Hyperboloidal precession of a dynamically symmetric satellite. Construction of normal forms of the Hamiltonian

The Norma specialized program package, intended for normalization of autonomous Hamiltonian systems by means of computer algebra, is used in studies of small-amplitude periodic motions in the neighbourhood of regular precessions of a dynamically symmetric satellite on a circular orbit. The case of hyperboloidal precession is considered. Analytical expressions for normal forms and generating functions depending on frequencies of the system as on parameters are derived. Possible resonances are considered in particular. The 6th order of normalization is achieved. Though the intermediate analytical expressions occupy megabytes of computer's main memory, final ones are quite compact. Obtained analytical expressions are applied to the analysis of stability of small-amplitude periodic motions in the neighbourhood of hyperboloidal precession.     

Mild-slope equation for water waves propagating over non-uniform currents and uneven bottoms

I~IOXThe interaction between surface waves and ambient currents and nearshore topography lies atone of the heat of morphological medelling. Accurate predictions of how wave propagates overcurrents and topography, and of the consequent erosion and dePOSition of sand on a beach or tidalflat are vital when assessing how a coastline may be affected by changing conditions.The mild-slope equation was introduced by Berkhoff (1972) as a way of approximating therefraction-diffraction of linearized s…     

用Hamilton方法计算滞弹地球自转的潮汐变化

在日月引潮力势作用下地球产生弹性形变．地幔粘滞性子致这个形变对于引潮力滞后,成为引起地球自转长期减慢的原因之一．地幔滞弹性也使有效洛夫数k增加,并使自转变化的周期项位相滞后,即产生反常位相项．本文首先用Hamilton方法计算了地球的形变．然后考虑到地幔的滞弹性,计算了在日月引潮力作用下的地球自转长期减慢和滞弹性对周期（带谐）变化的影响．     

地震波传播的哈密顿表述及辛几何算法

地震波传播过程本质上是能量在传播过程中逐步损耗直至殆尽的过程,而在实际应用中,常在无能量损耗假设下,用弹性波动方程或标量波动方程描述它.在哈密顿(Hamilton)体系表述下,地震波传播过程即为一个无限维的哈密顿系统随时间的演化过程.若不计能量损耗,波场演化过程实质上为一个单参数连续的辛变换,因而对应的数值算法应为辛几何算法.本文首先从地震波标量方程出发,给出哈密顿体系下地震波传播的表述,即任意两个时刻的波场是通过辛变换联系起来的.随后,把波场在时间和相空间离散化后,给出了用于波场计算的一些辛格式,如显式辛格式、隐式辛格式和蛙跳辛格式.并进一步讨论了有限差分格式和辛格式的异同.然后,应用显式辛格式和同阶的有限差分方法给出了同一理论速度模型下的波场和Marmousi速度模型下的单炮记录.数值结果表明,辛算法是一类可行的波场模拟的数值算法.在时间步长较小时,有限差分方法是辛算法的一个很好近似.文中的理论和方法,为地震波传播理论及实际应用研究提供了新的途径.     

Continuation of normal doubly symmetric orbits in conservative reversible systems

In this paper we introduce the concept of a quasi-submersive mapping between two finite-dimensional spaces, we obtain the main properties of such mappings, and we introduce “normality conditions” under which a particular class of so-called “constrained mappings” are quasi-submersive at their zeros. Our main application is concerned with the continuation properties of normal doubly symmetric orbits in time-reversible systems with one or more first integrals. As examples we study the continuation of the figure-eight and the supereight choreographies in the N-body problem.     

