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June  2021, 13(2): 209-246. doi: 10.3934/jgm.2021011

Matched pair analysis of the Vlasov plasma

1. 

Department of Mathematics, Gebze Technical University, 41400 Gebze-Kocaeli, Turkey

2. 

Department of Mathematics, Işık University, 34980 Şile-İstanbul, Turkey

* Corresponding author: Serkan Sütlü

Received  July 2020 Revised  January 2021 Published  June 2021 Early access  May 2021

We present the Hamiltonian (Lie-Poisson) analysis of the Vlasov plasma, and the dynamics of its kinetic moments, from the matched pair decomposition point of view. We express these (Lie-Poisson) systems as couplings of mutually interacting (Lie-Poisson) subdynamics. The mutual interaction is beyond the well-known semi-direct product theory. Accordingly, as the geometric framework of the present discussion, we address the matched pair Lie-Poisson formulation allowing mutual interactions. Moreover, both for the kinetic moments and the Vlasov plasma cases, we observe that one of the constitutive subdynamics is the compressible isentropic fluid flow, and the other is the dynamics of the kinetic moments of order $ \geq 2 $. In this regard, the algebraic/geometric (matched pair) decomposition that we offer, is in perfect harmony with the physical intuition. To complete the discussion, we present a momentum formulation of the Vlasov plasma, along with its matched pair decomposition.

Citation: Oǧul Esen, Serkan Sütlü. Matched pair analysis of the Vlasov plasma. Journal of Geometric Mechanics, 2021, 13 (2) : 209-246. doi: 10.3934/jgm.2021011
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show all references

References:
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R. Abraham and J. E. Marsden, Foundations of Mechanics, Benjamin/Cummings Publishing Co., Inc., Advanced Book Program, Reading, Mass., 1978.

[2]

A. L. Agore and G. Militaru, Extending structures for Lie algebras, Monatsh. Math., 174 (2014), 169-193.  doi: 10.1007/s00605-013-0537-7.

[3]

V. I. Arnol'd, Mathematical Methods of Classical Mechanics, Graduate Texts in Mathematics, 60, Springer-Verlag, New York, 1989. doi: 10.1007/978-1-4757-2063-1.

[4]

E. Binz, J. Śniatycki and H. Fischer, Geometry of Classical Fields, North-Holland Mathematics Studies, 154, Mathematical Notes, 123, North-Holland Publishing Co., Amsterdam, 1988.

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F. J. Bloore and M. Assimakopoulos, A natural one-form for the Schouten concomitant, Internat. J. Theoret. Phys., 18 (1979), 233-238.  doi: 10.1007/BF00671759.

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M. G. Brin, On the Zappa-Szép product, Comm. Algebra, 33 (2005), 393-424.  doi: 10.1081/AGB-200047404.

[7]

T. Brzeziński, Crossed products by a coalgebra, Comm. Algebra, 25 (1997), 3551-3575.  doi: 10.1080/00927879708826070.

[8]

C. Cercignani, V. I. Gerasimenko and D. Y. Petrina, Many-Particle Dynamics and Kinetic Equations, Mathematics and its Applications, 420, Kluwer Academic Publishers Group, Dordrecht, 1997. doi: 10.1007/978-94-011-5558-8.

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S. S. Chern, W. H. Chen and K. S. Lam, Lectures on Differential Geometry, Series on University Mathematics, 1, World Scientific Publishing Co. Inc., River Edge, NJ, 1999. doi: 10.1142/3812.

[11]

M. de León and P. R. Rodrigues, Methods of Differential Geometry in Analytical Mechanics, North-Holland Mathematics Studies, 158, North-Holland Publishing Co., Amsterdam, 1989.

[12]

O. Esen, M. Grmela, H. Gümral and M. Pavelka, Lifts of symmetric tensors: Fluids, plasma, and Grad hierarchy, Entropy, 21 (2019), 33pp. doi: 10.3390/e21090907.

[13]

O. Esen, P. Guha and S. Sütlü, Bicocycle double cross constructions, preprint, arXiv: 2104.08973.

[14]

O. Esen and H. Gümral, Geometry of plasma dynamics II: Lie algebra of Hamiltonian vector fields, J. Geom. Mech., 4 (2012), 239-269.  doi: 10.3934/jgm.2012.4.239.

[15]

O. Esen and H. Gümral, Lifts, jets and reduced dynamics, Int. J. Geom. Methods Mod. Phys., 8 (2011), 331-344.  doi: 10.1142/S0219887811005166.

[16]

O. Esen and H. Gümral, Tulczyjew's triplet for Lie groups I: Trivializations and reductions, J. Lie Theory, 24 (2014), 1115-1160.

[17]

O. Esen and H. Gümral, Tulczyjew's triplet for Lie groups II: Dynamics, J. Lie Theory, 27 (2017), 329-356.

[18]

O. Esen, M. Kudeyt, and S. Sütlü, Second order Lagrangian dynamics on double cross product groups, J. Geom. Phys. 159 (2021), 18pp. doi: 10.1016/j.geomphys.2020.103934.

[19]

O. EsenM. Pavelka and M. Grmela, Hamiltonian coupling of electromagnetic field and matter, Int. J. Adv. Eng. Sci. Appl. Math., 9 (2017), 3-20.  doi: 10.1007/s12572-017-0179-4.

[20]

O. Esen and S. Sütlü, Discrete dynamical systems over double cross-product Lie groupoids, Int. J. Geom. Methods Mod. Phys., 18 (2021), 40pp. doi: 10.1142/S0219887821500572.

