Oğul ESEN, Serkan SÜTLÜ, Gökhan ÖZCAN

Sönümlemeli Sistemlerin Eşlenmesi Üzerine

Bu makalede karşılıklı etki tepki içindeki iki sönümlemeli sistemin beraber (kollektif- eşlenmiş) hareketinin analizi üzerine bir yöntem öneriyoruz. Aşikardır ki; eşlenmiş hareketi kontrol eden diferansiyel denklemler iki sistemin denklemlerini bir arada yazmak dışında karşılıklı etki tepkinin doğurduğu fazladan terimler içerecektir. Karşılıklı etkiyi belirleyen ilave terimler, Lie cebirlerinin karşılıklı etkisi ile üretilecektir ve bu şekilde pür geometrik/cebirsel bir inşa gerçekleştirilecektir. Sonrasında elde ettiğimiz sonuçları 3 ve 4 boyutlu örneklerde göstereceğiz.

On Matched Pair of Dissipation Systems

In this work we propose a method to analyse the collective motion of two mutually interactingdissipative systems. It is obvious that; the differential equations controlling the matched motion willinclude extra terms generated by the mutual interactions, apart from writing the equations of the twosystems together. These additional terms will be produced in this work by the mutual actions of Liealgebras, and in this way a pure geometric / algebraic construction will be realized. We shall thenillustrate our results on 3 and 4 dimensional examples.

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Bloch, A., Krishnaprasad, P. S., Marsden, J. E. and Ratiu, T. S., 1996. The Euler-Poincaré equations and double bracket dissipation. Communications in Mathematical Physics., 175, 1, 1-42.
Esen, O. and Sütlü, S. 2016. Hamiltonian dynamics on matched pairs. International Journal of Geometric Methods in Modern Physics, 13, 10, 1650128.
Esen, O. and Sütlü, S. 2017. Lagrangian dynamics on matched pairs. Journal of Geometry and Physics, 111, 142-157.
Esen, O., Kudeyt, M. and Sütlü, S. 2021. Second order Lagrangian dynamics on double cross product groups. Journal of Geometry and Physics, 159, 103934.
Esen, O. and Sütlü, S. 2021. Discrete dynamical systems over double cross-product Lie groupoids. International Journal of Geometric Methods in Modern Physics. https://doi.org/10.1142/S0219887821500572
Grmela, M., 1984a. Bracket formulation of dissipative fluid mechanics equations. Physics Letters A, 102, 8, 355–358.
Grmela, M., 1984b. Particle and bracket formulations of kinetic equations. Contemp. Math, 28, 125-132.
Grmela, M. and Öttinger, H. C., 1997. Dynamics and thermodynamics of complex fluids. I. Development of a general formalism. Physical Review E, 56, 6, 6620.
Holm, D. D. 2011. Geometric Mechanics Part I: Dynamics and Symmetry. World Scientific Publishing Company.
Kaufman, A. N. 1984. Dissipative Hamiltonian systems: a unifying principle. Physics Letters A, 100, 8, 419-422.
de León, M. and Rodrigues, P. R., 2011. Methods of differential geometry in analytical mechanics. Elsevier.
Majid, S., 1990. Matched pairs of Lie groups associated to solutions of the Yang-Baxter equations. Pacific Journal of Mathematics, 141, 2, 311-332.
Majid, S., 2000. Foundations of Quantum Group Theory. Cambridge University Press.
Marsden, J. E. and Ratiu, T. S., 2013. Introduction to mechanics and symmetry: a basic exposition of classical mechanical systems. Springer Science & Business Media, 17.
Mielke, A., 2011. Formulation of thermoelastic dissipative material behavior using GENERIC. Continuum Mechanics and Thermodynamics, 23, 3, 233-256.
Morrison, P. J., 1984. Bracket formulation for irreversible classical fields. Physics Letters A, 100, 8, 423-427.
Morrison, P. J., 1986. A paradigm for joined Hamiltonian and dissipative systems. Physica D: Nonlinear Phenomena, 18, 1-3, 410-419.
Morrison, P. J., 2009. Thoughts on brackets and dissipation: old and new. Journal of Physics: Conference Series IOP Publishing, 169, 1.
Pavelka, M., Klika, V. and Grmela, M., 2018. Multiscale thermo-dynamics: introduction to GENERIC. Walter de Gruyter.
Shannon, C. E., 1948. A mathematical theory of communication. 27, J. L. Doob, The Bell System Technical Journal , 379-423.
Şuhubi, E. S. 2008. Dış Form Analizi. Türkiye Bilimler Akademisi.