Related papers: Differential geometry using quaternions
While real Hamiltonian mechanics and Hermitian quantum mechanics can both be cast in the framework of complex canonical equations, their complex generalisations have hitherto been remained tangential. In this paper quaternionic and…
We study differential geometric properties of cuspidal edges with boundary. There are several differential geometric invariants which are related with the behavior of the boundary in addition to usual differential geometric invariants of…
In a recent paper [J.Math.Phys. vol42, 2236-2265 (2001)], we discussed differential operators within a quaternionic formulation of quantum mechanics. In particular, we proposed a practical method to solve quaternionic and complex linear…
The fibre bundles adjoint to generalized almost quaternionic structures are studied. The most important classes of generalized almost quaternionic manifolds are considered.
The Lagrangian formalism is used to derive covariant equations that are suitable for use in continuously distributed matter in curved spacetime. Special attention is given to theoretical representation, in which the Lagrangian and its…
The combination of words ``discrete curvature'' is only an apparent contradiction. In this survey we describe curvature notions associated with polygons, polyhedral surfaces, and with abstract polyhedral manifolds. Several theorems about…
Consider a surface described by a Hamiltonian which depends only on the metric and extrinsic curvature induced on the surface. The metric and the curvature, along with the basis vectors which connect them to the embedding functions defining…
In this article a relation between curvature functionals for surfaces in the Euclidean space and area functionals in relative differential geometry will be given. Relative differential geometry can be described as the geometry of surfaces…
There exists a well established differential topological theory of singularities of ordinary differential equations. It has mainly studied scalar equations of low order. We propose an extension of the key concepts to arbitrary systems of…
Differential geometric formulation of quantum gauge theory of gravity is studied in this paper. The quantum gauge theory of gravity which is proposed in the references hep-th/0109145 and hep-th/0112062 is formulated completely in the…
This text is a support for different courses of the master of Mechanics of the University Paris-Saclay. The content of this text is an introduction, for graduate students, to tensor algebra and analysis. Far from being exhaustive, the text…
The known Complex Step Derivative (CSD) method allows easy and accurate differentiation up to machine precision of real analytic functions by evaluating them a small imaginary step next to the real number line. The current paper proposes…
We explain the use of dual quaternions to represent poses, twists, and wrenches.
We introduce fractional flat space, described by a continuous geometry with constant non-integer Hausdorff and spectral dimensions. This is the analogue of Euclidean space, but with anomalous scaling and diffusion properties. The basic tool…
The development of mathematics has been characterized by the increasing interconnectivity of seemingly separate disciplines. Such interplay has been facilitated by a massive development in formalism; category theory has provided a common…
This paper continues the author's previous work on a limit-free algebraic-geometric construction of the derivative in the class of polynomial functions and extends the proposed framework to elementary functions. Derivatives of rational…
Hopf algebra structures on the extended q-superplane and its differential algebra are defined. An algebra of forms which are obtained from the generators of the extended q-superplane is introduced and its Hopf algebra structure is given
We give a simple and self contained introduction to quaternions and their practical usage in dynamics. The rigid body dynamics are presented in full details. In the appendix, some more exotic relations are given that allow to write more…
A systematic theory is introduced for calculating the derivatives of quaternion matrix function with respect to quaternion matrix variables. The proposed methodology is equipped with the matrix product rule and chain rule and it is able to…
We study geometric structures arising from Hermitian forms on linear spaces over real algebras beyond the division ones. Our focus is on the dual numbers, the split-complex numbers, and the split-quaternions. The corresponding geometric…