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In studies of the elliptic restricted three-body problem, the true anomaly of the motion of the primaries is often used as the independent variable. The equations of motion then show invariancy in form from the circular case. It is of interest whether other independent variables exist, such that the invariant form of the equations is maintained. It is found that true anomaly is the only such variable.
Attempt to develop a gauge-invariant theory of the motion of singularities in an n-dimensional Riemannian space. A gauge-invariant theory is developed for a nongeodetic particle represented by a time-like world line in the linearized field theory. The solution of the basic algebraic and differential equations is to be constructed from the source line, the future null cones emanating from it, a scale factor, and nothing else, and the leading term of the solution is the Schwarzschild field. A class of possible solutions is examined, and it is shown that this class contains just one acceptable member - namely, a particle in constant acceleration.
Two procedures are presented for the tracking problem whose desired trajectory is given as an output of a known dynamical system. Time-invariant, single-input, single-output, controllable and observable systems are considered. It is shown that the time-invariant overall structure with zero steady-state error can be obtained by choosing an appropriate dynamic controller and an appropriate cost functional.
The curvilinear invariant quaternion formalism is examined for curved space time. Einstein's gravitation equation is shown to have a simple and natural form in this notation. The hypermass generalization of particle mass, which was generated in our studies of the Dirac equation, is incorporated in gravitation by generalizing Einstein's equation. Covariance requires that the gravitational constant be generalized to an invariant quaternion when the mass is. The modification appears minor and of no importance cosmologically, unless one begins considering time and mass dependence of G.
A general approach to the analysis and design of closed-loop control for slowly time-varying linear systems is presented. The approach uses the method of generalized multiple scales in which slow and fast dynamics are systematically separated by employing different 'clocks' which measure time at varying rates. The clocks, which are necessarily nonlinear functions of time, are chosen in such a manner that the system dynamics are asymptotically invariant with respect to the new time scale. A transfer function relating the output to the input for general linear slowly time-varying systems is developed and represents the actual system under certain conditions. The clock function is shown to satisfy an algebraic characteristic equation and can be determined in terms of the coefficients. The approach can be viewed as an extension of time-invariant linear feedback control theory.
A review of recent data on and conceptions of processes in which high energy particles are transported across magnetic shells in the earth's magnetosphere is presented. The processes discussed include adiabatic processes (with conservation of all three adiabatic invariants of charged particle motion in the magnetic field), diffusion processes (with violation of only the third or third and second invariants), and the so-called fast nonadiabatic processes, which appear to result from the action of strong electric fields.
Data was obtained from instrumentation on Explorers 34 and 41 on cosmic-ray anisotropy and magnetic field vectors during five solar flare events. The analysis was conducted in the energy range from 0.7 to 7.6 MeV, of the late decay phase, to evaluate the dependence of net cosmic-ray anisotropy vector amplitude and direction on the magnetic field azimuth. Results showed that in the late decay phase the direction of the net cosmic-ray anisotropy vector was invariant in relation to the direction of the magnetic field, particle energy, and species. Within the statistical error of the available data the invariant direction was perpendicular to the mean magnetic field direction.
The calculus of differential forms is used to discuss the local integration theory of a general set of autonomous first order ordinary differential equations. Geometrically, such a set is a vector field V in the space of dependent variables. Integration consists of seeking associated geometric structures invariant along V: scalar fields, forms, vectors, and integrals over subspaces. It is shown that to any field V can be associated a Hamiltonian structure of forms if, when dealing with an odd number of dependent variables, an arbitrary equation of constraint is also added. Families of integral invariants are an immediate consequence. Poisson brackets are isomorphic to Lie products of associated CT-generating vector fields. Hamilton's variational principle follows from the fact that the maximal regular integral manifolds of a closed set of forms must include the characteristics of the set.
By using new invariants in the theory of finite elasticity an expression is obtained for the stored energy function of slightly compressible materials in which the effects of the distortional change (change of shape) and of the volume change are clearly separated. The volume-related terms are expressed as a function of the third invariant, the classical compressibility, and an induced anisotropy of the effective compressibility which is due to the large deformations. After evaluating the terms, using data on pressure, volume, uniaxial strain, and fractional volume change vs strain data on natural rubber from the literature, it is shown that the volume change contribution to the total stress observed in a simple tensile experiment can be clearly separated from the distortional contribution, even at finite strains.
Four design problems are considered: two on the optimal minimum order observer-based compensator design and two on the optimal limited state variable feedback controller. The problem of designing an optimal discrete time linear time-invariant observer-based compensator for the regulation of an n dimensional linear discrete time time-invariant plant with m independent outputs is considered. This is a stochastic design problem to the extent that the initial plant state is assumed to be a random vector with known first and second order statistics. The compensator parameters are obtained by minimizing the expectation, with respect to the initial conditions, of the standard cost, quadratic in the state and control vectors with the inclusion of cross terms.
