On Resonant Cases of Degeneracy in the Problem of Orbital Stability of Periodic Solutions of a Hamiltonian System with Two Degrees of Freedom

In this paper a method is proposed for constructing a nonlinear canonical change of variables, which makes it possible to avoid the singularity when introducing local coordinates in the neighborhood of periodic orbit of the autonomous Hamiltonian system with two degrees of freedom. As an application of this method, orbital stability of pendulum-like oscillations of a heavy rigid body with a fixed point in the Bobylev-Steklov case is investigated. In particular, it is performed a nonlinear study of the orbital stability for the so-called case of degeneracy, when it is necessary to take into account terms of order six in the Hamiltonian expansion in a neighborhood of the unperturbed periodic orbit.

The present paper deals with equations, which generalize the known Euler-Poisson equations for the motion of a heavy rigid body about a fixed point. These equations arise in dynamics of systems of coupled rigid bodies. In these equations the generalized inertia tensor depends upon components of vertical vector, i.e. it is not constant. Our aim is to analyze Lyapunov stability of stationary solutions and orbital stability of periodic solutions of the equations under study.

In this paper the motion of a rigid body with a fixed point in a uniform gravity field is considered. It is assumed that the main moments of inertia of the body correspond to the case of Goryachev–Chaplygin, i.e., they are in the ratio of 1:1:4. In contrast to the integrable case of Goryachev–Chaplygin, there are no restrictions imposed on the position of the center of mass of the body. The problem of the orbital stability of periodic pendulum motions of a body (oscillations and rotations) is investigated. The equations of perturbed motion were obtained and the problem of orbital stability was reduced to the stability problem of the equilibrium position of a second-order linear system with 2π-periodic coefficients, the right-hand sides of which depend on two parameters. Based on the analysis of the linearized system, it has been established that the rotations are orbitally unstable for all possible values of the parameters. Moreover, a diagram of stability of pendulum oscillations has been constructed, on which the regions of orbital instability (parametric resonance) and regions of orbital stability in the linear approximation are indicated. At small values of oscillations amplitudes, a nonlinear analysis was performed. Based on the analysis of the coefficients of the normalized Hamilton function using theorems of the KAM theory, rigorous conclusions on the orbital stability of pendulum oscillations with small amplitudes were obtained.

A method is presented of constructing a nonlinear canonical change of variables which makes it possible to introduce local coordinates in a neighborhood of periodic motions of an autonomous Hamiltonian system with two degrees of freedom. The problem of the orbital stability of pendulum-like oscillations of a heavy rigid body with a fixed point in the Bobylev – Steklov case is discussed as an application. The nonlinear analysis of orbital stability is carried out including terms through degree six in the expansion of the Hamiltonian function in a neighborhood of the unperturbed periodic motion. This makes it possible to draw rigorous conclusions on orbital stability for the parameter values corresponding to degeneracy of terms of degree four in the normal form of the Hamiltonian function of equations of perturbed motion.

The effect of swimming activity on bone architecture in growing rats

Recently Liu & Chao (1991) (hereafter cited as ‘L&C’) published a research note stating that prior studies about the orientation of the principal axes of the Earth ‘fell short in providing explicit derivation and expressions’ and that discussions on the subject matter were made, ‘usually in passing’. One can only interpret these remarks as inferring that no previous investigator has derived and published values for the angle between the first principal axis of the Earth and a ‘geographical’ (terrestrial would be a more appropriate word) reference frame. This inquiry appears to be the final objective of their analysis from which everything else evolves, as their summary emphatically demonstrates. The basic theory (equations 2a-2e in L&C) to which the authors parenthetically add ‘complete detailed derivation of the above with explicit normalization factors’ while directing the reader to another reference written by one of the co-authors, was fully elaborated in the seminal book by Hotine (1969, p. 161), well before the availability of reliable artificial satellite data. The coverage of the subject in this treatise does not fall under the author’s imprecise criticism of ‘skimpy, if treated at all’. Hotine arrives at very general equations which are presented, coincidentally, in complex form, the notation ultimately favoured by L&C. Later Bursa (1977) formulated, rather exhaustively, the mathematical problem addressed by L&C with complete explicit derivations and final angular expressions to determine the Earth’s principal directions. This was followed by another short and revealing contribution (Bursa 1983a) (hereafter referred to as ‘B83A’) which is crucial in this context. In fact, equation (6) in B83A [and also in Bursa (1983b)l is practically identical to the central equation in L&C (number Sb), except for an elementary switch to complex notation. It is difficult to understand why L&C also fail to mention the most obvious and rigorous alternative for determining this and other related dynamical parameters: diagonalization of the Earth’s 3 x 3 symmetric inertia tensor by using well known eigentheory procedures. At the core of the problem is the straightforward relationship between the inertia tensor of any arbitrary body and the normalized coefficients of degree 2 and order 2 (i.e. C,,, S,,, n=2, m = 0, . . . , n) of its spherical harmonic expansion of the gravitational potential. By diagonalizing the inertia matrix, not only the orientation of the principal axes (normalized eigenvectors) with respect to a coefficient-implied terrestrial coordinate system can be obtained, but also, as a by-product, the three principal moments of inertia

Cross-sectional structural parameters from densitometry

The orbital asymptotic stability of the periodic solutions of autonomous systems with an impulse effect is investigated. An analogue of the Andronov–Witt theorem is proved.

In rigid dynamics theory the gyroscope has a special role, as one of the bodies with the most complex motion, having three degrees of freedom. As consequence of the complexity of the motion is the fact that only for three situations, the equations of motion can be analytically integrated. For all the three situations, it is assumed that the principal moments of inertia (PMI) with respect to the principal axes that are normal to the rotation axes, are equal, condition fulfilled by the shape of body of revolution of the gyroscope. The paper analyses the case when the two principal moments of inertia are unequal. The equations of motion are numerically integrated for different values of the ratio between the two PMI. The effect of the disparity between the two PMI upon the motion of the gyroscope is proved by comparison with the motion of the axisymmetric gyroscope, both obtained by numerical integration.

The principal moments of the Earth's inertia and their differences have been computed and their actual accuracy estimated on the basis of the most recent values of the 2nd degree geopotential parameters (model GEM-T2) and of the parameter H in the precession constant. The contributions due to the zero frequency zonal term in the tidal potential have been determined.

Existence and orbital stability of periodic solutions for small-parameter nonlinear differential systems

The orbital asymptotic stability of the periodic solutions of autonomous systems with an impulse effect is investigated. An analogue of the Andronov–Witt theorem is proved.

Basic concepts for dynamical systems are introduced. The Poincare–Bendixson theorem is proved and used to study the existence and orbital stability of periodic solutions for planar equations. Invariant manifolds for n-dimensional nonlinear equations are investigated.

Sufficient Conditions for Orbital Stability of Periodic Solutions of Time Delay Systems Containing Hysteresis Nonlinearities

An impulsive system with a linear continuous-time part and a nonlinear discrete-time part is considered. A criterion for exponential orbital stability of its periodic solutions is obtained. The proof is based on linearization by the first approximation of an auxiliary discrete-time system. The formulation of the criterion depends significantly on a number of impulses per period of the solution. The paper provides a mathematical rationale for some results previously examined in mathematical biology by computer simulations.