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We present and compare different numerical schemes for the integration of the variational equations of autonomous Hamiltonian systems whose kinetic energy is quadratic in the generalized momenta and whose potential is a function of the generalized positions. We apply these techniques to Hamiltonian systems of various degrees of freedom, and investigate their efficiency in accurately reproducing well-known properties of chaos indicators like the Lyapunov Characteristic Exponents (LCEs) and the Generalized Alignment Indices (GALIs). We find that the best numerical performance is exhibited by the textit{`tangent map (TM) method}, a scheme based on symplectic integration techniques which proves to be optimal in speed and accuracy. According to this method, a symplectic integrator is used to approximate the solution of the Hamiltons equations of motion by the repeated action of a symplectic map $S$, while the corresponding tangent map $TS$, is used for the integration of the variational equations. A simple and systematic technique to construct $TS$ is also presented.
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The reader can find in the literature a lot of different techniques to study the dynamics of a given system and also, many suitable numerical integrators to compute them. Notwithstanding the recent work of Maffione et al. (2011a) for mappings, a detailed comparison among the widespread indicators of chaos in a general system is still lacking. Such a comparison could lead to select the most efficient algorithms given a certain dynamical problem. Furthermore, in order to choose the appropriate numerical integrators to compute them, more comparative studies among numerical integrators are also needed. This work deals with both problems. We first extend the work of Maffione et al. (2011) for mappings to the 2D Henon & Heiles (1964) potential, and compare several variational indicators of chaos: the Lyapunov Indicator (LI); the Mean Exponential Growth Factor of Nearby Orbits (MEGNO); the Smaller Alignment Index (SALI) and its generalized version, the Generalized Alignment Index (GALI); the Fast Lyapunov Indicator (FLI) and its variant, the Orthogonal Fast Lyapunov Indicator (OFLI); the Spectral Distance (D) and the Dynamical Spectras of Stretching Numbers (SSNs). We also include in the record the Relative Lyapunov Indicator (RLI), which is not a variational indicator as the others. Then, we test a numerical technique to integrate Ordinary Differential Equations (ODEs) based on the Taylor method implemented by Jorba & Zou (2005) (called taylor), and we compare its performance with other two well-known efficient integrators: the Prince & Dormand (1981) implementation of a Runge-Kutta of order 7-8 (DOPRI8) and a Bulirsch-Stoer implementation. These tests are run under two very different systems from the complexity of their equations point of view: a triaxial galactic potential model and a perturbed 3D quartic oscillator.
We present numerical simulations for the three-body problem, in which three particles lie at rest at the vertex of a perturbed equilateral triangle. In the unperturbed problem, the three particles fall towards the center of mass of the system to form a three-body collision, or singularity, where the particles overlap in space and time. By perturbing the initial positions of the particles, we are able to study chaos in the vicinity of the singularity. Here we cover the full range in parameter space for binary formation due to three-body interactions of isolated single stars, covering the singular region corresponding to an equilateral triangle and extending to sufficiently deformed triangles that we enter the binary-single scattering regime (i.e., one side of the triangle is very short and the other two are very long). We make phase space plots to study the regular and ergodic subsets of our simulations independently and derive the expected properties of the left-over binaries from three-body binary formation in isotropic cluster environments. We further provide fits to the ergodic subset to characterize the properties of the left-over binaries. We identify the discrepancy between the statistical theory and the simulations to the regular subset of interactions, which exhibit only weak chaos. As we decrease the scale of the perturbations in the initial positions, the phase space becomes entirely dominated by regular interactions, according to our metric for chaos.
The gravitational interaction between a protoplanetary disc and planetary sized bodies that form within it leads to the exchange of angular momentum, resulting in migration of the planets and possible gap formation in the disc for more massive planets. In this article, we review the basic theory of disc-planet interactions, and discuss the results of recent numerical simulations of planets embedded in protoplanetary discs. We consider the migration of low mass planets and recent developments in our understanding of so-called type I migration when a fuller treatment of the disc thermodynamics is included. We discuss the runaway migration of intermediate mass planets (so-called type III migration), and the migration of giant planets (type II migration) and the associated gap formation in the disc. The availability of high performance computing facilities has enabled global simulations of magnetised, turbulent discs to be computed, and we discuss recent results for both low and high mass planets embedded in such discs.
In many problems in particle cosmology, interaction rates are dominated by ${2}leftrightarrow{2}$ scatterings, or get a substantial contribution from them, given that ${1}leftrightarrow{2}$ and ${1}leftrightarrow{3}$ reactions are phase-space suppressed. We describe an algorithm to represent, regularize, and evaluate a class of thermal ${2}leftrightarrow{2}$ and ${1}leftrightarrow{3}$ interaction rates for general momenta, masses, chemical potentials, and helicity projections. A key ingredient is an automated inclusion of virtual corrections to ${1}leftrightarrow{2}$ scatterings, which eliminate logarithmic and double-logarithmic IR divergences from the real ${2}leftrightarrow{2}$ and ${1}leftrightarrow{3}$ processes. We also review thermal and chemical potential induced contributions that require resummation if plasma particles are ultrarelativistic.
This paper is a continuation of our earlier study on the integrability of the Friedmann equations in the light of the Chebyshev theorem. Our main focus will be on a series of important, yet not previously touched, problems when the equation of state for the perfect-fluid universe is nonlinear. These include the generalized Chaplygin gas, two-term energy density, trinomial Friedmann, Born--Infeld, and two-fluid models. We show that some of these may be integrated using Chebyshevs result while other are out of reach by the theorem but may be integrated explicitly by other methods. With the explicit integration, we are able to understand exactly the roles of the physical parameters in various models play in the cosmological evolution. For example, in the Chaplygin gas universe, it is seen that, as far as there is a tiny presence of nonlinear matter, linear matter makes contribution to the dark matter, which becomes significant near the phantom divide line. The Friedmann equations also arise in areas of physics not directly related to cosmology. We provide some examples ranging from geometric optics and central orbits to soap films and the shape of glaciated valleys to which our results may be applied.
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