Quaternionic Equation Research Articles

Quaternion Models and Algorithms of Solving the General Problem of Energetically Optimal Spacecraft Orbit Reorientation

The problem of optimal reorientation of the spacecraft orbit is considered in quaternion formulation. Control (vector of the acceleration of the jet thrust) is limited in magnitude. It is required to determine the optimal orientation of the vector of the acceleration in space to solve the problem. It is necessary to minimize the energy consumption of the process of reorientation of the spacecraft orbit. We used quaternion differential equation of the orientation of the spacecraft orbit to describe the motion of the center of mass of the spacecraft. The problem was solved using the maximum principle of L. S. Pontryagin. We simplified the differential equations of the problem using known partial solution of the equation for the variable conjugated to true anomaly. The problem of optimal reorientation of the spacecraft orbit was reduced to a boundary value problem with a moving right end of the trajectory described by a system of nonlinear differential equations of fifteenth order. For the numerical solution of the obtained boundary value problem the transition to dimensionless variables was carried out. At the same time a characteristic dimensionless parameter of the problem appeared in the phase and conjugate equations. We constructed an original numerical algorithm for finding unknown initial values of conjugate variables. The algorithm is a combination of Runge-Kutta 4th order method and two methods for solving boundary value problems: modified Newton method and gradient descent method. The using of these two methods for solving boundary value problems has improved the accuracy of the solution of the investigated boundary value problem of optimal control. Examples of numerical solution of the problem are given for the cases when the difference (in angular measure) between initial and final orientations of the spacecraft orbit is equals to a few (or tens of) degrees. Graphs of changes component of the quaternion of the spacecraft orbit orientation; variables characterizing the shape and dimensions of the spacecraft orbit; optimal control are plotted. The analysis of the obtained solutions is given. The features and regularities of the process of optimal reorientation of the spacecraft orbit are established. We found that when the difference between initial and final spacecraft orbits is small there is a one point of extremum for the eccentricity of the spacecraft orbit and for modulo of the vector of orbital velocity moment of the spacecraft. And there are a few points of local extremum for these functions when the difference between initial and final spacecraft orbits is large.

Coupling of matter field with the electromagnetic field

A novel mechanism is found in which the matter field is coupled to the electromagnetic ones without invoking the minimal coupling recipe. A quanto-electromagnetic field is derived that can induce charge and current to a quantum particle. The governing equations of the two fields can be extended to uniformly moving charged particle. New conservation equations governing the two interacting fields are derived. When the electromagnetic field interacts with itself, the obtained conservation equations are found to describe the conservation of spin and helicity of the electromagnetic field. The multiplication of the quaternionic Maxwell's equation by a quaternionic matter field yields the quantized Maxwell's equations but with interacting electromagnetic field. These new fields reflect the particle-wave nature exhibited by a quantum particle. Various kinds of currents are obtained reflecting the interaction between the matter and the electromagnetic fields.

Inertial Navigation in Space Using the Regular Quaternion Equations of Astrodynamics

Quaternion equations are proposed for the ideal operation of spatial inertial navigation systems with an azimuthally stabilized platform and a gyrostabilized platform that keeps its orientation invariant in inertial space, quaternion equations for the ideal operation of strapdown inertial navigation systems in the regular four-dimensional Kustaanheimo-Stiefel variables with consideration of zonal, tesseral, and sectorial harmonics of the Earth's gravitational field. The equations are dynamically similar to the regular equations of perturbed spatial two-body problem in the Kustaanheimo-Stiefel variables, which enables to use the results obtained in regular celestial mechanics and astrodynamics theory in inertial astronavigation. The development of operational algorithms for these navigation systems using these equations is considered.