Zeros of the transfer matrix relating the outputs to the disturbances ('disturbance zeros') of a linear time-invariant system are defined. It is shown that these zeros are invariant under output feedback to control inputs but not under state feedback to control inputs, and the effect of state feedback on disturbance zeros is studied. The results are used to develop an algorithm for assigning disturbance zeros and system poles by means of state feedback. As an application of the algorithm, it is shown that the disturbance zeros can be positioned such that the effect of a class of disturbances at the outputs is eliminated in the steady state. An example is given to illustrate the main results of the paper.
A modern control theory design package (ORACLS) for constructing controllers and optimal filters for systems modeled by linear time-invariant differential or difference equations is described. Numerical linear-algebra procedures are used to implement the linear-quadratic-Gaussian (LQG) methodology of modern control theory. Algorithms are included for computing eigensystems of real matrices, the relative stability of a matrix, factored forms for nonnegative definite matrices, the solutions and least squares approximations to the solutions of certain linear matrix algebraic equations, the controllability properties of a linear time-invariant system, and the steady state covariance matrix of an open-loop stable system forced by white noise. Subroutines are provided for solving both the continuous and discrete optimal linear regulator problems with noise free measurements and the sampled-data optimal linear regulator problem. For measurement noise, duality theory and the optimal regulator algorithms are used to solve the continuous and discrete Kalman-Bucy filter problems. Subroutines are also included which give control laws causing the output of a system to track the output of a prescribed model.
AE-C and AE-D observations at low altitudes in the high-latitude region near magnetic local noon have revealed properties of low energy electrons and protons that are similar to those determined from earlier particle and field-aligned current observations. The principal feature determined in the present study is a well-defined boundary (less than a few km or 0.01 deg in invariant latitude) which appears only in the postnoon high-latitude region. Energy spectra of low energy electrons and protons show a distinctive softening at latitudes above this boundary. The region of soft particle precipitation above this boundary (located near 78 deg invariant latitude during moderately disturbed to quiet geomagnetic conditions) is identified as the dayside cusp.
A theory is developed in which a perfectly conducting cylindrical wall is used to describe incompressible MHD turbulence. Dynamic equations of incompressible MHD and its associated fields are expanded in a system of vector eigenfunctions of the curl. The total energy, magnetic helicity, and cross helicity have simple quadratic expressions in the expansion coefficients. The invariants are temporally constant for a truncation at a large, but finite, number of coefficients. The invariants are used to construct canonical distributions. It is noted that for all initial conditions, including quiescent ones, the mean-square velocity fields have finite values. The stability problem is transformed into a search for values of the integral variants which minimize the mean-square velocity fields. This leads to an extremal helicity principle.
The valid and invalid application of the Mikhailov criterion to linear, time-invariant systems with time delays is discussed. The Mikhailov criterion is a graphical procedure which was developed to examine the stability of linear, time-invariant systems with no time delays. Two equivalent formulations of the criterion are discussed. Results indicate that the first formulation remains valid for time-delayed systems of the retared type, with the understanding that the Mikhailov curve need not necessarily always rotate in the counterclockwise direction for a stable system. Erroneous results in the second formulation are formed when there are time delays in the systems.
A method for determining the weak statistical stationarity of a random process is presented. The core of this testing procedure consists of generating an equivalent ensemble which approximates a true ensemble. Formation of an equivalent ensemble is accomplished through segmenting a sufficiently long time history of a random process into equal, finite, and statistically independent sample records. The weak statistical stationarity is ascertained based on the time invariance of the equivalent-ensemble averages. Comparison of these averages with their corresponding time averages over a single sample record leads to a heuristic estimate of the ergodicity of a random process. Specific variance tests are introduced for evaluating the statistical independence of the sample records, the time invariance of the equivalent-ensemble autocorrelations, and the ergodicity. Examination and substantiation of these procedures were conducted utilizing turbulent velocity signals.
The present paper deals with the characteristics of low-energy electrons measured onboard the Atmosphere Explorer C (AE-C) satellite above the south polar cusp during the intense geomagnetic storm of September 21, 1977. The low-energy electron fluxes measured with the Photoelectron Spectrometer experiment indicate that the dayside polar cusp was displaced down to 69-72 deg invariant latitude during the storm. A region of intense fluxes of precipitating electrons was observed in the region near 1700 MLT between 66 and 69 deg invariant latitude, which statistically coincides with that of ascending field-alignment currents in the disturbed afternoon auroral region.
In the present paper, Liouville's theorem is used in combination with approximate, but extremely accurate, expressions which reflect the invariance of the magnetic moment and the longitudinal invariant to determine analytically the evolution of an adiabatically convecting energetic particle distribution. Features of the convecting distribution, reproduced by this model, include positive pitch angle anisotropy, regions in velocity space where the nonmonotonic perpendicular energy distribution is greater than zero, and the energy dependence of the degree of particle injection. The energy dependence of the injection yields upper and lower cutoffs to the distribution within the plasmasphere, and only an upper cutoff outside. This approach is used to study the evolution of ion cyclotron waves in a convecting particle distribution.