Matching of Dynamical Systems

Year 2017, Volume: 21 Issue: 3, 469 - 480, 01.06.2017

Oğul Esen

https://doi.org/10.16984/saufenbilder.298954

Cited By: 1

Abstract

The equations (matched Lie-Poisson and matched Euler-Poincaré) are written for a couple of mutually interacting
physical systems. It is shown that the matched dynamics is a generalization of the well-developed semi-direct
product theory. Two examples are provided. The first one is to write the matched equations for the matched pair of
upper and lower triangular matrix groups whose diagonal entries are 1. The second example is to write the matched
equations for the Lie group obtained by matching a nilpotent group of class two by itself. Two new open problems
are presented. One of these is to write pure geometric relation between the plasma and fluid in the framework of the
matched dynamics. The other is to match two discrete systems under mutual interaction.  

Keywords

Matched pairs, Lie-Poisson equations, Euler-Poincaré equations, nilpotent group of class two, Vlasov equations, Euler equations, discrete systems

Dinamik Sistemlerin Eşlenmesi

Abstract

Karşılıklı etki-tepki içindeki iki fiziksel sistemin ortak hareketlerini veren denklemler (eşlenmiş Lie-Poisson ve
eşlenmiş Euler-Poincaré) elde edilecektir. Eşlenmiş denklemlerin literatürde çokça çalışılmış yarı-direkt çarpım
teorisinin genişlemesi olduğu gösterilecektir. İki örnek verilecektir. İlki, köşegen elemanları 1 olan alt ve üst
üçgensel matris gruplarının oluşturduğu eşlenmiş Lie grubu üzerinde eşlenmiş Lie-Poisson denklemlerinin
yazılmasıdır. İkinci örnek ise ikinci sınıf nilpotent grupların kendiyle eşlenmesi ile elde edilecek Lie grupları
üzerinde eşlenmiş hareket denklemlerinin yazılmasıdır. İki yeni açık problem sunulacaktır. Bunlardan ilki, plazma ve
akışkan arasında pür geometrik yapının eşlenmiş dinamik düzleminde ele alınması, diğeri ise karşılıklı etki-tepki
içindeki iki kesikli sistemin eşlenmesidir.  

Keywords

Eşlenmiş çiftler, Lie-Poisson denklemleri, Euler-Poincaré denklemleri, ikinci sınıf nilpotent grup, Vlasov denklemi, Euler denklemi, kesikli sistemler