Quaternion Equations of Disturbed Motion of an Artificial Earth Satellite

The quaternion equations of disturbed motion of an artificial Earth satellite (AES) in Earth’s gravitational field (with regard to its zonal, tesseral, and sectorial harmonics) are obtained in four-dimensional Kustaanheimo–Stiefel variables and in modified four-dimensional variables. In the latter variables the equations of satellite motion have a simpler and more symmetric structure, as compared to the equations of motion in the Kustaanheimo–Stiefel variables. The obtained equations are linear for undisturbed Keplerian motion and, in general, have the form of equations of motion of a disturbed four-dimensional oscillator. These equations, in contrast to classical equations, are regular (they do not contain specific points of singularity) for satellite motion in the central gravitational field of Earth under an effect of disturbing forces, the description of which does not contain negative, higher-than-first powers of the satellite distance to Earth’s center. In these equations, the basic variables are Kustaanheimo–Stiefel variables, or modified four-dimensional variables proposed by the author, as well as satellite motion energy and time. The new independent variable is associated with time by the differential relation containing the satellite distance to Earth’s center of mass. The equations are convenient for applying nonlinear mechanics methods and for high-precision numerical calculations. In the case of artificial satellite motion in Earth’s gravitational field, the description of which does not take into account the tesseral and sectorial harmonics, but takes into account its zonal harmonics, the first integrals of the equations of satellite motion are found, and the replacements of variables and transformations of these equations are proposed, which allowed one to get, for studying satellite motion, closed systems of differential equations of lower dimension, in particular, the system of equations of fourth and third orders.

Compactness results for triholomorphic maps

We consider triholomorphic maps from an almost hyper-Hermitian manifold $\mathcal{M}^{4m}$ into a hyperKahler manifold $\mathcal{N}^{4n}$. This means that $u \in W^{1,2}$ satisfies a quaternionic del-bar equation. We work under the assumption that $u$ is locally strongly approximable in $W^{1,2}$ by smooth maps: then such maps are almost stationary harmonic (when $\mathcal{M}$ is hyperKahler as well, then stationary harmonic). We show that in this more general situation the classical $\epsilon$-regularity result still holds. We then address compactness issues for a weakly converging sequence $u_\ell \rightharpoonup u_\infty$ of strongly approximable triholomorphic maps $u_\ell:\mathcal{M} \to \mathcal{N}$ with uniformly bounded Dirichlet energies. The blow up analysis leads, as in the usual stationary setting, to the existence of a rectifiable blow-up set $\Sigma$ of codimension $2$, away from which the sequence converges strongly. The defect measure $\Theta(x) {\mathcal{H}}^{4m-2} \llcorner \Sigma$ encodes the loss of energy in the limit; we prove that for a.e. point on $\Sigma$ the value of $\Theta$ is given by the sum of energies of a (finite) number of smooth non-constant holomorphic bubbles (here the holomorphicity is understood w.r.t. a complex structure on $\mathcal{N}$ that depends on the chosen point on $\Sigma$). In the case that $\mathcal{M}$ is hyperKahler this result was established by C. Y. Wang (2003) with a different proof; we rely on Lorentz space estimates. By means of a calibration and a homological argument we further prove that for each portion of $\Sigma \setminus \text{Sing}_{u_\infty}$ contained in a Lipschitz graph we find a unique alm. compl. st. on $\mathcal{M}$ that makes the portion pseudoholomorphic and smooth, with $\Theta$ constant; moreover the bubbles originating at points of such a smooth piece are holomorphic for a common complex structure.

Finite-time Attitude Tracking Control Method of Microsatellite Based on Adaptive Robustness and Neural Network Approximation

Abstract Aiming at the attitude tracking control of microsatellites, a finite-time attitude tracking control method based on adaptive robustness and neural network approximation is proposed. Specifically, based on quaternion and Newton-Euler equation, the model of microsatellite attitude tracking control is established. Then, a terminal sliding surface is proposed. All terms except control torque are regarded as total disturbance, which is estimated by three groups of adaptive variables and approximated by Chebyshev neural network. Adaptive robustness and neural network approximation are combined by a switching function and the controller is designed based on it. After that, the finite-time stability of the controller is proved by finite-time Lyapunov stability theorem. Finally, the numerical simulations are carried out and have verified the correctness and effectiveness of the method, which indicates that the controller has quick convergence speed and excellent control precision.

Algebraic Techniques for Linear Equations over Quaternions and Split Quaternions: A Unified Approach in Quaternionic and Split Quaternionic Mechanics

This paper aims to present, in a unified manner, algebraic techniques for linear equations which are valid on both the algebras of quaternions and split quaternions. This paper, introduces a concept of v-quaternion, studies the problem of v-quaternionic linear equations by means of a complex representation and a real representation of v-quaternion matrices, and gives two algebraic methods for solving v-quaternionic linear equations. This paper also gives a unification of algebraic techniques for quaternionic and split quaternionic linear equations in quaternionic and split quaternionic mechanics.

