On the number of isolating integrals in systems with three degrees of freedom

Dynamical systems with three degrees of freedom can be reduced to the study of a fourdimensional mapping. We consider here, as a model problem, the mapping given by the following equations: $$\left\{ \begin{gathered} x_1 = x_0 + a_1 {\text{ sin (}}x_0 {\text{ + }}y_0 {\text{)}} + b{\text{ sin (}}x_0 {\text{ + }}y_0 {\text{ + }}z_{\text{0}} {\text{ + }}t_{\text{0}} {\text{)}} \hfill \\ y_1 = x_0 {\text{ + }}y_0 \hfill \\ z_1 = z_0 + a_2 {\text{ sin (}}z_0 {\text{ + }}t_0 {\text{)}} + b{\text{ sin (}}x_0 {\text{ + }}y_0 {\text{ + }}z_{\text{0}} {\text{ + }}t_{\text{0}} {\text{) (mod 2}}\pi {\text{)}} \hfill \\ t_1 = z_0 {\text{ + }}t_0 \hfill \\ \end{gathered} \right.$$ We have found that as soon asb≠0, i.e. even for a very weak coupling, a dynamical system with three degrees of freedom has in general either two or zero isolating integrals (besides the usual energy integral).     

On the disappearance of isolating integrals in dynamical systems with more than two degrees of freedom

We continue to study the number of isolating integrals in dynamical systems with three and four degrees of freedom, using as models the measure preserving mappingsT already introduced in preceding papers (Froeschlé, 1973; Froeschlé and Scheidecker, 1973a).Thus, we use here a new numerical method which enables us to take as indicator of stochasticity the variation withn of the two (respectively three) largest eigenvalues-in absolute magnitude-of the linear tangential mappingT ⁿ * ofT ⁿ . This variation appears to be a very good tool for studying the diffusion process which occurs during the disappearance of the isolating integrals, already shown in a previous paper (Froeschlé, 1971). In the case of systems with three degrees of freedom, we define and give an estimation of the diffusion time, and show that the gambler's ruin model is an approximation of this diffusion process.     

Disappearance of integrals in systems of more than two degrees of freedom

The disappearance of some integrals of motion when two or more resonance conditions are approached at the same time is explained. As an example a Hamiltonian of three degrees of freedom is considered in action-angle variables which in zero order represents three harmonic oscillators, while the perturbation contains two trigonometric terms. One integral disappears if two appropriate resonant conditions are approached sufficiently closely.     

Bifurcations in systems of three degrees of freedom

We study the bifurcations of families of double and quadruple period orbits in a simple Hamiltonian system of three degrees of freedom. The bifurcations are either simple or double, depending on whether a stability curve crosses or is tangent to the axis b=–2. We have also generation of a new family whenever a given family has a maximum or minimum or .The double period families bifurcate from simple families of periodic orbits. We construct existence diagrams to show where any given family exists in the control space (, ) and where it is stable (S), simply unstable (U), doubly unstable (DU), or complex unstable (), We construct also stability diagrams that give the stability parameters b₁ and b₂ as functions of (for constant ), or of (for constant ).The quadruple period orbits are generated either from double period orbits, or directly from simple period orbits (at double bifurcations). We derive several rules about the various types of bifurcations. The most important phenomenon is the collision of bifurcations. At any such collision of bifurcations the interconnections between the various families change and the general character of the dynamical system changes.     

Some models with three isolating integrals of motion

The existence of a third isolating (nonclassical) integral of motion is, in a certain sense, related to a nongaussian velocity distribution in stellar systems. Based on a previously found series of models with St?ckel potentials, several characteristics of these models are generalized by linear superposition to describe some observed properties of galactic systems. Corresponding formulas for the circular rotation velocity and surface density are obtained. The variations of these characteristics are plotted graphically for two interesting cases.     

On the number of effective integrals in galactic models

Three different numerical techniques are tested to determine the number of integrals of motion in dynamical systems with three degrees of freedom.

1. The computation of the whole set of Lyapunov Characteristic Exponents (LCE).
2. The triple sections in the configurations space.
3. The Stine-Noid box-counting technique.

These methods are applied to a triple oscillator with coupling terms of the third order. Cases are found for which one effective integral besides the Hamiltonian subsists during a very long time. Such orbits display simultaneously chaotic and quasi-periodic motion, according to which coordinates are considered. As an application, the LCE procedure is applied to a triaxial elliptical galaxy model. Contrary to similar 2-dimensional systems, this 3-dimensional one presents noticeable zones in the phase space without any non-classical integral.     

