The theory of canonical perturbations applied to attitude dynamics and to the Earth rotation. Osculating and nonosculating Andoyer variables

In the method of variation of parameters we express the Cartesian coordinates or the Euler angles as functions of the time and six constants. If, under disturbance, we endow the “constants” with time dependence, the perturbed orbital or angular velocity will consist of a partial time derivative and a convective term that includes time derivatives of the “constants”. The Lagrange constraint, often imposed for convenience, nullifies the convective term and thereby guarantees that the functional dependence of the velocity on the time and “constants” stays unaltered under disturbance. “Constants” satisfying this constraint are called osculating elements. Otherwise, they are simply termed orbital or rotational elements. When the equations for the elements are required to be canonical, it is normally the Delaunay variables that are chosen to be the orbital elements, and it is the Andoyer variables that are typically chosen to play the role of rotational elements. (Since some of the Andoyer elements are time-dependent even in the unperturbed setting, the role of “constants” is actually played by their initial values.) The Delaunay and Andoyer sets of variables share a subtle peculiarity: under certain circumstances the standard equations render the elements nonosculating. In the theory of orbits, the planetary equations yield nonosculating elements when perturbations depend on velocities. To keep the elements osculating, the equations must be amended with extra terms that are not parts of the disturbing function [Efroimsky, M., Goldreich, P.: J. Math. Phys. 44, 5958–5977 (2003); Astron. Astrophys. 415, 1187–1199 (2004); Efroimsky, M.: Celest. Mech. Dyn. Astron. 91, 75–108 (2005); Ann. New York Acad. Sci. 1065, 346–374 (2006)]. It complicates both the Lagrange- and Delaunay-type planetary equations and makes the Delaunay equations noncanonical. In attitude dynamics, whenever a perturbation depends upon the angular velocity (like a switch to a noninertial frame), a mere amendment of the Hamiltonian makes the equations yield nonosculating Andoyer elements. To make them osculating, extra terms should be added to the equations (but then the equations will no longer be canonical). Calculations in nonosculating variables are mathematically valid, but their physical interpretation is not easy. Nonosculating orbital elements parameterise instantaneous conics not tangent to the orbit. (A nonosculating i may differ much from the real inclination of the orbit, given by the osculating i.) Nonosculating Andoyer elements correctly describe perturbed attitude, but their interconnection with the angular velocity is a nontrivial issue. The Kinoshita–Souchay theory tacitly employs nonosculating Andoyer elements. For this reason, even though the elements are introduced in a precessing frame, they nevertheless return the inertial velocity, not the velocity relative to the precessing frame. To amend the Kinoshita–Souchay theory, we derive the precessing-frame-related directional angles of the angular velocity relative to the precessing frame. The loss of osculation should not necessarily be considered a flaw of the Kinoshita–Souchay theory, because in some situations it is the inertial, not the relative, angular velocity that is measurable [Schreiber, K. U. et al.: J. Geophys. Res. 109, B06405 (2004); Petrov, L.: Astron. Astrophys. 467, 359–369 (2007)]. Under these circumstances, the Kinoshita–Souchay formulae for the angular velocity should be employed (as long as they are rightly identified as the formulae for the inertial angular velocity).     

基于DEVS形式理论和面向对象方法学的DEVS系统建模仿真框架

描述了一个基于离散事件系统（ＤＥＶＳ）形式理论和面向对象方法学的 ＤＥＶＳ系统建模仿真框架ＳｉｍｕＣｌａｓｓ,它不但在 ＶｉｓｕａｌＣ＋＋环境下实现了 ＤＥＶＳ形式理论及其相关的层次仿真算法,而且结合了两种强大的系统开发框架：ＤＥＶＳ形式理论和面向对象范形,ＳｉｍｕＣｌａｓｓ支持在Ｃ＋＋面向对象环境中运用层次模块化方法进行离散事件模型的开发。     

Seismic Hazard of Honduras

In this paper we have described the proceduresused, input data applied and results achieved in ourefforts to develop seismic hazard maps of Honduras.The probabilistic methodology of Cornell is employed.Numerical calculations were carried out by making useof the computer code SEISRISK III. To examine theimpact of uncertainties in seismic and structuralcharacteristics, the logic tree formalism has beenused. We compiled a de-clustered earthquake cataloguefor the region comprising 1919 earthquakes occurringduring the period from 1963 to 1997. Unified momentmagnitudes were introduced. Definition of aseismotectonic model of the whole region under review,based on geologic, tectonic and seismic information,led to the definition of seven seismogenetic zones forwhich seismic characteristics were determined. Fourdifferent attenuation models were considered. Resultsare expressed in a series of maps of expected PGA for60% and 90% probabilities of nonexceedence in a50-year interval which corresponds to return periodsof 100 and 475 years, respectively. The highest PGAvalues of about 0.4g (90% probability ofnon-exceedence) are expected along the borders withGuatemala and El Salvador.     

