arXiv subjects
Publications and source records attributed to Giuseppe Pucacco.
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.
We investigate autonomous perturbations on the orbits of LISA, namely the effects produced by fields that can be expressed only in terms of the position, but not of time in the Hill frame. This first step in the study of the LISA orbits has been the subject of recent papers which implement analytical techniques based on a "post-epicyclic" approximation in the Hill frame to find optimal unperturbed orbits. The natural step forward is to analyze the perturbations to purely Keplerian orbits. In the present work a particular emphasis is put on the tidal field of the Earth assumed to be stationary in the Hill frame. An accurate interpretation of the global structure of the perturbed solution sheds light on possible implications on injection in orbit when the time base-line of the mission is longer than that assumed in previous papers. Other relevant classes of autonomous perturbations are those given by the corrections to the Solar field responsible for a slow precession and a global stationary field, associated to sources like the interplanetary dust or a local dark matter component. The inclusion of simple linear contributions in the expansion of these fields produces secular solutions that can be compared with the measurements and possibly used to evaluate some morphological property of the perturbing components.
We discuss pseudo-Riemannian metrics on 2-dimensional manifolds such that the geodesic flow admits a nontrivial integral quadratic in velocities. We construct local normal forms of such metrics. We show that these metrics have certain useful properties similar to those of Riemannian Liouville metrics, namely: 1) they admit geodesically equivalent metrics; 2) one can use them to construct a big family of natural systems admitting integrals quadratic in momenta; 3) the integrability of such systems can be generalized to the quantum setting; 4) these natural systems are integrable by quadratures.
Analytic methods to investigate periodic orbits in galactic potentials. To evaluate the quality of the approximation of periodic orbits in the logarithmic potential constructed using perturbation theory based on Hamiltonian normal forms. 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 that looking for approximate rational solutions (Prendergast method). It is shown that with a normal form truncated at the lowest order incorporating the relevant resonance it is possible to construct quite accurate solutions both for normal modes and periodic orbits in general position.
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.
In this paper we explore the general conditions in order that a 2-dimensional natural Hamiltonian system possess a second invariant which is a polynomial in the momenta and is therefore Liouville integrable. We examine the possibility that the invariant 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 energy (strong integrability). Using null complex coordinates, we show that the leading order coefficient of the polynomial is an arbitrary holomorphic function in the case of weak integrability and a polynomial in the coordinates in the strongly integrable one. We review the results obtained so far with strong invariants up to degree four and provide some new examples of weakly integrable systems with linear and quadratic invariants.
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.
The phase-space structure of two families of galactic potentials is approximated with a resonant detuned normal form. The normal form series is obtained by a Lie transform of the series expansion around the minimum of the original Hamiltonian. Attention is focused on the quantitative predictive ability of the normal form. We find analytical expressions for bifurcations of periodic orbits and compare them with other analytical approaches and with numerical results. The predictions are quite reliable even outside the convergence radius of the perturbation and we analyze this result using resummation techniques of asymptotic series.
We construct local normal forms of pseudo-Riemannian projectively equivalent 2-dimensional metrics.
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.
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.
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.
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.
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.
The study of diffusion in Hamiltonian systems has been a problem of interest for a number of years. In this paper we explore the influence of self-consistency on the diffusion properties of systems described by coupled symplectic maps. Self-consistency, i.e. the back-influence of the transported quantity on the velocity field of the driving flow, despite of its critical importance, is usually overlooked in the description of realistic systems, for example in plasma physics. We propose a class of self-consistent models consisting of an ensemble of maps globally coupled through a mean field. Depending on the kind of coupling, two different general types of self-consistent maps are considered: maps coupled to the field only through the phase, and fully coupled maps, i.e. through the phase and the amplitude of the external field. The analogies and differences of the diffusion properties of these two kinds of maps are discussed in detail.
Two-dimensional Hamiltonian systems admitting second invariants which are quartic in the momenta are investigated using the Jacobi geometrization of the dynamics. This approach allows for a unified treatment of invariants at both arbitrary and fixed energy. In the differential geometric picture, the quartic invariant corresponds to the existence of a fourth rank Killing tensor. Expressing the Jacobi metric in terms of a Kahler potential, the integrability condition for the existence of the Killing tensor at fixed energy is a non-linear equation involving the Kahler potential. At arbitrary energy, further conditions must be imposed which lead to an overdetermined system with isolated solutions. We obtain several new integrable and superintegrable systems in addition to all previously known examples.