Non-integrability of first order resonances in Hamiltonian systems in three degrees of freedom

The normal forms of the Hamiltonian 1:2:ω resonances to degree three for ω = 1, 3, 4 are studied for integrability. We prove that these systems are non-integrable except for the discrete values of the parameters which are well known. We use the Ziglin–Morales–Ramis method based on the differential Galois theory.     

Intrinsic variational equations for conservative dynamical systems withN degrees of freedom

For a conservative dynamical system withn deg. of freedom we show that the equations of variation along an orbit may be written with respect to an orthonormal moving frame (a generalized Frenet frame) in which the tangential variation is given by a quadrature and the normal andn-2 binormal variations are solutions ofn-1 coupled second order equations of the form of Hill's equation.     

Generalized limits to the number of light particle degrees of freedom from big bang nucleosynthesis

We compute the big bang nucleosynthesis limit on the number of light neutrino degrees of freedom in a model-independent likelihood analysis based on the abundances of ⁴He and ⁷Li. We use the two-dimensional likelihood functions to simultaneously constrain the baryon-to-photon ratio and the number of light neutrinos for a range of ⁴He abundances Y_p = 0.225–0.250, as well as a range in primordial ⁷Li abundances from (1.6 to 4.1) ×10⁻¹⁰. For (⁷Li/H)_p = 1.6 × 10⁻¹⁰, as can be inferred from the ⁷Li data from Population II halo stars, the upper limit to N_ν based on the current best estimate of the primordial ⁴He abundance of Y_p = 0.238 is N_ν < 4.3 and varies from N_ν < 3.3 (at 95% C.L.) when Y_p = 0.225 to N_ν < 5.3 when Y_p = 0.250. If ⁷Li is depleted in these stars the upper limit to N_ν is relaxed. Taking (⁷Li/H)_p = 4.1 × 10⁻¹⁰, the limit varies from N_ν < 3.9 when Y_p = 0.225 to N_ν 6 when Y_p = 0.250. We also consider the consequences on the upper limit to N_ν if recent observations of deuterium in high-redshift quasar absorption-line systems are confirmed.     

A problem of libration with two degrees of freedom

Moore (1983) presented a theory of resonance with two degrees of freedom based on the Bohlin-von Zeipel procedure. This procedure is now applied to librational motion with all constants re-evaluated in terms of values of the momenta given either by the initial conditions, or, in the case of the momentumy ₁ conjugate to the critical argument x₁, by its value at the libration centre. Numerical results are presented for a resonant satellite in a near 12 hr orbit and for a geosynchronous satellite. The theory is further developed to include near-circular orbits by recasting the problem in terms of the Poincaré eccentric variables.     

On the construction of dynamic systems from given integrals

In this communication we propose a new approach for studying a particular type of inverse problems in mechanics related to the construction of a force field from given integrals.An extension of the Danielli problem is obtained. The given results are applied to the Suslov problem, and illustrated in specific examples.     

Periodic orbits in a dynamical system with three degrees of freedom

We use the analytical method of Lindstedt to make an inventory of the regular families of periodic orbits and to obtain approximate analytical solutions in a three-dimensional harmonic oscillator with perturbing cubic terms. We compare these solutions to the results of numerical computations at a specific orbital resonance.     

Stellar population synthesis with more degrees of freedom than observables

In order to derive the stellar population of a galaxy or a star cluster, it is a common practice to fit its spectrum by a combination of spectra extracted from a data base (e.g. a library of stellar spectra). If the data to be fitted are equivalent widths, the combination is a non-linear one and the problem of finding the 'best' combination of stars that fits the data becomes complex. It is probably because of this complexity that the mathematical aspects of the problem did not receive a satisfying treatment; the question of the uniqueness of the solution , for example, was left in uncertainty. In this paper we complete the solution of the problem by considering the underdetermined case where there are fewer equivalent widths to fit than stars in the data base (the overdetermined case was treated previously). The underdetermined case is interesting to consider because it leaves space for the addition of supplementary astrophysical constraints. In fact, it is shown in this paper that when a solution exists it is generally not unique. There are infinitely many solutions, all of them contained within a convex polyhedron in the solutions vector space. The vertices of this polyhedron are extremal solutions of the stellar population synthesis. If no exact solution exists, an approximate solution can be found using the method described for the overdetermined case. Also provided is an algorithm able to solve the problem numerically; in particular all the vertices of the polyhedron are found.     

