Split Quaternions and Particles in (2+1)-Space

It is known that quaternions represent rotations in 3D Euclidean and Minkowski spaces. However, product by a quaternion gives rotation in two independent planes at once and to obtain single-plane rotations one has to apply by half-angle quaternions twice from the left and on the right (with its inverse). This 'double cover' property is potential problem in geometrical application of split quaternions, since (2+2)-signature of their norms should not be changed for each product. If split quaternions form proper algebraic structure for microphysics, representation of boosts in (2+1)-space leads to the interpretation of the scalar part of quaternions as wavelength of particles. Invariance of space-time intervals and some quantum behavior, like noncommutativity and fundamental spinor representation, probably also are algebraic properties. In our approach the Dirac equation represents the Cauchy-Riemann analyticity condition and the two fundamental physical parameters (speed of light and Planck's constant) appear from the requirement of positive definiteness of quaternionic norms.Comment: The version published in Eur. Phys. J.

Нетрадиційні кватерніони і пентаніони в задачах інерціальної орієнтації

The article considers the nonclassical quaternions and pentanions of helf-rotations of solidbody and their application in problems of control and orientation of moving objects. In contrast to classicalrationed Hamiltonian quaternions of complete rotations the nonclassical quaternions of helf-rotations maybe null, they have variable rates, depending on the angle of Euler finite rotationРассматрены неклассические кватернионы и пентанионы полувращений твердого тела и их применение в задачахуправления и ориентации движущихся объектов. В отличие от классических нормированных гамильтоновых кватернионов полных вращений неклассические кватернионы полувращений могут быть нулевыми, они имеют переменные нормы, зависящие от угла эйлерова конечного вращенияРозглянуто некласичні кватерніони і пентаніони напівобертання твердого тіла та їх застосування в задачах керування і орієнтації рухомих об’єктів. На відміну від класичних нормованих гамільтонових кватерніонів повних обертань некласичні кватерніони напівобертання можуть бути нульовими, вони мають змінні норми, що залежать від кута ейлерового кінцевого обертанн

The quaternion-based three-dimensional beam theory

This paper presents the equations for the implementation of rotational quaternions in the geometrically exact three-dimensional beam theory. A new finite-element formulation is proposed in which the rotational quaternions are used for parametrization of rotations along the length of the beam. The formulation also satisfies the consistency condition that the equilibrium and the constitutive internal force and moment vectors are equal in its weak form. A strict use of the quaternion algebra in the derivation of governing equations and for the numerical solution is presented. Several numerical examples demonstrate the validity, performance and accuracy of the proposed approach. (C) 2009 Elsevier B.V. All rights reserved

Lagrangian particle paths and ortho-normal quaternion frames

Experimentalists now measure intense rotations of Lagrangian particles in turbulent flows by tracking their trajectories and Lagrangian-average velocity gradients at high Reynolds numbers. This paper formulates the dynamics of an orthonormal frame attached to each Lagrangian fluid particle undergoing three-axis rotations, by using quaternions in combination with Ertel's theorem for frozen-in vorticity. The method is applicable to a wide range of Lagrangian flows including the three-dimensional Euler equations and its variants such as ideal MHD. The applicability of the quaterionic frame description to Lagrangian averaged velocity gradient dynamics is also demonstrated.Comment: 9 pages, one figure, revise

Time-like definition of quaternions in exterior algebra

A formal description of quaternions by means of exterior calculus is provided. Considering a three-dimensional space-time characterized by having three time coordinates, we have been able to consistently recover a suitable formulation of quaternions by means of the properties arising from exterior algebra and calculus. As an application, it is also illustrated how rotations may be written in terms of quaternions according to the exterior-algebraic notation.Comment: 7 page