Pulsed Optimal Spacecraft Orbit Reorientation by Means of Reactive Thrust Orthogonal to the Osculating Orbit. II

The first part of the article provides an overview of the work on the differential equations of the spacecraft (SC) orbit orientation and the problem of optimal reorientation of a spacecraft orbit in an inertial coordinate system by means of reactive acceleration orthogonal to the osculating plane of the spacecraft. The theory of solving the problem of the optimal reorientation of the orbit of the spacecraft using the quaternionic differential equation for the orientation of the orbital coordinate system in a non-linear continuous formulation (using limited (small) thrust) is presented. As a minimized quality functional, a combined functional is used equal to the weighted sum of the reorientation time and thrust impulse (characteristic speed) during the reorientation of the orbit of the spacecraft (special cases of this functional are the speed response case and the characteristic speed minimization separately).

The perturbed three-dimensional two body problem: regular quaternion equations of relative motion

The perturbed three-dimensional two body problem: regular quaternion equations of relative motion

Inertial navigation in space using regular quaternion equation of astrodynamics

Inertial navigation in space using regular quaternion equation of astrodynamics

A new development in quantum field equations of dyons

In this study, we describe a novel approach to quantum phenomena of the generalized electromagnetic fields of dyons with quaternionic analysis. Starting with quaternionic quantum wave equations, we have established a quantized condition for time coordinate that transforms microscopic to macroscopic fields. In view of the classical electromagnetic field equations, we propose a new set of quantized Proca–Maxwell’s equations for dyons. Furthermore, a quantized form of four-current densities and the quantized Lorentz gauge conditions for electric and magnetic potentials, respectively, of dyons are obtained. We have established the new quantized continuity equations for electric and magnetic densities of dyons, which are associated with a torque density result from the two spin states. The quantized Klein–Gordon-like field equations and the unified quaternionic electromagnetic potential wave equations for massive dyons are demonstrated. Moreover, we investigate the quaternionic quantized relativistic Dirac field equations for massive dyons, which indicate that the antiparticle of dyons will exist, called antidyons.

Quaternionic Approach to Dual Magnetohydrodynamics of Dyonic Cold Plasma

The dual magnetohydrodynamics of dyonic plasma describes the study of electrodynamics equations along with the transport equations in the presence of electrons and magnetic monopoles. In this paper, we formulate the quaternionic dual fields equations, namely, the hydroelectric and hydromagnetic fields equations which are an analogous to the generalized Lamb vector field and vorticity field equations of dyonic cold plasma fluid. Further, we derive the quaternionic Dirac-Maxwell equations for dual magnetohydrodynamics of dyonic cold plasma. We also obtain the quaternionic dual continuity equations that describe the transport of dyonic fluid. Finally, we establish an analogy of Alfven wave equation which may generate from the flow of magnetic monopoles in the dyonic field of cold plasma. The present quaternionic formulation for dyonic cold plasma is well invariant under the duality, Lorentz, and CPT transformations.

Basis-free solution to general linear quaternionic equation

ABSTRACTA linear quaternionic equation in one quaternionic variable is of the form , where the are given quaternionic coefficients. If introducing basis elements of pure quaternions, then the quaternionic equation becomes four linear equations in four unknowns over the reals, and solving such equations is trivial. On the other hand, finding a quaternionic rational function expression of the solution that involves only the input quaternionic coefficients and their conjugates, called a basis-free solution, is non-trivial.In 1884, Sylvester initiated the study of basis-free solution to linear quaternionic equation. He considered the three-termed equation , and found its solution by successive left and right multiplications. In 2013, Schwartz extended the technique to the four-termed equation, and obtained the basis-free solution in explicit form.This paper solves the general problem for arbitrary number of terms in the non-degenerate case.

Analytic Controlling Reorientation of a Spacecraft Using a Combined Criterion of Optimality

We consider the problem of optimally controlling the reorientation of a spacecraft (SC) from an arbitrary initial angular state into a given final angular position. We study the case when the minimized functional joins, in the given proportion, the time spent and the integral of the squared modulus of the angular momentum on the reorientation of a SC. The problem is solved in a kinematic setting. We consider two versions of the problem of the optimal rotation of a SC, with bounded and unbounded control. Using the necessary optimality conditions in the form of the Pontryagin maximum principle and the quaternion method for solving control problems on the motion of spacecrafts, we obtain an analytical solution of the posed problem. The solution of the problem is based on the quaternionic differential equation relating the angular momentum vector of a SC with the orientation quaternion of the related coordinate system. We present formalized equations and give computational expressions for constructing the optimal control program. We state the control law as an explicit dependence of the control variables on the phase coordinates. Using the transversality condition as a necessary optimality condition, we determine the maximal value of the modulus of the angular momentum for the optimal motion. For a dynamically symmetric SC, the problem of reorientation in space is solved completely: we obtain the dependences for the optimal law of the change of the angular momentum vector as explicit time functions. We give the results of the mathematical modeling of the motion for optimal control which demonstrate the practical realizability of designed algorithm for controlling the spatial orientation of a SC.

