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

Gravothermal Catastrophe in Anisotropic Spherical Systems

In this paper we investigate the gravothermal instability of spherical stellar systems endowed with a radially anisotropic velocity distribution. We focus our attention on the effects of anisotropy on the conditions for the onset of the instability and in particular we study the dependence of the spatial structure of critical models on the amount of anisotropy present in a system. The investigation has been carried out by the method of linear series which has already been used in the past to study the gravothermal instability of isotropic systems. We consider models described by King, Wilson and Woolley-Dickens distribution functions. In the case of King and Woolley-Dickens models, our results show that, for quite a wide range of amount of anisotropy in the system, the critical value of the concentration of the system (defined as the ratio of the tidal to the King core radius of the system) is approximately constant and equal to the corresponding value for isotropic systems. Only for very anisotropic systems the critical value of the concentration starts to change and it decreases significantly as the anisotropy increases and penetrates the inner parts of the system. For Wilson models the decrease of the concentration of critical models is preceded by an intermediate regime in which critical concentration increases, it reaches a maximum and then it starts to decrease. The critical value of the central potential always decreases as the anisotropy increases.Comment: 7pages, 5figures, to appear in MNRAS (figures have been replaced with their corrected versions

(1+1)-dimensional separation of variables

In this paper we explore general conditions which guarantee that the geodesic flow on a 2-dimensional manifold with indefinite signature is locally separable. This is equivalent to showing that a 2-dimensional natural Hamiltonian system on the hyperbolic plane possesses a second integral of motion which is a quadratic polynomial in the momenta associated with a 2nd-rank Killing tensor. We examine the possibility that the integral is preserved by the Hamiltonian flow on a given energy hypersurface only (weak integrability) and derive the additional requirement necessary to have conservation at arbitrary values of the Hamiltonian (strong integrability). Using null coordinates, we show that the leading-order coefficients of the invariant are arbitrary functions of one variable in the case of weak integrability. These functions are quadratic polynomials in the coordinates in the case of strong integrability. We show that for $(1+1)$-dimensional systems there are three possible types of conformal Killing tensors, and therefore, three distinct separability structures in contrast to the single standard Hamilton-Jacobi type separation in the positive definite case. One of the new separability structures is the complex/harmonic type which is characterized by complex separation variables. The other new type is the linear/null separation which occurs when the conformal Killing tensor has a null eigenvector.Comment: To appear on Journal of Mathematical Physic

Resonances and bifurcations in systems with elliptical equipotentials

We present a general analysis of the orbit structure of 2D potentials with self-similar elliptical equipotentials by applying the method of Lie transform normalization. We study the most relevant resonances and related bifurcations. We find that the 1:1 resonance is associated only to the appearance of the loops and leads to the destabilization of either one or the other normal modes, depending on the ellipticity of equipotentials. Inclined orbits are never present and may appear only when the equipotentials are heavily deformed. The 1:2 resonance determines the appearance of bananas and anti-banana orbits: the first family is stable and always appears at a lower energy than the second, which is unstable. The bifurcation sequence also produces the variations in the stability character of the major axis orbit and is modified only by very large deformations of the equipotentials. Higher-order resonances appear at intermediate or higher energies and can be described with good accuracy.Comment: Accepted for publication on MNRA

Equivariant singularity analysis of the 2:2 resonance

We present a general analysis of the bifurcation sequences of 2:2 resonant reversible Hamiltonian systems invariant under spatial $\Z_2\times\Z_2$ symmetry. The rich structure of these systems is investigated by a singularity theory approach based on the construction of a universal deformation of the detuned Birkhoff normal form. The thresholds for the bifurcations are computed as asymptotic series also in terms of physical quantities for the original system

Optimizing the Earth-LISA "rendez-vous"

We present a general survey of heliocentric LISA orbits, hoping it might help in the exercise of rescoping the mission. We try to semi-analytically optimize the orbital parameters in order to minimize the disturbances coming from the Earth-LISA interaction. In a set of numerical simulations we include nonautonomous perturbations and provide an estimate of Doppler shift and breathing as a function of the trailing angle.Comment: 18 pages, 16 figures. Submitted on CQ

Evolution and stability of Laplace-like resonances under tidal dissipation

AbstractThe Laplace resonance is a configuration that involves the commensurability between the mean motions of three small bodies revolving around a massive central one. This resonance was first observed in the case of the three inner Galilean satellites, Io, Europa, and Ganymede. In this work the Laplace resonance is generalised by considering a system of three satellites orbiting a planet that are involved in mean motion resonances. These Laplace-like resonances are classified in three categories: first-order (2:1&2:1, 3:2&3:2, 2:1&3:2), second-order (3:1&3:1) and mixed-order resonances (2:1&3:1). In order to study the dynamics of the system we implement a model that includes the gravitational interaction with the central body, the mutual gravitational interactions of the satellites, the effects due to the oblateness of the central body and the secular interaction of a fourth satellite and a distant star. Along with these contributions we include the tidal interaction between the central body and the innermost satellite. We study the survival of the Laplace-like resonances and the evolution of the orbital elements of the satellites under the tidal effects. Moreover, we study the possibility of capture into resonance of the fourth satellite

On the Orbit Structure of the Logarithmic Potential

We investigate the dynamics in the logarithmic galactic potential with an analytical approach. The phase-space structure of the real system is approximated with resonant detuned normal forms constructed with the method based on the Lie transform. Attention is focused on the properties of the axial periodic orbits and of low order `boxlets' that play an important role in galactic models. Using energy and ellipticity as parameters, we find analytical expressions of several useful indicators, such as stability-instability thresholds, bifurcations and phase-space fractions of some orbit families and compare them with numerical results available in the literature.Comment: To appear on the Astrophysical Journa

Stability of axial orbits in galactic potentials

We investigate the dynamics in a galactic potential with two reflection symmetries. The phase-space structure of the real system is approximated with a resonant detuned normal form constructed with the method based on the Lie transform. Attention is focused on the stability properties of the axial periodic orbits that play an important role in galactic models. Using energy and ellipticity as parameters, we find analytical expressions of bifurcations and compare them with numerical results available in the literature.Comment: 20 pages, accepted for publication on Celestial Mechanics and Dynamical Astronom

The 3-dimensional cored and logarithm potencials: Periodic orits

Agraïments: The first author is partially supported by CNPq grant 201802/2012-0.We study analytically families of periodic orbits for the cored and logarithmic Hamiltonians H(x, y, z, px, py, pz) = (p2x +p2y +p2z/q)/2+ (1+x2 +(y2 +z2)/q2)1/2, and H(x, y, z, px, py, pz) = (p2x +p2y +p2z/q)/2+ (log(1+x2 +(y2 + z2)/q2))/2, with 3 degrees of freedom, which are relevant in the analysis of the galactic dynamics. First, after introducing a scale transformation in the coordinates and momenta with a parameter ε, we show that both systems give essentially the same set of equations of motion up to first order in ε. Then the conditions for finding families of periodic orbits, using the averaging theory up to first order in ε, apply equally to both systems in every energy level H = h > 0. The averaging method used proves the existence of at most three periodic orbits, for ε small enough, and gives an analytic approximation for the initial conditions of these periodic orbits