辛算法在动力天文中的应用（Ⅲ）

文［１］和文［２］从哈密顿系统的整体结构保持一角度阐明了辛算法［３－６］的主要功能，本文将从定量的角度进一步表明辛算法的另一独特优点－可以控制天体运动沿迹误差的快速增长，并对可分离哈密顿系统的显式辛差分格式稍加改进，推广应用到一般动力系统，该系统含有小耗散项或小的不可分离项，计算结果表明，效果极佳，因此，辛算法与传统的数值解法相比，确有很多优点。     

The Lissajous transformation I. Basics

A new canonical transformation is proposed to handle elliptic oscillators, that is, Hamiltonian systems made of two harmonic oscillators in a 1-1 resonance. Lissajous elements pertain to the ellipse drawn with a light pen whose coordinates oscillate at the same frequency, hence their name. They consist of two pairs of angle-action variables of which the actions and one angle refer to basic integrals admitted by an elliptic oscillator, namely, its energy, its angular momentum and its Runge-Lenz vector. The Lissajous transformation is defined in two ways: explicitly in terms of Cartesian variables, and implicitly by resolution of a partial differential equation separable in polar variables. Relations between the Lissajous variables, the common harmonic variables, and other sets of variables are discussed in detail.     

Solute transport in a single fracture with negligible matrix permeability: 2. mathematical formalism

This report provides an overview of the mathematical expressions for modeling fundamental solute transport mechanisms at the fracture scale. It focuses on low-permeability rocks where advection in the matrix is negligible as compared to that in fractures. The following processes are considered: (1) advective transport in fractures, (2) hydrodynamic dispersion along the fracture axis, (3) molecular diffusion from the fracture to the porous matrix, (4) sorption reactions on the fracture walls and within the matrix, and (5) decay reactions. The aim of this review is to gather in a single article the transport equations and their analytical solutions, using a homogeneous notation to facilitate comparison and exploitation.

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Resumen Este informe proporciona una revisión de las expresiones matemáticas para modelar los mecanismos fundamentales de transporte de solutos a escala de fracturas. Se centra en rocas de baja permeabilidad donde la advección en la matriz es despreciable en comparación con la de las fracturas. Se considera los procesos siguientes: (1) transporte advectivo en las fracturas; (2) dispersión hidrodinámica a lo largo del eje de la fractura; (3) difusión molecular desde la fractura a la matriz porosa; (4) reacciones de sorción en la pared de la fractura y en la matriz; y (5) reacciones de degradación. El propósito de esta revisión es reunir en un único artículo las ecuaciones de transporte y sus soluciones analíticas con una notación homogénea para facilitar su comparación y aplicación.

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Résumé Ce rapport présente une vue d'ensemble des expressions mathématiques proposées dans la littérature pour modéliser le transport de soluté à l'échelle de la fracture. L'accent est mis sur les roches faiblement perméables où le transport par convection dans la matrice rocheuse peut être considéré comme négligeable devant le transport dans les fractures. Les processus considérés sont les suivants: (1) transport par convection dans les fractures, (2) dispersion hydrodynamique, (3) échanges de soluté par diffusion moléculaire entre les fractures et la matrice rocheuse, (4) réactions de sorption sur les parois des fractures et à l'intérieur de la matrice et (5) réactions de décroissance. L'objectif premier de cette revue est de rassembler les équations de transport et leurs solutions analytiques en utilisant une notation homogène pour faciliter leur utilisation et les comparaisons.

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Discrete Virial Theorem

We reexamine the classical virial theorem for bounded orbits of arbitrary autonomous Hamiltonian systems possessing both regular and chaotic orbits. New and useful forms of the virial theorem are obtained for natural Hamiltonian flows of arbitrary dimension. A discrete virial theorem is derived for invariant circles and periodic orbits of natural symplectic maps. A weak and a strong form of the virial theorem are proven for both flows and maps. While the Birkhoff Ergodic Theorem guarantees the existence of the relevant time averages for both regular and chaotic orbits, the convergence is very rapid for the former and extremely slow for the latter. This circumstance leads to a simple and efficient measure of chaoticity. The results are applied to several problems of current physical interest, including the Hénon–Heiles system, weak chaos in the standard map, and a 4D Froeschlé map.     

Hamiltonian Stability of Spin–Orbit Resonances in Celestial Mechanics

The behaviour of ‘resonances’ in the spin-orbit coupling in celestial mechanics is investigated in a conservative setting. We consider a Hamiltonian nearly-integrable model describing an approximation of the spin-orbit interaction. The continuous system is reduced to a mapping by integrating the equations of motion through a symplectic algorithm. We study numerically the stability of periodic orbits associated to the above mapping by looking at the eigenvalues of the matrix of the linearized map over the full cycle of the periodic orbit. In particular, the value of the trace of the matrix is related to the stability character of the periodic orbit. We denote by ε_* (p/q) the value of the perturbing parameter at which a given elliptic periodic orbit with frequency p/q becomes unstable. A plot of the critical function ε_* (p/q) versus the frequency at different orbital eccentricities shows significant peaks at the synchronous resonance (for low eccentricities) and at the synchronous and 3:2 resonances (at higher eccentricities) in good agreement with astronomical observations. This revised version was published online in July 2006 with corrections to the Cover Date.     

Optimization of the shape of Gaussian beams

The applicability and accuracy of the Gaussian beam method depend on the proper choice of the shape of beams. Gaussian beams become inaccurate solutions of the elastodynamic equation if the velocity field changes considerably within the beam width. We present a procedure of determining the optimum initial shape of Gaussian beams based on minimizing the average squared widths of Gaussian beams and smoothing the distribution of the optimum parameters of Gaussian beams on the Hamiltonian hypersurface in the phase-space. The original method of smoothing represents an essential part of the algorithm, which is designed particularly for the optimization of the shape of Gaussian beams for Gaussian beam or packet migrations.