Finding how many isolating integrals of motion an orbit obeys

The correlation dimension, that is the dimension obtained by computing the correlation function of pairs of points of a trajectory in phase space, is a numerical technique introduced in the field of non-linear dynamics in order to compute the dimension of the manifold in which an orbit moves, without the need of knowing the actual equations of motion that give rise to the trajectory. This technique has been proposed in the past as a method to measure the dimension of stellar orbits in astronomical potentials, that is the number of isolating integrals of motion the orbits obey. Although the algorithm can in principle yield that number, some care has to be taken in order to obtain good results. We studied the relevant parameters of the technique, found their optimal values, and tested the validity of the method on a number of potentials previously studied in the literature, using the Smaller Alignment Index (SALI), Lyapunov exponents and spectral dynamics as gauges.     

Rational approximations for a Hamiltonian System with three degrees of freedom

We develop a new method to find solutions of the equations of motion in Hamiltonian Dynamical Systems. The idea of the method is to express the solution of the nonlinear ODE in the formx=N/D ⁿ , whereN andD are Fourier series andn is an appropriate constant. We apply this method to a galactic potential with three degrees of freedom.Paper presented at the 11th European Regional Astronomical Meetings of the IAU on New Windows to the Universe, held 3–8 July, 1989, Tenerife, Canary Islands, Spain.     

A resonance problem of two degrees of freedom

An analytical theory is presented for determining the motion described by a Hamiltonian of two degrees of freedom. Hamiltonians of this type are representative of the problem of an artificial Earth satellite in a near-circular orbit or a near-equatorial orbit and in resonance with a longitudinal dependent part of the geopotential. Using the classical Bohlin-von Zeipel procedure the variation of the elements is developed through a generating function expressed as a trigonometrical series. The coefficients of this series, determined in ascending powers of an auxiliary parameter, are the solutions of paired sets of ordinary differential equations and involve elliptic functions and quadrature. The first order solution accounts for the full variation of the resonance terms with the second coordinate.     

Numerical study of periodic orbit properties in a dynamical system with three degrees of freedom

The locations and stability features of the main symmetrical periodic orbits in the potential $$V = \tfrac{1}{2}\left( {Ax^2 + By^2 + Cz^2 } \right) - \varepsilon xz^2 - \eta yz^2 with \sqrt {A:} \sqrt {B:} \sqrt C = 6:4:3$$ are calculated. Two resonant 1-periodic orbits reveal themselves to be the most important of the system. The third dimension and the additional coupling term have a large effect upon the emergence and stability of p.o. prolongated from the bi-dimensional cases 4∶3 and 2∶1. The existence of three main instability types leads to behaviours much more complicated than in systems with two degrees of freedom. Particularly the presence of complex instability, a new feature with respect to bi-dimensional problems, may produce large instability regions in the set of initial conditions. Some asymptotic curves emanating from unstable orbits are calculated in the four-dimensional space of section. The aspect of such curves is considerably modified when a perturbation is added in the third dimension. The neighbourhood of orbits suffering from complex instability is studied in the space of section and by means of the maximum Lyapunov Characteristic Number technique. It is shown that the motion can deviate far from the vicinity of the p.o. representative point as soon as the orbit is of complex instability. When the perturbation is large enough, the stochasticity produced by this type of instability can be very important.     

On the 3/1 planar and circular resonant problem

The simplest model of a resonant problem of second order is the planar and circular case. Simplification like this is very old and for 3/1 resonance, several authors have studied this problem with different purposes. In this work, we test this model for the available asteroids, by applying Hori's perturbation method. Explicit solutions of the intermediate orbit are obtained. In the plane of two constants of the problem, all types of motion are described. By testing the model, it is shown that, in general, one can confirm results of numerical integrations indicating libration for a few number of asteroids and circulation for most of them. However, agreement in numerical values for amplitude and period of librations seems to be not possible mainly if Jupiter's eccentricity is neglected. On the other hand, even though there might be some physical reasons determining that only asteroids with high eccentricity may librate, it is shown that, from mathematical point of view, libration may occur even in the case of small eccentricities provided that some relations are satisfied.