Existence and stability results for quaternion fuzzy fractional differential equations in the sense of Hilfer

Existence and stability results for quaternion fuzzy fractional differential equations in the sense of Hilfer

A relativistic quantum theory of dyons wave propagation

Beginning with the quaternionic generalization of the quantum wave equation, we construct a simple model of relativistic quantum electrodynamics for massive dyons. A new quaternionic form of unified relativistic wave equation consisting of vector and scalar functions is obtained, and also satisfy the quaternionic momentum eigenvalue equation. Keeping in mind the importance of quantum field theory, we investigate the relativistic quantum structure of electromagnetic wave propagation of dyons. The present quantum theory of electromagnetism leads to generalized Lorentz gauge conditions for the electric and magnetic charge of dyons. We also demonstrate the universal quantum wave equations for two four-potentials as well as two four-currents of dyons. The generalized continuity equations for massive dyons in case of quantum fields are expressed. Furthermore, we concluded that the quantum generalization of electromagnetic field equations of dyons can be related to analogous London field equations (i.e., current to electromagnetic fields in and around a superconductor).

Explicit determinantal representation formulas for the solution of the two-sided restricted quaternionic matrix equation

Weighted singular value decomposition of a quaternion matrix and with its help determinantal representations of the quaternion weighted Moore–Penrose inverse have been derived recently by the author. In this paper, using these determinantal representations, explicit determinantal representation formulas for the solution of the restricted quaternionic matrix equations, $$\mathbf{A}{} \mathbf{X}{} \mathbf{B}=\mathbf{D}$$ , and consequently, $$\mathbf{A}{} \mathbf{X}=\mathbf{D}$$ and $$\mathbf{X}{} \mathbf{B}=\mathbf{D}$$ are obtained within the framework of the theory of column–row determinants. We consider all possible cases depending on weighted matrices.

On a three-dimensional Riccati differential equation and its symmetries

A three-dimensional Riccati differential equation of complex quaternion-valued functions is studied. Many properties similar to those of the ordinary differential Riccati equation such that linearization and Picard theorem are obtained. Lie point symmetries of the quaternionic Riccati equation are calculated as well as the form of the associated three-dimensional potential of the Schrödinger equation. Using symmetry reductions and relations between the three-dimensional Riccati and the Schrödinger equation, examples are given to obtain solutions of both equations.

General Solution of the Inhomogeneous Div-Curl System and Consequences

We consider the inhomogeneous div-curl system (i.e. to find a vector field with prescribed div and curl) in a bounded star-shaped domain in $$\mathbb {R}^3$$ . An explicit general solution is given in terms of classical integral operators, completing previously known results obtained under restrictive conditions. This solution allows us to solve questions related to the quaternionic main Vekua equation $$DW=(Df/f)\overline{W}$$ in $$\mathbb {R}^3$$ , such as finding the vector part when the scalar part is known. In addition, using the general solution to the div-curl system and the known existence of the solution of the inhomogeneous conductivity equation, we prove the existence of solutions of the inhomogeneous double curl equation, and give an explicit solution for the case of static Maxwell’s equations with only variable permeability.

Quaternionic formulation of the two-component Kohn-Sham equations and efficient exploitation of point group symmetry.

The quaternionic formulation of the time-reversal invariant quasirelativistic Kohn-Sham equations with exact Hartree-Fock exchange leads to hypercomplex one-component equations with half of the dimension compared to the original two-component problem. The combination of the quaternionic equations with point group symmetry exploitation for D2h and its subgroups by construction of corepresentations leads to quaternionic, complex, or real algorithms depending on the structure of the point group. In this work, the quaternionic approach with point group symmetry exploitation of the relativistic four-component Dirac-Hartree-Fock theory by Saue and Jensen [J. Chem. Phys. 111, 6211 (1999)] will be adopted to the quasirelativistic two-component Kohn-Sham scheme for closed-shell systems. The implementation in the program system TURBOMOLE is applied to the large lead cluster Pb56 as a test case.