References

• [1] R. Abraham ve J. E. Marsden, Foundation of Mechanics, Massachusetts: Benjamin/Cummings Publishing Company, 1978.
• [2] R. Abraham, J. E. Marsden ve T. Ratiu, Manifolds, Tensor Analysis, and Applications, Springer Science Business Media, 1998.
• [3] M. Adams, T. Ratiu ve R. Schmidt, «The Lie group structure of diffeomorphism groups and invertible Fourier integral operators with applications,» içinde Infinite dimensional Groups with Applications, New York, Springer, 1985, pp. 1-69.
• [4] V. I. Arnold, Mathematical Methods of Classical Mechanics, SpringerVerlag, 1978.
• [5] A. Banyaga, The Structure of Classical Diffeomorphism Groups, Dortrecht: Kluwer, 1997.
• [6] A. Bobenko ve Y. B. Suris, «Discrete Lagrangian Redcution, discrete Euler-Poincaré Eqautions, and Semidirect Products,» Letters in Mathematical Physics, cilt 491, pp. 79-93, 1999.
• [7] M. Brouveris, F. Gay-Balmaz, D. D. Holm ve T. S. Ratiu, «The Momentum Map Representation of Images,» Journal of Nonlinear Science, cilt 21, pp. 115-150, 2011.
• [8] L. Colombo ve D. M. d. Diego, «Higher-order variational problems on Lie groups and optimal control applications,» J. Geom. Mech, no. 64, pp. 451-478., 2014.
• [9] L. Colombo ve H. O. Jacobs, «Lagrangian Mechanics on Centered Semi-direct Products,» %1 içinde Geometry, Mechanics, and Dynamics, New York, Springer, 2015, pp. 167-184.
• [10] H. Cendra, D. D. Holm, J. E. Marsden ve T. S. Ratiu, «Lagrangian reduction, the Euler-Poincaré equations, and semidirect products,» Translations of the American Mathematical Society-Series, cilt 2, no. 186, pp. 1-26, 1998.
• [11] D. C. Ellis, F. Gay-Balmaz, D. D. Holm, V. Putkaradze ve T. S. Ratiu, «Symmetry reduced dynamics of charged molecular strands,» Archive for rational mechanics and analysis, cilt 3, no. 197, pp. 811-902, 2010.
• [12] E. I. Khukhro, Nilpotent Groups and Their Automorphisms, Walter de Gruyter, 1993.
• [13] O. Esen, M. Pavelka ve M. Grmela, «Hamiltonian Coupling of Electromagnetic Field and Matter.,» arXiv preprint, p. arXiv:1607.02023, 2016.
• [14] O. Esen ve S. Sütlü, «Hamiltonian dynamics on matched pairs,» International Journal of Geometric Methods in Modern Physics, cilt 13, p. 24 sayfa, 2016.
• [15] O. Esen ve S. Sütlü, «Lagrangian dynamics on matched pairs,» Journal of Geometry and Physics, cilt 111, pp. 142-157, 2017.
• [16] F. Gay-Balmaz ve C. Tronci, «Vlasov moment flows and geodesics on the Jacobi group,» Journal of Mathematical Physics, cilt 53, p. 123502, 2012.
• [17] D. D. Holm ve B. A. Kupershmidt, «Noncanonical Hamiltonian-formulation of ideal magnetohydrodynamics.,» Physica D, cilt 7, p. 330–333, 1983.
• [18] P. J. Morrison ve J. M. Greene, «Noncanonical hamiltonian density formulation of hydrodynamics and ideal magnetohydrodynamics,» Phys. Rev. Lett., cilt 48, p. 569–569, 1982.
• [19] T. Ratiu, «Euler-Poisson equations on Lie algebras and the N-dimensional heavy rigid body,» American journal of mathematics, pp. 409-448, 1982.
• [20] J. E. Marsden, T. Ratiu ve A. Weinstein, «Semidirect products and reduction in mechanics,» Transactions of the American Mathematical Society, cilt 281, no. 1, pp. 147-177, 1984.
• [21] J. E. Marsden, T. Ratiu ve A. Weinstein, «Reduction and Hamiltonian structures on duals of semidirect product Lie algebras,» Cont. Math. AMS, cilt 28, pp. 55-100, 1984.
• [22] D. D. Holm, J. E. Marsden ve T. Ratiu, «The Euler–Poincaré equations and semidirect products with applications to continuum theories,» Advances in Mathematics, cilt 137, no. 1, pp. 1-81, 1998.
• [23] J. Gibbons, D. D. Holm ve C. Tronci, «Geometry of Vlasov kinetic moments: a bosonic Fock space for the symmetric Schouten bracket,» Physics Letters A, cilt 37223, pp. 4184-4196, 2008.
• [24] J. Gibbons, D. D. Holm ve C. Tronci, «Vlasov moments, integrable systems and singular solutions,» Physics Letters A, cilt 3727, pp. 1024-1033, 2008.
• [25] O. Gonzalez, «Time integration and discrete Hamiltonian systems,» Journal of Nonlinear Science, cilt 65, pp. 449-467, 1996.
• [26] V. Guillemin ve S. Sternberg, Symplectic Techniques in Physics, Cambridge University Press, 1990.
• [27] S. Helgason, Differential Geometry, Lie Groups, and Symmetric Spaces, Academic press, 1979.
• [28] D. D. Holm, Geometric Mechanics, Imperial College Press, 2008.
• [29] D. D. Holm ve C. Tronci, «Geodesic Vlasov Equations And Their Integrable Moment Closures,» Journal Of Geometric Mechanics, cilt 1, pp. 181-208, 2009.
• [30] S. M. Jalnapurkar, M. Leok, J. E. Marsden ve M. West, «Discrete Routh reduction,» Journal of Physics A: Mathematical and General, cilt 39, no. 19, p. 5521, 2006.
• [31] A. Kreigl ve P. W. Michor, The Convenient Setting of Global Analysis, American Mathematical Soc, 1997.
• [32] Y. Kosmann-Schwarzbach ve F. Magri, «Poisson-Lie groups and complete integrability. I. Drinfeld bigebras, dual extensions and their canonical representations,» Annales de l'IHP Physique théorique, cilt 49, no. 4, pp. 433-460, 1988.