[21]

O. Esen and S. Sütlü, Hamiltonian dynamics on matched pairs, Int. J. Geom. Methods Mod. Phys., 13 (2016), 24pp. doi: 10.1142/S0219887816501280.

[22]

O. Esen and S. Sütlü, Lagrangian dynamics on matched pairs, J. Geom. Phys., 111 (2017), 142-157.  doi: 10.1016/j.geomphys.2016.10.005.

[23]

D. B. Fuks, Cohomology of Infinite-Dimensional Lie Algebras, Contemporary Soviet Mathematics, Consultants Bureau, New York, 1986.

[24]

I. M. Gel'fandD. I. Kalinin and D. B. Fuks, The cohomology of the Lie algebra of Hamiltonian formal vector fields, Funkcional. Anal. i Priložen., 6 (1972), 25-29. 

[25]

J. Gibbons, Collisionless Boltzmann equations and integrable moment equations, Phys. D, 3 (1981), 503-511.  doi: 10.1016/0167-2789(81)90036-1.

[26]

J. GibbonsD. D. Holm and C. Tronci, Geometry of Vlasov kinetic moments: A bosonic Fock space for the symmetric Schouten bracket, Phys. Lett. A, 372 (2008), 4184-4196.  doi: 10.1016/j.physleta.2008.03.034.

[27]

J. GibbonsD. D. Holm and C. Tronci, Vlasov moments, integrable systems and singular solutions, Phys. Lett. A, 372 (2008), 1024-1033.  doi: 10.1016/j.physleta.2007.08.054.

[28]

K. Grabowska and M. Zając, The Tulczyjew triple in mechanics on a Lie group, J. Geom. Mech., 8 (2016), 413-435.  doi: 10.3934/jgm.2016014.

[29]

H. Grad, On Boltzmann's $H$-theorem, J. Soc. Indust. Appl. Math., 13 (1965), 259-277.  doi: 10.1137/0113016.

[30]

M. Grmela, L. Hong, D. Jou, G. Lebon and M. Pavelka, Hamiltonian and Godunov structures of the Grad hierarchy, Phys. Rev. E, 95 (2017). doi: 10.1103/PhysRevE.95.033121.

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D. D. Holm and B. A. Kupershmidt, Noncanonical Hamiltonian formulation of ideal magnetohydrodynamics, Phys. D, 7 (1983), 330-333.  doi: 10.1016/0167-2789(83)90136-7.

[35]

D. D. Holm, T. Schmah and C. Stoica, Geometric Mechanics and Symmetry. From Finite to Infinite Dimensions, Oxford Texts in Applied and Engineering Mathematics, 12, Oxford University Press, Oxford, 2009.

[36]

D. D. Holm and C. Tronci, Geodesic Vlasov equations and their integrable moment closures, J. Geom. Mech., 1 (2009), 181-208.  doi: 10.3934/jgm.2009.1.181.

[37]

B. Janssens and C. Vizman, Central extensions of Lie algebras of symplectic and divergence free vector fields, in Geometry of Jets and Fields, Banach Center Publ., 110, Polish Acad. Sci. Inst. Math., Warsaw, 2016,105-114. doi: 10.4064/bc110-0-7.

[38]

G. I. Kac, Extensions of groups to ring groups, Math. USSR Sb., 5 (1968), 451-474.  doi: 10.1070/SM1968v005n03ABEH003627.

[39]

M. Kikkawa, Geometry of homogeneous Lie loops, Hiroshima Math. J., 5 (1975), 141-179.  doi: 10.32917/hmj/1206136626.

[40]

M. K. Kinyon and A. Weinstein, Leibniz algebras, Courant algebroids, and multiplications on reductive homogeneous spaces, Amer. J. Math., 123 (2001), 525-550.  doi: 10.1353/ajm.2001.0017.

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I. Kolář, P. W. Michor and J. Slovák, Natural Operations in Differential Geometry, Springer-Verlag, Berlin, 1993. doi: 10.1007/978-3-662-02950-3.

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Y. Kosmann-Schwarzbach and F. Magri, Poisson-Lie groups and complete integrability. I. Drinfel'd bialgebras, dual extensions and their canonical representations, Ann. Inst. H. Poincaré Phys. Théor., 49 (1988), 433-460. 

[43]

C. D. Levermore, Moment closure hierarchies for kinetic theories, J. Statist. Phys., 83 (1996), 1021-1065.  doi: 10.1007/BF02179552.

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P. Libermann and C.-M. Marle, Symplectic Geometry and Analytical Mechanics, Mathematics and its Applications, 35, Reidel Publishing Co., Dordrecht, 1987. doi: 10.1007/978-94-009-3807-6.

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J.-H. Lu and A. Weinstein, Poisson Lie groups, dressing transformations, and Bruhat decompositions, J. Differential Geom., 31 (1990), 501-526.  doi: 10.4310/jdg/1214444324.

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S. Majid, Matched pairs of Lie groups associated to solutions of the Yang-Baxter equations, Pacific J. Math., 141 (1990), 311-332.  doi: 10.2140/pjm.1990.141.311.

[48]

S. Majid, Physics for algebraists: Non-commutative and non-cocommutative Hopf algebras by a bicrossproduct construction, J. Algebra, 130 (1990), 17-64.  doi: 10.1016/0021-8693(90)90099-A.

[49]

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