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Sönümlemeli Sistemlerin Eşlenmesi Üzerine

Year 2021, , 273 - 282, 30.04.2021
https://doi.org/10.35414/akufemubid.803443

Abstract

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.

Supporting Institution

tübitak

Project Number

117F426

Thanks

Bu çalışma TÜBİTAK’ın (Türkiye Bilimsel ve Teknolojik Araştırma Kurumu) 117F426 proje numarasıyla “Eşlenmiş Lagrange Ve Hamilton Sistemleri” başlıklı proje kapsamında yapılmıştır. Tüm yazarlar desteği için TÜBİTAK’a müteşekkirdir.

References

  • 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.
  • 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.

On Matched Pair of Dissipation Systems

Year 2021, , 273 - 282, 30.04.2021
https://doi.org/10.35414/akufemubid.803443

Abstract

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

Project Number

117F426

References

  • 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.
  • 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.
There are 21 citations in total.

Details

Primary Language Turkish
Subjects Engineering
Journal Section Articles
Authors

Oğul Esen 0000-0002-6766-0287

Gökhan Özcan 0000-0001-6260-4641

Serkan Sütlü 0000-0003-0925-8668

Project Number 117F426
Publication Date April 30, 2021
Submission Date October 1, 2020
Published in Issue Year 2021

Cite

APA Esen, O., Özcan, G., & Sütlü, S. (2021). Sönümlemeli Sistemlerin Eşlenmesi Üzerine. Afyon Kocatepe Üniversitesi Fen Ve Mühendislik Bilimleri Dergisi, 21(2), 273-282. https://doi.org/10.35414/akufemubid.803443
AMA Esen O, Özcan G, Sütlü S. Sönümlemeli Sistemlerin Eşlenmesi Üzerine. Afyon Kocatepe Üniversitesi Fen Ve Mühendislik Bilimleri Dergisi. April 2021;21(2):273-282. doi:10.35414/akufemubid.803443
Chicago Esen, Oğul, Gökhan Özcan, and Serkan Sütlü. “Sönümlemeli Sistemlerin Eşlenmesi Üzerine”. Afyon Kocatepe Üniversitesi Fen Ve Mühendislik Bilimleri Dergisi 21, no. 2 (April 2021): 273-82. https://doi.org/10.35414/akufemubid.803443.
EndNote Esen O, Özcan G, Sütlü S (April 1, 2021) Sönümlemeli Sistemlerin Eşlenmesi Üzerine. Afyon Kocatepe Üniversitesi Fen Ve Mühendislik Bilimleri Dergisi 21 2 273–282.
IEEE O. Esen, G. Özcan, and S. Sütlü, “Sönümlemeli Sistemlerin Eşlenmesi Üzerine”, Afyon Kocatepe Üniversitesi Fen Ve Mühendislik Bilimleri Dergisi, vol. 21, no. 2, pp. 273–282, 2021, doi: 10.35414/akufemubid.803443.
ISNAD Esen, Oğul et al. “Sönümlemeli Sistemlerin Eşlenmesi Üzerine”. Afyon Kocatepe Üniversitesi Fen Ve Mühendislik Bilimleri Dergisi 21/2 (April 2021), 273-282. https://doi.org/10.35414/akufemubid.803443.
JAMA Esen O, Özcan G, Sütlü S. Sönümlemeli Sistemlerin Eşlenmesi Üzerine. Afyon Kocatepe Üniversitesi Fen Ve Mühendislik Bilimleri Dergisi. 2021;21:273–282.
MLA Esen, Oğul et al. “Sönümlemeli Sistemlerin Eşlenmesi Üzerine”. Afyon Kocatepe Üniversitesi Fen Ve Mühendislik Bilimleri Dergisi, vol. 21, no. 2, 2021, pp. 273-82, doi:10.35414/akufemubid.803443.
Vancouver Esen O, Özcan G, Sütlü S. Sönümlemeli Sistemlerin Eşlenmesi Üzerine. Afyon Kocatepe Üniversitesi Fen Ve Mühendislik Bilimleri Dergisi. 2021;21(2):273-82.


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