Non-Integrability of a weakly integrable Hamiltonian system

The geometric approach to mechanics based on the Jacobi metric allows to easily construct natural mechanical systems which are integrable (actually separable) at a fixed value of the energy. The aim of the present paper is to investigate the dynamics of a simple prototype system outside the zero-energy hypersurface. We find that the general situation is that in which integrability is not preserved at arbitrary values of the energy. The structure of the Hamiltonian in the separating coordinates at zero energy allows a perturbation treatment of this system at energies slightly different from zero, by which we obtain an analytical proof of non-integrability.Comment: 24 pages, accepted for publication on Celestial Mechanics and Dynamical Astronom

An energy-momentum map for the time-reversal symmetric 1:1 resonance with Z_2 X Z_2 symmetry

We present a general analysis of the bifurcation sequences of periodic orbits in general position of a family of reversible 1:1 resonant Hamiltonian normal forms invariant under $\Z_2\times\Z_2$ symmetry. The rich structure of these classical systems is investigated both with a singularity theory approach and geometric methods. The geometric approach readily allows to find an energy-momentum map describing the phase space structure of each member of the family and a catastrophe map that captures its global features. Quadrature formulas for the actions, periods and rotation number are also provided.Comment: 22 pages, 3 figures, 1 tabl

Integrable Hamiltonian systems with vector potentials

We investigate integrable 2-dimensional Hamiltonian systems with scalar and vector potentials, admitting second invariants which are linear or quadratic in the momenta. In the case of a linear second invariant, we provide some examples of weakly-integrable systems. In the case of a quadratic second invariant, we recover the classical strongly-integrable systems in Cartesian and polar coordinates and provide some new examples of integrable systems in parabolic and elliptical coordinates.Comment: 23 pages, Submitted to Journal of Mathematical Physic

Relevance of the 1:1 resonance in galactic dynamics

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.Comment: Corrected typos, to appear on the European Physical Journal Plus. arXiv admin note: text overlap with arXiv:0906.3138v

Invariants at fixed and arbitrary energy. A unified geometric approach

Invariants at arbitrary and fixed energy (strongly and weakly conserved quantities) for 2-dimensional Hamiltonian systems are treated in a unified way. This is achieved by utilizing the Jacobi metric geometrization of the dynamics. Using Killing tensors we obtain an integrability condition for quadratic invariants which involves an arbitrary analytic function $S(z)$. For invariants at arbitrary energy the function $S(z)$ is a second degree polynomial with real second derivative. The integrability condition then reduces to Darboux's condition for quadratic invariants at arbitrary energy. The four types of classical quadratic invariants for positive definite 2-dimensional Hamiltonians are shown to correspond to certain conformal transformations. We derive the explicit relation between invariants in the physical and Jacobi time gauges. In this way knowledge about the invariant in the physical time gauge enables one to directly write down the components of the corresponding Killing tensor for the Jacobi metric. We also discuss the possibility of searching for linear and quadratic invariants at fixed energy and its connection to the problem of the third integral in galactic dynamics. In our approach linear and quadratic invariants at fixed energy can be found by solving a linear ordinary differential equation of the first or second degree respectively.Comment: Some misprints corrected with respect to the printed versio

Bifurcation sequences in the symmetric 1:1 Hamiltonian resonance

We present a general review of the bifurcation sequences of periodic orbits in general position of a family of resonant Hamiltonian normal forms with nearly equal unperturbed frequencies, invariant under $Z_2 \times Z_2$ symmetry. The rich structure of these classical systems is investigated with geometric methods and the relation with the singularity theory approach is also highlighted. The geometric approach is the most straightforward way to obtain a general picture of the phase-space dynamics of the family as is defined by a complete subset in the space of control parameters complying with the symmetry constraint. It is shown how to find an energy-momentum map describing the phase space structure of each member of the family, a catastrophe map that captures its global features and formal expressions for action-angle variables. Several examples, mainly taken from astrodynamics, are used as applications.Comment: 36 pages, 10 figures, accepted on International Journal of Bifurcation and Chaos. arXiv admin note: substantial text overlap with arXiv:1401.285

On the detuned 2:4 resonance

We consider families of Hamiltonian systems in two degrees of freedom with an equilibrium in 1:2 resonance. Under detuning, this "Fermi resonance" typically leads to normal modes losing their stability through period-doubling bifurcations. For cubic potentials this concerns the short axial orbits and in galactic dynamics the resulting stable periodic orbits are called "banana" orbits. Galactic potentials are symmetric with respect to the co-ordinate planes whence the potential -- and the normal form -- both have no cubic terms. This $\mathbb{Z}_2 \times \mathbb{Z}_2$-symmetry turns the 1:2 resonance into a higher order resonance and one therefore also speaks of the 2:4 resonance. In this paper we study the 2:4 resonance in its own right, not restricted to natural Hamiltonian systems where $H = T + V$ would consist of kinetic and (positional) potential energy. The short axial orbit then turns out to be dynamically stable everywhere except at a simultaneous bifurcation of banana and "anti-banana" orbits, while it is now the long axial orbit that loses and regains stability through two successive period-doubling bifurcations.Comment: 31 pages, 7 figures: On line first on Journal of Nonlinear Science (2020

Hamiltonian Normal Forms and Galactic Potentials

The study of self-gravitating stellar systems has provided important hints to develop tools of analytical mechanics. In the present contribution we review how to exploit detuned resonant normal forms to extract information on several aspects of the dynamics in systems with self-similar elliptical equipotentials. In particular, using energy and ellipticity as parameters, we compute the instability thresholds of axial orbits, bifurcation values of low-order boxlets and phase-space fractions pertaining to the families around them. We also show how to infer something about the singular limit of the potential.Comment: To be published in "Chaos in Astronomy", G. Contopoulos, P.A. Patsis (eds.), Springe

Halo orbits around the collinear points of the restricted three-body problem

We perform an analytical study of the bifurcation of the halo orbits around the collinear points $L_1$, $L_2$, $L_3$ for the circular, spatial, restricted three--body problem. Following a standard procedure, we reduce to the center manifold constructing a normal form adapted to the synchronous resonance. Introducing a detuning, which measures the displacement from the resonance and expanding the energy in series of the detuning, we are able to evaluate the energy level at which the bifurcation takes place for arbitrary values of the mass ratio. In most cases, the analytical results thus obtained are in very good agreement with the numerical expectations, providing the bifurcation threshold with good accuracy. Care must be taken when dealing with $L_3$ for small values of the mass-ratio between the primaries; in that case, the model of the system is a singular perturbation problem and the normal form method is not particularly suited to evaluate the bifurcation threshold.Comment: 35 pages, 3 figures, updated version accepted for publication on Physica

The dynamics of the de Sitter resonance

We study the dynamics of the de Sitter resonance, namely the stable equilibrium configuration of the first three Galilean satellites. We clarify the relation between this family of configurations and the more general Laplace resonant states. In order to describe the dynamics around the de Sitter stable equilibrium, a one-degree of freedom Hamiltonian normal form is constructed and exploited to identify initial conditions leading to the two families. The normal form Hamiltonian is used to check the accuracy in the location of the equilibrium positions. Besides, it gives a measure of how sensitive it is with respect to the different perturbations acting on the system. By looking at the phase-plane of the normal form, we can identify a \sl Laplace-like \rm configuration, which highlights many substantial aspects of the observed one.Comment: Accepted for publication on Celestial Mechanics and Dynamical Astronom