arXiv subjects
Publications and source records attributed to Robert M. Yamaleev.
The semi-velocity is defined as an exponential function of the rapidity. Physically the semi-velocity is interpreted as the relativistic analogue of the phase velocity, a geometrical interpretation done within the framework of Beltrami-Klein and Poincaré models. A kinematical wave equation related with the Fermat principle and the concept of the semi-velocity is derived.
Two alternative ways of description an evolution constrained by mass-shell equation are given by the hyperbolic and the periodic angles. In the both cases the angles are proportional to the mass. The differential operators with respect to these coordinates are not commute, the commutation relations coincide with commutation relations of the fermi-like oscillator. The derivative of the periodic angle with respect to the hyperbolic angle is equal to the velocity. This relationship prompts us to conclude that the hyperbolic angle $ϕ$ is the time-like parameter, whereas the periodic angle $θ$ is the space-like parameter. The evolution equations admit an extension to the case of three (or more) dimensions. It is proved, the energy and momentum defined in the space of four-rapidity obey Klein-Gordon equations constrained by the classical trajectory of a relativistic particle. It is conjectured, for small values of a proper mass influence of the constraint is weakened and the classical motion gains features of a wave motion.
The generalized Riccati equation defined as an equation between first order derivative and the cubic polynomial is named Riccati-Abel equation. Unlike solutions of ordinary Riccati equation, the solutions of Riccati-Abel equation do not admit an addition formula. In the present paper we explain a nature of this fault and elaborate a method of solution of this problem. We show that the addition formula for Riccati-Abel equation can be established only for pair of solutions. Furthermore, it is shown that analogously with ordinary Riccati equation, the relationships with linear differential equations and the general complex algebra of third order can be established only for the pair of solutions of Riccati-Abel equation.
The notion of difference between two quantities plays a basic role in mathematics, consequently in all branches of human activity where the mathematics is applied. However the long stand question is: what is the difference between three (or more) quantities? The binary operation [a,b]=(a-b) possesses the following principal feature: with respect to the third quantity (c) this operation is decomposed into a sum of the same operations between (a) and (c), and (c)and (b), i.e., [a,b]=[a,c]+[c,b]. Denote by [a,b,c] difference between three quantities (a,b,c). With respect to additional quantity (d) this definition of the difference has to possess with the following property [a,b,c]=[d,b,c]+[a,d,c]+[a,b,d]. We prove that this property of difference between three (or n>2) quantities is satisfied by one of the features of Vandermonde determinant.
The purpose the present paper is to construct the hyperbolic trigonometry on Euclidean plane without refereing to hyperbolic plane. In this paper we show that the concept of hyperbolic angle and its functions forming the hyperbolic trigonometry give arise on Euclidean plane in a natural way. The method is based on a key- formula establishing a relationship between exponential function and the ratio of two segments. This formula opens a straightforward pathway to hyperbolic trigonometry on the Euclidean plane. The hyperbolic law of cosines I and II and the hyperbolic law of sines are derived by using of the key-formula and the methods of Euclidean Geometry, only. It is shown that these laws are consequences of the interrelations between distances and radii of the intersecting semi-circles.
In this paper, we introduce a new concept of rapidity and define formulae for energy and momentum as functions of the new rapidity, we denominate "counter-rapidity". The correspoding velocity is measured from the state of light speed in direction to the rest state. This theory admits correct limit of Dirac equation at the zero-mass point and explain violation of the parity at this point. We show that the velocity of the particle near light-speed is quantized. With the new concept of rapidity are related a new group of transformations and new link between relativistic dynamics and hyperbolic geometry.
In this paper we suggest new classification of polynomials and evolution equations for the roots and the coefficients remaing the polynomials within proper class. In the basis of the developed evolution equations we built new dynamics generalizing the relativistic dynamics.