symmetric Perturbed Hamiltonians in 1:1:1 Resonance

et al. 2018

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We study the dynamics of a family of perturbed three-degreesof-freedom (3-DOF) Hamiltonian systems which are in 1:1:1 resonance. The perturbation consists of axially symmetric cubic and quartic arbitrary polynomials. Our analysis is performed by normalisation, reduction and KAM techniques. Firstly, the system is reduced by the axial symmetry and then, periodic solutions and KAM 3-tori of the full system are determined from the relative equilibria. Next, the oscillator symmetry is extended by normalisation up to terms of degree 4 in rectangular coordinates; after truncation of higher orders and reduction to the orbit space, some relative equilibria are established and periodic solutions and KAM 3-tori of the original system are obtained. As a third step, the reduction of the two symmetries leads to a one-degrees-offreedom system that is completely analysed in the twice reduced space. All the relative equilibria, together with the stability and parametric bifurcations are determined. Moreover the invariant 2-tori (related to the critical points of the twice reduced space), some periodic solutions and the KAM 3-tori, all corresponding to the full system, are established. Additionally, the bifurcations of equilibria occurring in the twice reduce space are reconstructed as quasi-periodic bifurcations involving 2-tori and periodic solutions of the full system.

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Symmetric periodic solutions of symmetric Hamiltonians in 1 : 1 resonance

Rothen

Vidal

2022

Math Methods in App Sciences

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The aim of this work is to prove analytically the existence of symmetric periodic solutions of the family of Hamiltonian systems with Hamiltonian function Hfalse(q1,q2,p1,p2false)=12false(q12+p12false)+12false(q22+p22false)+a0.1emq14+b0.1emq12q22+c0.1emq24$$ H\left({q}_1,{q}_2,{p}_1,{p}_2\right)=\frac{1}{2}\left({q}_1^2+{p}_1^2\right)+\frac{1}{2}\left({q}_2^2+{p}_2^2\right)+a\kern0.1em {q}_1^4+b\kern0.1em {q}_1^2{q}_2^2+c\kern0.1em {q}_2^4 $$ with three real parameters a,b$$ a,b $$, and c$$ c $$. Moreover, we characterize the stability of these periodic solutions as a function of the parameters. Also, we find a first‐order analytical approach of these symmetric periodic solutions. We emphasize that these families of periodic solutions are different from those that exist in the literature.

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Singular reduction of resonant Hamiltonians

Cited by 15 publications

References 59 publications

A first integral to the partially averaged Newtonian potential of the three-body problem

A first integral to the partially averaged Newtonian potential of the three-body problem

Dynamics of Axially Symmetric Perturbed Hamiltonians in 1:1:1 Resonance

Symmetric periodic solutions of symmetric Hamiltonians in 1 : 1 resonance

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