Discrete Lagrangian Dynamics on Matched Pair Lie Groups

Year 2021, Volume: 33 Issue: 2, 250 - 258, 31.03.2021

Oğul Esen , Mahmut Kudeyt , Serkan Sütlü

https://doi.org/10.7240/jeps.784138

Abstract

The Lagrangian formulation of the discrete dynamics on matched pair Lie groups is studied. As a result, the discrete matched Lagrange equations that manage the joint behavior of two discrete systems in mutual interaction are obtained. In particular, the discrete equations on the tangent group of a Lie group are presented. The results are illustrated on a matched pair group constructed on two copies of the Heisenberg group, and thus obtained discrete Lagrange equations are written with matrix notation.

Keywords

Discrete Lagrange equations, matched pair of Lie groups

Project Number

117F426

EŞLENMİŞ LİE GRUPLARI ÜZERİNDEKİ LAGRANGE FARK DENKLEMLERİ

Abstract

Sürekli olmayan dinamiğin Lagrange formülasyonu eşlenmiş Lie gruplar üzerinde çalışılmıştır. Sonuç olarak, karşılıklı etkileşim altındaki kesikli iki sistemin dinamiğini tarif eden eşlenmiş fark denklemleri elde edilmiştir. Özel olarak da, bir Lie grubunun teğet grubu üzerindeki fark denklemleri ifade edilmiştir. Elde edilen sonuçlar, Heisenberg grubunun iki kopyası üzerine bina edilen bir eşlenmiş Lie grubu özelinde çalışılmış, ve elde edilen Lagrange fark denklemleri matris formunda yazılmıştır.

Keywords

(Lagrange) fark denklemleri, eşlenmiş Lie grupları, eşlenmiş fark denklemleri

Supporting Institution

Tübitak

Project Number

117F426

Thanks

Bu çalışma OE ve SS’nin "Lagrange ve Hamilton Sistemlerinin Eşlenmesi (Matched pairs of Lagrangian and Hamiltonian Systems)" adlı ve 117F426 kodlu TÜBİTAK projesi kapsamındadır. Yazarlar TÜBİTAK’a desteği için teşekkürü bir borç bilir.

References

• Referans1 Marrero, J. C., Martín de Diego, D. ve Martínez, E. 2006. Discrete Lagrangian and Hamiltonian mechanics on Lie groupoids. Nonlinearity, 19(6):1313–1348.
• Referans2 Hairer, E., Lubich, C. ve Wanner, G. (2006). Geometric numerical integration, volume 31 of Springer Series in Computational Mathematics. Springer-Verlag, Berlin, second edition. Structure-preserving algorithms for ordinary differential equations.
• Referans3 Lee, T., Leok, M. ve McClamroch, N.H. (2007). Lie group variational integrators for the full body problem. Comput. Methods Appl. Mech. Engrg., 196(29-30):2907–2924.
• Referans4 Marsden, J. E., Pekarsky, S. ve Shkoller, S. (1999). Discrete Euler-Poincaré and Lie-Poisson equations. Nonlinearity, 12(6):1647– 1662.
• Referans5 Lu, J.-H. and Weinstein, A. (1990). Poisson Lie groups, dressing transformations, and Bruhat decompositions. J. Differential Geom., 31(2):501–526.
• Referans6 Majid, S. (1990). Matched pairs of Lie groups associated to solutions of theYang-Baxter equations. Pacific J. Math., 141(2):311–332.
• Referans7 Takeuchi, M. (1981). Matched pairs of groups and bismash products of Hopf algebras. Comm. Algebra, 9(8):841–882.
• Referans8 Esen, O. ve Sütlü, S. (2018). Matched pairs of discrete dynamical systems. arXiv preprint arXiv:1809.00521.
• Referans9 Knapp, A. W. (1988). Lie groups, Lie algebras, and cohomology, volume 34 of Mathematical Notes. Princeton University Press, Princeton, NJ.
• Referans10 Majid, S. (1995). Foundations of quantum group theory. Cambridge University Press, Cambridge.
• Referans11 Weinstein, A. (1996). Lagrangian mechanics and groupoids. In Mechanics day (Waterloo, ON, 1992), volume 7 of Fields Inst. Commun., pages 207–231. Amer. Math. Soc., Providence, RI.
• Referans12 Esen, O. (2017). Dinamik sistemlerin eşlenmesi. Sakarya University Journal of Science, 21(3):469–480.
• Referans13 Şuhubi, E. (2013). Exterior analysis: Using applications of differential forms. Elsevier.
• Referans14 Bobenko, A. I. ve Suris, Y. B. (1999). Discrete Lagrangian reduction, discrete Euler-Poincaré equations, and semidirect products. Lett. Math. Phys., 49(1):79–93.
• Referans15 Marsden, J. E., Pekarsky, S. ve Shkoller, S. (2000). Symmetry reduction of discrete Lagrangian mechanics on Lie groups. J. Geom. Phys., 36(1-2):140–151.
• Referans16 Esen, O. ve Sütlü, S. (2017). Lagrangian dynamics on matched pairs. J. Geom. Phys., 111:142–157.
• Referans17 Hindeleh, F. Y. (2006). Tangent and cotangent bundles, automorphism groups and representations of Lie groups. ProQuest LLC, Ann Arbor, MI.
• Referans18 Kolář, I., Michor, P. W. ve Slovák, J. (1993). Natural operations in differential geometry. Springer-Verlag, Berlin.
• Referans19 Michor, P. W. (2008). Topics in differential geometry, volume 93. American Mathematical Soc.
• Referans20 Vizman, C. (2013). The group structure for jet bundles over Lie groups. Journal of Lie Theory, 23.
• Referans21 Yano, K. ve Ishihara, S. (1973). Tangent and cotangent bundles: differential geometry, volume 16. Dekker.
• Referans22 Hall, B. C., (2003). Lie groups, Lie algebras, and representations. Springer-Verlag, New York.