Abstract
We consider a commutative part of the Generalized Clifford Algebras, denominated asalgebra of multicomplex numbers. By using the multicomplex algebra as underlying algebraic structure we construct oscillator model for the Nambu’s formulation of dynamics. We propose a new dynamicals principle which gives rise to two kinds of Hamilton-Nambu equations inD≥2-dimensional phase space. The first one is formulated with (D−1)-evolution parameter and a single Hamiltonian. The Haniltonian of the oscillator model in such dynamics is given byD-degree homogeneous form. In the second formulation, vice versa, the evolution of the system along a single evolution parameter is generated by (D−1) Hamiltonian.
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References
Keller Jaime, Dirac form of Maxell equations,Z n-graded algebras, in: “Spinors, Twistors, Clifford Algebras and Quantum Deformations”, 189–196, eds. Z. Oziewicz, B. Jancewicz, A. Borowiec, Kluwer Academic Publishers (1993).
Fleury N., M. Rausch de Traubenberg and R M. Yamaleev, Generalized Clifford Algebra and Hiperspin Manifold.Int. J. Theor. Phys.,32 (4), (1993); Preprint CRN/PHTH 92-12, Strasbourg, (1992).
Yamaleev R. M., Elements of Cubic Quantum Mechanics,JINR Comm., P2-88-147, Dubna, (1988); Model of Polylinear Oscillator in the Space of Fractional Quantum Numbers,JINR Comm., P2-88-871, Dubna, (1988); Cubic Forms and Quantum Mechanics,JINR Rapid Comm., (1) 34–89, Dubna, (1989); On Geometrical Form in Three-Dimensional Space with Cubic Metric,JINR Comm., P5-89-269, Dubna, (1989); On Construction of Quantum Mechanics on Cubic Forms,JINR Comm., E2-89-326, Dubna, (1989); Equation of Motion of Four Degree for Tetranions,JINR Comm., P2-91-460, Dubna, (1991); New Dynamical Equations for Many Particle System on the Basis of Multicomplex Algebra, in: “Clifford Algebras and Their Application in Math. Phys.”, eds. V. Dietrich et al., 433–441, (1998).
Kwasniewski A. K., On the Onsager Problem for Potts Models,J. Phys. A.: Math. Gen., 19 1460–1476, (1986).
Nambu Y., Generalized Hamiltonian Dynamics,Phys. Rev.,D7 (8), 2405 (1973).
Estabrook F. B.,Phys. Rev.,D8 2740, (1973); Bayen F. and M. Flato,Phys. Rev.,D11 3049, (1975); Mukunda N. and E. Sudarshan,Phys. Rev.,D13, 530–532 (1976); Hirayama M.,Phys. Rev.,D16 530–532, (1977); Flato M., A. Lichnerowicz and D. Sternheimer,J. Math. Phys.,17 1754, (1976); Takhtajan L.,Commun. Math. Phys.,160 295–315, (1994); Chatterjee R.,Lett Math. Phys.,36 117–126, (1996).
Yamaleev R. M., Generalized Newtonian Equations of Motion,Annals of Physics,277 1–18, (1999).
Yamaleev R. M., Introduction into Theory of N-unitary Group,JINR Comm., P2-90-129, Dubna (1990).
Fleury N., M. Rausch de Traubenberg and R. M. Yamaleev, Commutative Extended Complex Numbers and Connected Trigonometry,J. Math. Analysis and Appl.,180 (2), (1993); Preprint CRN/PHTH 91-07, Strasbourg (1991); Fleury N., M. Rausch de Traubenberg and R. M. Yamaleev, Extended Complex Number Analysis and Conformal-like Transformations.J. Math. Analysis and Appl.,191 118–136, (1995).
Bakai A. S., Yu. P. Stepanovski, Adiabatic Invariants. Kiev, Naukova Dumka (1981).
Landau L. D., E. M. Lifshitz, “Mechanics, 3-d”, Ed. Pergamon Press, Oxford, London, New York, (1976).
Strocchi F.,Rev. Mod. Phys. 38 (1), 36 (1966).
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Yamaleev, R.M. From Generalized Clifford Algebras to nambu’s formulation of dynamics. AACA 10, 15–38 (2000). https://doi.org/10.1007/BF03042006
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DOI: https://doi.org/10.1007/BF03042006