Hamiltonian dynamics of a rigid body in a central gravitational field

This paper concerns the dynamics of a rigid body of finite extent moving under the influence of a central gravitational field. A principal motivation behind this paper is to reveal the hamiltonian structure of the n-body problem for masses of finite extent and to understand the approximation inherent to modeling the system as the motion of point masses. To this end, explicit account is taken of effects arising because of the finite extent of the moving body. In the spirit of Arnold and Smale, exact models of spin-orbit coupling are formulated, with particular attention given to the underlying Lie group framework. Hamiltonian structures associated with such models are carefully constructed and shown to benon-canonical. Special motions, namely relative equilibria, are investigated in detail and the notion of anon-great circle relative equilibrium is introduced. Non-great circle motions cannot arise in the point mass model. In our analysis, a variational characterization of relative equilibria is found to be very useful. Thereduced hamiltonian formulation introduced in this paper suggests a systematic approach to approximation of the underlying dynamics based on series expansion of the reduced hamiltonian. The latter part of the paper is concerned with rigorous derivations of nonlinear stability results for certain families of relative equilibria. Here Arnold's energy-Casimir method and Lagrange multiplier methods prove useful. This work was supported in part by the AFOSR University Research Initiative Program under grant AFOSR-87-0073, by AFOSR grant 89-0376, and by the National Science Foundation's Engineering Research Centers Program: NSFD CDR 8803012. The work of P.S. Krishnaprasad was also supported by the Army Research Office through the Mathematical Sciences Institute of Cornell University.     

Linearization: Laplace vs. Stiefel

The method for processing perturbed Keplerian systems known today as the linearization was already known in the XVIII^th century; Laplace seems to be the first to have codified it. We reorganize the classical material around the Theorem of the Moving Frame. Concerning Stiefel's own contribution to the question, on the one hand, we abandon the formalism of Matrix Theory to proceed exclusively in the context of quaternion algebra; on the other hand, we explain how, in the hierarchy of hypercomplex systems, both the KS-transformation and the classical projective decomposition emanate by doubling from the Levi-Civita transformation. We propose three ways of stretching out the projective factoring into four-dimensional coordinate transformations, and offer for each of them a canonical extension into the moment space. One of them is due to Ferrándiz; we prove it to be none other than the extension of Burdet's focal transformation by Liouville's technique. In the course of constructing the other two, we examine the complementarity between two classical methods for transforming Hamiltonian systems, on the one hand, Stiefel's method for raising the dimensions of a system by means of weakly canonical extensions, on the other, Liouville's technique of lowering dimensions through a Reduction induced by ignoration of variables.     

Periodic orbits accumulating onto elliptic tori for the (<Emphasis Type="Italic">N</Emphasis> + 1)-body problem

We prove the existence of infinitely many periodic solutions, with larger and larger minimal period, accumulating onto elliptic invariant tori for (an “outer solar-system” model of) the planar (N + 1)-body problem.      

Stability of relative equilibria in arbitrary axisymmetric gravitational and magnetic fields

Relative equilibria occur in a wide variety of physical applications, including celestial mechanics, particle accelerators, plasma physics, and atomic physics. We derive sufficient conditions for Lyapunov stability of circular orbits in arbitrary axisymmetric gravitational (electrostatic) and magnetic fields, including the effects of local mass (charge) and current density. Particularly simple stability conditions are derived for source‐free regions, where the gravitational field is harmonic (∇²U = 0) or the magnetic field irrotational (∇ × B = 0). In either case the resulting stability conditions can be expressed geometrically (coordinate‐free) in terms of dimensionless stability indices. Stability bounds are calculated for several examples, including the problem of two fixed centers, the J₂ planetary model, galactic disks, and a toroidal quadrupole magnetic field. This revised version was published online in July 2006 with corrections to the Cover Date.     

The Lissajous transformation II. Normalization

Normalization of a perturbed elliptic oscillator, when executed in Lissajous variables, amounts to averaging over the elliptic anomaly. The reduced Lissajous variables constitute a system of cylindrical coordinates over the orbital spheres of constant energy, but the pole-like singularities are removed by reverting to the subjacent Hopf coordinates. The two-parameter coupling that is a polynomial of degree four admitting the symmetries of the square is studied in detail. It is shown that the normalized elliptic oscillator in that case behaves everywhere in the parameter plane like a rigid body in free rotation about a fixed point, and that it passes through butterfly bifurcations wherever its phase flow admits non isolated equilibria.