September  2012, 4(3): 239-269. doi: 10.3934/jgm.2012.4.239

Geometry of plasma dynamics II: Lie algebra of Hamiltonian vector fields

1. 

Department of Mathematics, Yeditepe University, 34755 Ataşehir, İstanbul, Turkey, Turkey

Received  November 2010 Revised  March 2012 Published  October 2012

We introduce natural differential geometric structures underlying the Poisson-Vlasov equations in momentum variables. First, we decompose the space of all vector fields over particle phase space into a semi-direct product algebra of Hamiltonian vector fields and its complement. The latter is related to dual space of the Lie algebra. We identify generators of homotheties as dynamically irrelevant vector fields in the complement. Lie algebra of Hamiltonian vector fields is isomorphic to the space of all Lagrangian submanifolds with respect to Tulczyjew symplectic structure. This is obtained as tangent space at the identity of the group of canonical diffeomorphisms represented as space of sections of a trivial bundle. We obtain the momentum-Vlasov equations as vertical equivalence, or representative, of complete cotangent lift of Hamiltonian vector field generating particle motion. Vertical representatives can be described by holonomic lift from a Whitney product to a Tulczyjew symplectic space. We show that vertical representatives of complete cotangent lifts form an integrable subbundle of this Tulczyjew space. We exhibit dynamical relations between Lie algebras of Hamiltonian vector fields and of contact vector fields, in particular; infinitesimal quantomorphisms on quantization bundle. Gauge symmetries of particle motion are extended to tensorial objects including complete lift of particle motion. Poisson equation is then obtained as zero value of momentum map for the Hamiltonian action of gauge symmetries for kinematical description.
Citation: Oǧul Esen, Hasan Gümral. Geometry of plasma dynamics II: Lie algebra of Hamiltonian vector fields. Journal of Geometric Mechanics, 2012, 4 (3) : 239-269. doi: 10.3934/jgm.2012.4.239
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show all references

References:
[1]

Springer-Verlag, 2nd edition New York, 1988. doi: 10.1007/978-1-4612-1029-0.  Google Scholar

[2]

Graduate Texts in Mathematics, 60, Springer-Verlag, 1989.  Google Scholar

[3]

Comment. Math. Helvetici, 53 (1978), 174-227. doi: 10.1007/BF02566074.  Google Scholar

[4]

Mathematics and its Applications, 400, Kluwer Academic Publishers Group, Dordrecht, 1997.  Google Scholar

[5]

Universitext v223. Springer, 2011. Google Scholar

[6]

Diff. Geom. Meth. in Math. Phys. (Proc. Conf., Aix-en-Provence/Salamanca, 1979), Lect. Notes in Math, Springer, Berlin, 836 (1980), 9-21.  Google Scholar

[7]

The Floer Memorial Volume, (Eds. H. Hofer, C. H. Taubes, A. Weinstein and E. Zehnder), Progress in Mathematics Birkauser, 133 (1995), 283-296.  Google Scholar

[8]

Cambridge University Press, Cambridge, 1986.  Google Scholar

[9]

North-Holland Mathematics Studies, 158, North-Holland, Amsterdam, 1989.  Google Scholar

[10]

J. Reine Angew. Math., 210 (1962), 73-88.  Google Scholar

[11]

Int. J. of Geom. Meth. in Modern Phys., 8 (2011), 331-344. doi: 10.1142/S0219887811005166.  Google Scholar

[12]

O. Esen and H. Gümral, Geometry of plasma dynamics III: Orbits of canonical diffeomorphisms,, work in progress., ().   Google Scholar

[13]

Proc. of the Conference (CSSR-GDR-Poland) on Diff. Geom. and its Appl., Nove Mesto na Morave, Sep. 1980, Univ. Praha, (1981), 89-102.  Google Scholar

[14]

F. Gay-Balmaz and C. Tronci, Vlasov moment flows and geodesics on the Jacobi group,, preprint, ().   Google Scholar

[15]

F. Gay-Balmaz, C. Tronci and C. Vizman, Geometric dynamics on the automorphism group of principal bundles: geodesic flows, dual pairs and chromomorphism groups,, preprint, ().   Google Scholar

[16]

Ann. Glob. Anal. Geom., 41 (2012), 1-24. doi: 10.1007/s10455-011-9267-z.  Google Scholar

[17]

Physica D, 3 (1981), 503-511. doi: 10.1016/0167-2789(81)90036-1.  Google Scholar

[18]

Ann. of Phys., 127 (1980), 220-253. doi: 10.1016/0003-4916(80)90155-4.  Google Scholar

[19]

Cambridge University Press (Cambridge), 1984.  Google Scholar

[20]

J. Math. Phys., 51 (2010), 23 pp. 083501.  Google Scholar

[21]

Moscow Math. J., 6 (2006), 307-315.  Google Scholar

[22]

Cont. Math. AMS, 28 (1984), 51-54. doi: 10.1090/conm/028/751974.  Google Scholar

[23]

Math. Boch., 116 (1991), 319-326.  Google Scholar

[24]

Interscience Tract, No. 15, 1963.  Google Scholar

[25]

Springer-Verlag, Berlin, Heidelberg, 1993.  Google Scholar

[26]

Geometry and differential geometry (Proc. Conf. Univ. Haifa, Isral, 1979), (eds. R. Artzy and I. Vaisman), Lecture Notes in Math., Springer-Verlag, Heidelberg, 792 (1980), 307-355.  Google Scholar

[27]

J. London Math. Soc., 40 (1963), 487-492. doi: 10.1112/jlms/s1-40.1.487.  Google Scholar

[28]

D. Reidel Publishing Company, Kluwer Academic Publishers Group, 1987.  Google Scholar

[29]

