Piero Cipriani

Geometry and chaos on Riemann and Finsler manifolds

Abstract In this paper we discuss some general aspects of the so-called geometrodynamical approach (GDA) to Chaos and present some results obtained within this framework. We firstly derive a naive but nevertheless a general geometrization procedure, and then specialize the discussion to the description of motion within the framework of two among the most representative implementations of the approach, namely the Jacobi and Finsler geometrodynamics. In order to support the claim that the GDA is not simply a mere re-transcription of the usual dynamics, but instead can give various hints on the understanding of the qualitative behaviour of dynamical systems (DSs), we then compare, from a formal point of view, the tools used within the framework of Hamiltonian dynamics to detect the presence of Chaos with the corresponding ones used within the GDA, i.e., the tangent dynamics and the geodesic deviation equations, respectively, pointing out their general inequivalence. Moreover, to advance the mathematical analysis and to highlight both the peculiarities and the analogies of the methods, we work out two concrete applications to the study of very different, yet typical in distinct contexts, dynamical systems. The first is the well-known Henon-Heiles Hamiltonian, which allows us to exploit how the Finsler GDA is well suited not only for testing the dynamical behaviour of systems with few degrees of freedom, but even to get deeper insights into the sources of instability. We show the effectiveness of the GDA, both in recovering fully satisfactory agreement with the most well-established outcomes and also in helping the understanding of the sources of Chaos. Then, in order to point out the general applicability of the method, we present the results obtained from the geometrical description of a General Relativistic DS, namely the Bianchi IX (BIX) cosmological model, whose peculiarity is well known as its very nature has been debated for a long time. Using the Finsler GDA, we obtain results with a built-in invariance, which give evidence to the non-chaotic behaviour of this system, excluding any global exponential instability in the evolution of the geodesic deviation.

FINSLER GEOMETRIC LOCAL INDICATOR OF CHAOS FOR SINGLE ORBITS IN THE HENON-HEILES HAMILTONIAN

Translating the dynamics of the Henon--Heiles hamiltonian as a geodesic flow on a Finsler manifold, we obtain a local and synthetic Geometric Indicator of Chaos (GIC) for two degrees of freedom continuous dynamical systems. It represents a link between local quantities and asymptotic behaviour of orbits giving a strikingly evident, one-to-one, correspondence between geometry and instability.

Dynamical instability and statistical behaviour of N-body systems

Abstract In this paper, we argue about a synthetic characterization of the qualitative properties of generic many-degrees-of-freedom (mdf) dynamical systems (DSs) by means of a geometric description of the dynamics [Geometro-Dynamical Approach (GDA)]. We exhaustively describe the mathematical framework needed to link geometry and dynamical (in)stability, discussing in particular which geometrical quantity is actually related to instability and why some others cannot give, in general, any indication of the occurrence of chaos. The relevance of the Schur theorem to select such Geometrodynamic Indicators (GDI) of instability is then emphasized, as its implications seem to have been underestimated in some of the previous works. We then compare the analytical and numerical results obtained by us and by Pettini and coworkers concerning the FPU chain, verifying a complete agreement between the outcomes of averaging the relevant GDIs over phase space (Casetti and Pettini, 1995) and our findings (Cipriani, 1993), obtained in a more conservative way, time-averaging along geodesics. Along with the check of the ergodic properties of GDIs, these results confirm that the mechanism responsible for chaos in realistic DSs largely depends on the fluctuations of curvatures rather than on their negative values, whose occurrence is very unlikely. On these grounds we emphasize the importance of the virialization process, which separates two different regimes of instability. This evolutionary path, predicted on the basis of analytical estimates, receives clear support from numerical simulations, which, at the same time, confirm also the features of the evolution of the GDIs along with their dependence on the number of degrees of freedom, N , and on the other relevant parameters of the system, pointing out the scarce relevance of negative curvature (for N ⪢ 1) as a source of instability. The general arguments outlined above, are then concretely applied to two specific N-body problems, obtaining some new insights into known outcomes and also some new results The comparative analysis of the FPU chain and the gravitational N-body system allows us to suggest a new definition of strong stochasticity, for any DS. The generalization of the concept of dynamical time-scale, tD, is at the basis of this new criterion. We derive for both the mdf systems considered the ( N , e)-dependence of tD (e being the specific energy) of the system. In light of this, the results obtained (Cerruti-Sola and Pettini, 1995), indeed turn out to be reliable, the perplexity there raised originating from the neglected N -dependence of tD, and not to an excessive degree of approximation in the averaged equations used. This points out also the peculiarities of gravitationally bound systems, which are always in a regime of strong instability; the dimensionless quantity L1 = γ1 · tD [γ1 is the maximal Lyapunov Characteristic Number (LCN)] being always positive and independent of e, as it happens for the FPU chain only above the strong stochasticity threshold (SST). The numerical checks on the analytical estimates about the ( N , e)-dependence of GDIs, allow us to single out their scaling laws, which support our claim that, for N ⪢ 1, the probability of finding a negative value of Ricci curvature is practically negligible, always for the FPU chain, whereas in the case of the Gravitational N-body system, this is certainly true when the virial equilibrium has been attained. The strong stochasticity of the latter DS is clearly due to the large amplitude of curvature fluctuations. To prove the positivity of Ricci curvature, we need to discuss the pathologies of mathematical Newtonian interaction, which have some implications also on the ergodicity of the GDIs for this DS. We discuss the Statistical Mechanical properties of gravity, arguing how they are related to its long range nature rather than to its short scale divergencies. The N -scaling behaviour of the single terms entering the Ricci curvature show that the dominant contribution comes from the Laplacian of the potential energy, whose singularity is reflected on the issue of equality between time and static averages. However, we find that the physical N-body system is actually ergodic where the GDIs are concerned, and that the Ricci curvature associated is indeed almost everywhere (and then almost always) positive, as long as N ⪢ 1 and the system is gravitationally bound and virialized. On these grounds the equality among the above mentioned averages is restored, and the GDA to instability of gravitating systems gives fully reliable and understandable results. Finally, as a by-product of the numerical simulations performed, for both the DSs considered, it emerges that the time averages of GDIs quickly approach the corresponding canonical ones, even in the quasi-integrable limit, whereas, as expected, their fluctuations relax on much longer timescales, in particular below the SST.