• [33] B. A. Kupershmidt ve T. Ratiu, «Canonical maps between semidirect products with applications to elasticity and superfluids,» Communications in Mathematical Physics, cilt 902, pp. 235-250, 1983.
• [34] C. Laurent-Gengoux, A. Pichereau ve P. Vanhaecke, Poisson Structures, Springer Science & Business Media, 2012.
• [35] P. Libermann ve C. M. Marle, Symplectic Geometry and Analytical Mechanics, Springer Science & Business Media, 1987.
• [36] J. H. Lu ve A. Weinstein, «Poisson Lie groups, dressing transformations, and Bruhat decompositions,» Journal of Differential Geometry, cilt 312, pp. 501-526, 1990.
• [37] S. Majid, «Physics for algebraists: Non-commutative and non-cocommutative Hopf algebras by a bicrossproduct construction,» Journal of Algebra, cilt 1301, pp. 17-64, 1990.
• [38] S. Majid, «Matched pairs of Lie groups associated to solutions of the Yang-Baxter equations,» Pacific Journal of Mathematics, cilt 1412, pp. 311-332, 1990.
• [39] S. Majid, Foundations of Quantum Group Theory, Cambridge University Press, 2000.
• [40] J. C. Marrero, D. M. d. Diego ve E. Martínez, «Discrete Lagrangian and Hamiltonian mechanics on Lie groupoids,» Nonlinearity, cilt 19, no. 6, p. 1313, 2006.
• [41] J. E. Marsden, G. Misiolek, J. P. Ortega, M. Perlmutter ve T. Ratiu, Hamiltonian Reduction by Stages, Berlin: Springer-Verlag, 2007.
• [42] J. E. Marsden, S. Pekarsky ve S. Shkoller, «Discrete Euler-Poincaré and Lie-Poisson equations,» Nonlinearity, cilt 12, no. 6, p. 1647, 1999.
• [43] J. E. Marsden, S. Pekarsky ve S. Shkoller, «Symmetry reduction of discrete Lagrangian mechanics on Lie groups,» Journal of geometry and physics, cilt 361, pp. 140-151, 2000.
• [44] J. E. Marsden ve T. Ratiu, «Reduction of Poisson manifolds,» Letters in Mathematical Physics,, cilt 112, pp. 161-169, 1986.
• [45] J. E. Marsden ve T. Ratiu, Introduction to Mechanics and Symmetry, Springer Science & Business Media, 1999.
• [46] J. E. Marsden ve J. Scheurle, «Lagrangian reduction and the double spherical pendulum,» Zeitschrift für angewandte Mathematik und Physik ZAMP, cilt 441, pp. 17-43, 1993.
• [47] J. E. Marsden ve J. Scheurle, «The reduced Euler-Lagrange equations,» Fields Institute Comm, cilt 1, pp. 139-164, 1993.
• [48] J. Marsden ve A. Weinstein, «Reduction of symplectic manifolds with symmetry”,,» Reports on mathematical physics, cilt 51, pp. 121-130, 1974.
• [49] J. E. Marsden ve A. Weinstein, «The Hamiltonian structure of the Maxwell-Vlasov equations,» Physica D, cilt 43, pp. 394-406, 1982.
• [50] J. E. Marsden, A. Weinstein, T. S. Ratiu, R. Schmid ve R. G. Spencer, «Hamiltonian systems with symmetry, coadjoint orbits and plasma physics,» içinde IUTAM-ISIMM symposium on modern developments in analytical mechanics, Torino, 1982.
• [51] J. E. Marsden ve M. West, «Discrete mechanics and variational integrators,» Acta Numerica, cilt 10, pp. 357-514, 2001.
• [52] K. R. Meyer, «Symmetries and integrals in mechanics,» içinde Dynamical systems, 1973, pp. 259-273.
• [53] T. Mokri, «Matched pairs of Lie algebroids,» Glasgow Mathematical Journal, cilt 39, no. 2, pp. 167-181, 1997.
• [54] P. J. Morrison, «The Maxwell-Vlasov equations as a continuous Hamiltonian system,» Physics Letters A, cilt 805, pp. 383-386, 1980.
• [55] P. J. Morrison ve J. M. Greene, «Noncanonical Hamiltonian density formulation of hydrodynamics and ideal magnetohydrodynamics,» Physical Review Letters, cilt 4501, p. 790, 1980.
• [56] H. Moscovici ve B. Rangipour, «Hopf algebras of primitive Lie pseudogroups and Hopf cyclic cohomology,» Advances in Mathematics, cilt 2203, pp. 706-790, 2009.
• [57] J. Moser ve A. P. Veselov, «Discrete versions of some classical integrable systems and factorization of matrix polynomials,» Communications in Mathematical Physics, cilt 139, no. 2, pp. 217-243, 1991.
• [58] P. J. Olver, Applications of Lie groups to Differential Equations, Springer Science & Business Media, 2000.
• [59] L. Polterovich, The Geometry of the Group of Symplectic Diffeomorphism, Birkhäuser, 2001.
• [60] M. Takeuchi, «Matched pairs of groups and bismash products of Hopf algebras”,,» Communications in Algebra, cilt 98, pp. 841-882, 1981.
• [61] W. M. Tulczyjew, «The Legendre transformation,» In Annales de l'IHP Physique théorique, cilt 27, no. 1, pp. 101-114, 1977.
• [62] I. Vaisman, Lectures On the Geometry of Poisson Manifolds, Birkhäuser, 1994.
• [63] A. P. Veselov, «Integrable discrete-time systems and difference operators”,» Functional Analysis and its Applications, cilt 22, no. 2, pp. 83-93, 1988.
• [64] A. Weinstein, «Lagrangian mechanics and groupoids,» Fields Inst. Commun, cilt 7, pp. 207-231, 1996.
• [65] A. Weinstein, Lectures on Symplectic Manifolds, American Mathematical Soc, 1977.
• [66]
• [67] A. Weinstein, «The local structure of Poisson manifolds,» Journal of differential geometry, cilt 183, pp. 523-557, 1983.
• K. Mackenzie, Lie groupoids and Lie algebroids in differential geometry, Cambridge university press, 1987.