Can. Math. Bull., 10 (1967), 247-250. doi: 10.4153/CMB-1967-023-x.  Google Scholar

[30]

Texts in Applied Mathematics, 17, Springer-Verlag, New York, 1994.  Google Scholar

[31]

Physica D, 4 (1982), 394-406. doi: 10.1016/0167-2789(82)90043-4.  Google Scholar

[32]

Canad. Math. Bull., 25 (1982), 129-142. doi: 10.4153/CMB-1982-019-9.  Google Scholar

[33]

Proc. IUTAM-ISIMM Symposium on Modern Developments in Analytical Mechanics, Atti della Academia della Scienze di Torino, 117 (1983), 289-340.  Google Scholar

[34]

Comtemp. Math., AMS, 28 (1984), 115-124.  Google Scholar

[35]

Second edition. Oxford Mathematical Monographs. The Clarendon Press, Oxford University Press, 1998.  Google Scholar

[36]

Cahiers Top. Geo. Diff., 19 (1978), 47-78.  Google Scholar

[37]

Phys. Lett. A, 80 (1980), 383-386. doi: 10.1016/0375-9601(80)90776-8.  Google Scholar

[38]

Phys. Rev. Lett., 45 (1980), 790-794. doi: 10.1103/PhysRevLett.45.790.  Google Scholar

[39]

PPPL-1788, 1981. Google Scholar

[40]

(La Jolla Institute, 1981) AIP Conf. Proc., (eds. M. Taber and Y. Treve), (AIP, New York), 88 (1982), 13-46. Google Scholar

[41]

Graduate Texts in Mathematics, 107. Springer-Verlag, New York, 1986.  Google Scholar

[42]

Suppl. ai Rendiconti del Circolo Mat. di Palermo, 21 (1989), 279-283.  Google Scholar

[43]

Infinite dimensional Groups with Applications, (ed. V. Kac), Math. Sci. Res. Inst. Publ., Springer, New York, 4 (1985), 1-69.  Google Scholar

[44]

Mathematische Zeitschrift, 177 (1981), 81-100. doi: 10.1007/BF01214340.  Google Scholar

[45]

Tohoku Math. J. (2), 10 (1958), 338-354. doi: 10.2748/tmj/1178244668.  Google Scholar

[46]

London Math. Soc., Lecture Notes Series, 142, Cambridge Univ. Press, 1989.  Google Scholar

[47]

Comm. Pure Appl. Math., 47 (1994), 683-709. doi: 10.1002/cpa.3160470505.  Google Scholar

[48]

Indiana Univ. Math. J., 22 (1972), 267-275. doi: 10.1512/iumj.1972.22.22021.  Google Scholar

[49]

Journal of Geometry and Physics, 35 (2000), 183-192. doi: 10.1016/S0393-0440(00)00008-5.  Google Scholar

[50]

Prentice-Hall, Inc., Englewood Cliffs, N. J., 1964  Google Scholar

[51]

Tohoku Math. J. (2), 17 (1965), 7-15. doi: 10.2748/tmj/1178243616.  Google Scholar

[52]

C. R. Acad. Sci. Paris, 254 (1962), 407-408.  Google Scholar

[53]

Ann. Inst. Henri PoincarSec. A: Phys. Thr., XXVII (1977), 101-114.  Google Scholar

[54]

Acta Physica Polonica B, 8 (1977), 431-447.  Google Scholar

[55]

Differential Geometric Methods in Mathematical Physics, Lect. Notes in Math., 570 (1977), 457-463. doi: 10.1007/BFb0087795.  Google Scholar

[56]

Ann. Mat. Pura Appl. (4), 130 (1981), 177-195.  Google Scholar

[57]

Differential geometrical methods in mathematical physics (Proc. Conf., Aix-en-Provence/Salamanca, 1979), 22-48, Lecture Notes in Math., 836, Springer, Berlin, 1980.  Google Scholar

[58]

Istituto Nazionale di Alta Matematica, Symposia Mathematica, 14 (1974), 247-258.  Google Scholar

[59]

Acta Phys. Polon. B, 30 (1999), 2909-2978.  Google Scholar

[60]

Rend. Circ. Mat. Palermo (2), 12 (1963), 211-228. doi: 10.1007/BF02843966.  Google Scholar

[61]

J. Math. Soc. Japan, 18 (1966), 194-210. doi: 10.2969/jmsj/01820194.  Google Scholar

[62]

J. London Math. Soc., 39 (1964), 495-500. doi: 10.1112/jlms/s1-39.1.495.  Google Scholar

[63]

J. Math. Soc. Japan, 19 (1967), 91-113. doi: 10.2969/jmsj/01910091.  Google Scholar

[64]

Int. J. Math. and Math. Sci., 8 (1985), 521-536. doi: 10.1155/S0161171285000564.  Google Scholar

[65]

Acad. Roy. Belgique. Bull. Cl. Sci. (5), 37 (1951), 610-620.  Google Scholar

[66]

Proc. of the Third International Workshop on Diff. Geom., Sibiu, Romania, Gen. Math., 5 (1997), 393-399.  Google Scholar

[67]

Annals of Global Analysis and Geometry, 39 (2011), 361-386.  Google Scholar

[68]

Exp. lec. from the CBMS, Regional Conference Series in Mathematics, No. 29. A.M.S., Providence, 1977.  Google Scholar

[69]

Advan. in Math., 6 (1971), 329-346. doi: 10.1016/0001-8708(71)90020-X.  Google Scholar

[70]

Annals of Math., 2 (1973), 377-410. doi: 10.2307/1970911.  Google Scholar

[71]

Phys. Lett. A., 86 (1981), 235-236. doi: 10.1016/0375-9601(81)90496-5.  Google Scholar

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