Some critical remarks on relaxation in N -body systems

SummaryWe analyse in detail the origin of some controversial points concerning the relationships between thechaotic behaviour of dynamics and therelaxation properties of Hamiltonian systems with many degrees of freedom (m.d.f.), showing that most claims existing in the literature and applied, in particular, to theN-body self-gravitating systems, are, at least, unjustified.

Jacobi Geometry and Chaos in N-Body Systems

One of the most ambitious tasks of analytical dynamics is to assess the long-time behaviour of an Hamiltonian system and to try to make predictions about its stability. In the realm of Celestial Mechanics this is related to the fundamental problem of N bodies interacting via the gravitational force (leaving away, for simplicity, dissipative effects) as a model for the Solar System. To the celestial mechanician N ≃ 10 appears to be already a big number, at least since it is quite bigger than 3. The reasons for this attitude are justified broadly with theoretical and practical arguments, the most compelling being that, when N is really big, let us say N ≃ 105 (the order required to treat a globular cluster) or N ≃ 1010 (that required to treat a galaxy), we are leaving the field of analytical mechanics and are entering the field of statistical mechanics. Easing the burden of such big systems is therefore accomplished by delivering the matter to another branch of physics. The problem is that the statistical mechanics of non equilibrium is still not a well grounded discipline and, worst of it, the statistical mechanics of large systems interacting via long-range forces cannot be carried out. So, even if many attempts were made to frame the matter into a stellar dynamical context, many points still remain obscure. Aim of this contribution is to signal the power of the technique based on the geometrization of dynamics to shed light on the behaviour of Hamiltonian systems with many degrees of freedom and to fill the gap of our understanding between them and small N systems.

The Many Faces of Dynamical Instability

Recently, [6, 7, 3, 8], we proposed a generalization to non Riemannian manifolds of the so-called Geometro-Dynamical Approach (GDA) to Chaos, [11, 2], able to widen the applicability of the method to a considerably larger class of dynamical systems (DS’s). Here, we carry on our efforts on a pathway directed towards a synthetic and a priori characterization of the qualitative properties of generic DS’s. Although being aware that this goal is very ambitious, and that, up to now, many of the trials have been discouraging, we shouldn’t forget the theoretical as well practical relevance held by a possible successful attempt. Indeed, if only it would be concevaible to single out a synthetic indicator of (in)stability, we will be able to avoid all the consuming computations needed to empirically discover the nature of a particular orbit, perhaps noticeably different with respect to another one very nearby. Besides this, going beyond the semi-phenomenological mere recognition of the occurrence of dynamical instability, this approach could give deeper hints on its sources, even in those situations where the boundary between Order and Chaos tends to become more and more nuanced, and different tools seem to give conflicting answers. Lately, a renewed interest towards a concise description of dynamical instability1 has grown, together with the feeling of the need to look deeply at the intermingled structures underlving the transition from quasi-integrable to stochastic motions.

Geometrodynamics, Chaos and Statistical Behaviour of N-Body Systems

We use a geometrical description of the dynamics of Hamiltonian systems to single out the sources of instability (and of Chaos, if any). We show that: A) the instability is driven by the fluctuations of some geometrical invariants, rather than by their average values; B) the most commonly used invariant has in general nothing to do with dynamic instability of realistic many degrees of freedom systems; C) in order to evaluate correctly the relevant quantities entering these geometric invariants, it is necessary the system settles down to a global vinal equilibrium, and that for this the number of degrees of freedom is crucial; D) the gravitating N-body system is peculiar for what it concerns both the dynamical properties and the possibility of a statistical description. So, all the claims that a geometric description of dynamics, in particular for the stellar dynamical problem, gives a direct estimate of some relaxation time are unjustified. Nevertheless, we point out that the geometrical transcription of hamiltonian systems, if carefully employed, can give deep informations about the degree of stochasticity in the dynamics, and very interesting insights on its sources. To overcome some of the limitations of the approach, for systems with few degrees of freedom and/or with time-dependent lagrangians, we introduce an extension, discussed in detail in the companion contribution[3].

Geometrodynamics on Finsler Spaces

We have developed a gauge invariant approach to study the dynamical behaviour of Lagrangian systems whose potential depends on both coordinates and velocities (possibly, on time), using a geometrical description. The manifold in which the dynamical systems live is a Finslerian space in which the conformai factor is a positively homogeneous function of first degree in the velocities (the homogeneous Lagrangian of the system). This method is a generalization of the standard geometrodynamical ones which use a Riemannian manifold (see also [11]), as it permits to study a wider class of dynamical systems. Moreover, it is well suited to treat conservative systems with few degrees of freedom and peculiar dynamical systems whose Lagrangian is not ”standard”, such as the one describing the so-called Mixmaster Universe.