Chapter 4. A Damped Oscillator Model of Relativistic Dynamics in Rapidity Space

$39.50

R.M. Yamaleev
Joint Institute for Nuclear Research, LIT, Dubna, Russia

Part of the book: Future Relativity, Gravitation, Cosmology

Abstract

The second order polynomial relationship between energy, mass and momentum, the mass-shell equation, is considered as the characteristic polynomial of the second order ordinary differential equation describing the damped oscillator model. It is shown, the quotient of two fundamental solutions of the differential equation describes an evolution of the energy and momentum of the relativistic particle with respect to hyperbolic and periodic angles. A geometrical interpretation is done within the framework of hyperbolic and elliptic geometries. Transitions between different kinds of the evolution parameters are interpreted as passes from one system of references onto the other. Evolution equations covariant with respect to these transitions are formulated within the framework of Jacobi elliptic functions.


References


[1] Yamaleev, R. M. Complex algebras on N-order polynomials and generalizations of
trigonometry, oscillator model and Hamilton dynamics. Advances in Applied Clifford
algebras, 15(1), (2005), 123.
[2] Dattoli, G., Di Palma, E., Nguen, F., Sabia, E. Generalized trigonometric functions
and elementary applications. Int. J. Appl. Comp. Math. (2017) 3(2) 445-458.
[3] Yamaleev, R. M. Multicomplex algebras on polynomials and generalized Hamilton
dynamics. J. Math. Anal. Appl. 322 (2006) 815-824.
[4] Yamaleev, R. M. Geometrical and physical interpretation of evolution governed by
general complex algebra. J. Math. Anal. Appl. 340 (2008) 1046-1057.
[5] Yamaleev, R. M. Extended relativistic mechanics of charged particle. Nova Science
Publishers, Inc. ISBN 1-59033-981-9. In: Relativity, Gravitation, Cosmology. Eds. V.
Dvoeglazov, A. Espinoza Garrido. 2004, pp.1-17.
[6] Yamaleev, R. M. Representation of solution of n-order Riccati equation via generalized trigonometric functions.
J. Math. Anal. Appl. (2014) 420(1) 334-347.
[7] Klein, F. On the geometric foundations of the Lorentz group. In: New Ideas in Math ematics [in Russian], No. 5, St. Peterburg (1914), pp. 144174. [Supplied by translator: This is the Russian translation of: F. Klein, ber die geometrischen Grundlagen
der Lorentzgruppe, Jahresber. Deutsch. Math.-Verein., Bd. 19 (1910). It can be found
also in: R. Fricke and A. Ostrowski (eds.), Felix Klein: Gesammelte Mathematische
Abhandlungen, Erster Band, Verlag von Julius Springer, Berlin (1921), pp. 533552;
reprinted, Springer-Verlag, Berlin-Heidelberg-New York (1973).]
[8] Chernikov, N. A. Introduction of Lobachevskii geometry into the theory of gravitation.
Sov. J. Pert. Nucl. 23 (5),pp. 507-520. 1992.
[9] Yamaleev, R. M., Rodriguez-Dominguez, A. R. As regard the speed in medium of the
electromagnetic radiatio field. J. Mod. Phys. 7 (2016) 1320-1330.
http://dx.doi.org/10.4236/jmp.2016.711118
[10] Kagan, V. F. Historical Foundations of Geometry. (in Russian) Moscow, 1949.
[11] Akhiezer, N. I. Elements of the Theory of Elliptic Functions, M. Nauka, 1970

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