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Author Yamaleev, Robert M.
Source SpringerLink
Content type Text
Publisher Birkhäuser-Verlag
File Format PDF
Copyright Year ©2003
Language English
Subject Domain (in DDC) Natural sciences & mathematics ♦ Serial publications
Subject Keyword Analysis ♦ Applications of Mathematics ♦ Popular Science in Mathematics/Natural Science/Technology
Abstract The central idea in the present paper is that corresponding to some increment of the particle’s energy there should correspond an extension of the degrees of freedom in the description. We then suggest to extend the formulations of Newtonian and relativistic mechanics. We start from the relativistic Lorentz-force equations, explore an algorithm of extension and use the latter to build an extension of the Newtonian equations of motion. The mapping between momenta of the Extended Newtonian and relativistic mechanics is built on the basis of Vieta’s formulae of a quadratic polynomial. The equations of motion in the external e.m. fields are presented in the basis of the quaternion algebra. Further, the algorithm of extension is used a second time and leads to Doubly Extended Newtonian Mechanics (DENM). On making use of Vieta’s formulae on the cubic polynomial from DENM equations we derive equations of the Extended Relativistic Mechanics (ERM). In the polar representation the dynamic equations of DENM are given by Jacobi equations for elliptic functions, whereas equations of motion of ERM are represented by Weierstrass equations for elliptic functions. The equations of particle motion under e.m. fields are given in the basis of the algebra of quaternions. The algorithm of extension is repeated n-times to obtain Extended Newtonian mechanics of (n+1)-order and the corresponding mapping onto the hyper-relativistic dynamics is constructed. The mapping is given by Vieta’s formulae between roots and coefficients of (n+1)-degree polynomial. The equations of motion in the external e.m. fields are given within the algebra of quaternions.
ISSN 01887009
Age Range 18 to 22 years ♦ above 22 year
Educational Use Research
Education Level UG and PG
Learning Resource Type Article
Publisher Date 2003-01-01
Publisher Place Basel
e-ISSN 16614909
Journal Advances in Applied Clifford Algebras
Volume Number 13
Issue Number 2
Page Count 36
Starting Page 183
Ending Page 218

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Source: SpringerLink