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
References:
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R. Abraham, J. E. Marsden and T. S. Ratiu, "Manifolds, Tensor Analysis, and Applications,", Springer-Verlag, (1988). doi: 10.1007/978-1-4612-1029-0.

[2]

V. I. Arnold, "Mathematical Methods of Classical Mechanics,", Graduate Texts in Mathematics, 60 (1989).

[3]

A. Banyaga, Sur la structure du groupe des diffmorphismes qui preservent une forme symplectique,, Comment. Math. Helvetici, 53 (1978), 174. doi: 10.1007/BF02566074.

[4]

A. Banyaga, "The Structure of Classical Diffeomorphism Groups,", Mathematics and its Applications, 400 (1997).

[5]

S. Benenti, "Hamiltonian Structures and Generating Families,", Universitext v223. Springer, (2011).

[6]

S. Benenti and W. M. Tulczyjew, The geometrical meaning and globalization of the Hamilton-Jacobi method,, Diff. Geom. Meth. in Math. Phys. (Proc. Conf., 836 (1980), 9.

[7]

M. Chaperon, On generating families,, The Floer Memorial Volume, 133 (1995), 283.

[8]

M. Crampin and F. A. E. Pirani, "Applicable Differential Geometry,", Cambridge University Press, (1986).

[9]

M. De Le and P. R. Rodrigues, "Methods of Differential Geometry in Analytical Mechanics,", North-Holland Mathematics Studies, 158 (1989).

[10]

P. Dombrowski, On the geometry of the tangent bundle,, J. Reine Angew. Math., 210 (1962), 73.

[11]

O. Esen and H. Gümral, Lifts, jets and reduced dynamics,, Int. J. of Geom. Meth. in Modern Phys., 8 (2011), 331. doi: 10.1142/S0219887811005166.

[12]

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

[13]

J. Gancarzewicz, Liftings of functions and vector fields to natural bundles,, Proc. of the Conference (CSSR-GDR-Poland) on Diff. Geom. and its Appl., (1981), 89.

[14]

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

[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, ().

[16]

F. Gay-Balmaz and C. Vizman, Dual pairs in fluid dynamics,, Ann. Glob. Anal. Geom., 41 (2012), 1. doi: 10.1007/s10455-011-9267-z.

[17]

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

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V. Guillemin and S. Sternberg, The moment map and collective motion,, Ann. of Phys., 127 (1980), 220. doi: 10.1016/0003-4916(80)90155-4.

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V. Guillemin and S. Sternberg, "Symplectic Techniques in Physics,", Cambridge University Press (Cambridge), (1984).

[20]

H. Gümral, Geometry of plasma dynamics I: Group of canonical diffeomorphisms,, J. Math. Phys., 51 (2010).

[21]

R. S. Ismagilov, M. Losik and P. W. Michor, A 2-cocycle on a group of symplectomorphisms,, Moscow Math. J., 6 (2006), 307.

[22]

A. N. Kaufman and R. L. Dewar, Canonical derivation of the Vlasov-Coulomb noncanonical Poisson structure,, Cont. Math. AMS, 28 (1984), 51. doi: 10.1090/conm/028/751974.

[23]

P. Kobak, Natural liftings of vector fields to tangent bundles of bundles of 1-forms,, Math. Boch., 116 (1991), 319.

[24]

S. Kobayashi and K. Nomizu, "Foundations of Differential Geometry Vol.1,", Interscience Tract, (1963).

[25]

I. Kolar, P. W. Michor and J. Slovak, "Natural Operations in Differential Geometry,", Springer-Verlag, (1993).

[26]

Y. Kosmann-Schwarzbach, Vector fields and generalized vector fields on fibered manifolds,, Geometry and differential geometry (Proc. Conf. Univ. Haifa, 792 (1980), 307.

[27]

A. J. Ledger and K. Yano, The tangent bundle of a locally symmetric space,, J. London Math. Soc., 40 (1963), 487. doi: 10.1112/jlms/s1-40.1.487.

[28]

P. Libermann and C. M. Marle, "Symplectic Geometry and Analytical Mechanics,", D. Reidel Publishing Company, (1987).

[29]

J. E. Marsden, A correspondence principle for momentum operators,, Can. Math. Bull., 10 (1967), 247. doi: 10.4153/CMB-1967-023-x.

[30]

J. E. Marsden and T. S. Ratiu, "Introduction to Mechanics and Symmetry,", Texts in Applied Mathematics, 17 (1994).

[31]

J. E. Marsden and A. Weinstein, The Hamiltonian structure of the Maxwell-Vlasov equations,, Physica D, 4 (1982), 394. doi: 10.1016/0167-2789(82)90043-4.

[32]

J. E. Marsden, A group theoretic approach to the equations of plasma physics,, Canad. Math. Bull., 25 (1982), 129. doi: 10.4153/CMB-1982-019-9.

[33]

J. E. Marsden, A. Weinstein, T. S. Ratiu, R. Schmid and R. G. Spencer, Hamiltonian systems with symmetry, coadjoint orbits and plasma physics,, Proc. IUTAM-ISIMM Symposium on Modern Developments in Analytical Mechanics, 117 (1983), 289.

[34]

J. E. Marsden, P. J. Morrison and A. Weinstein, Hamiltonian structure of the BBGKY hierarchy equations,, Comtemp. Math., 28 (1984), 115.

[35]

D. McDuff and D. Salamon, "Introduction to Symplectic Topology,", Second edition. Oxford Mathematical Monographs. The Clarendon Press, (1998).

[36]

P. Michor, Manifolds of smooth maps,, Cahiers Top. Geo. Diff., 19 (1978), 47.

[37]

P. J. Morrison, The Maxwell-Vlasov equations as a continuous Hamiltonian system,, Phys. Lett. A, 80 (1980), 383. doi: 10.1016/0375-9601(80)90776-8.

[38]

P. J. Morrison and J. M. Greene, Noncanonical Hamiltonian density formulation of hydrodynamics and magnetohydrodynamics,, Phys. Rev. Lett., 45 (1980), 790. doi: 10.1103/PhysRevLett.45.790.

[39]

P. J. Morrison, "Hamiltonian Field Description of One-Dimensional Poisson-Vlasov Equations,", PPPL-1788, (1981).

[40]

P. J. Morrison, Poisson brackets for fluids and plasmas, in Mathematical Methods in Hydrodynamics and Integrability in Dynamical Systems,, (La Jolla Institute, 88 (1982), 13.

[41]

P. J. Olver, "Applications of Lie Groups to Differential Equations,", Graduate Texts in Mathematics, 107 (1986).

[42]

Z. Pogoda, Horizontal lifts and foliations,, Suppl. ai Rendiconti del Circolo Mat. di Palermo, 21 (1989), 279.

[43]

T. S. Ratiu, R. Schmid and M. R. Adams, The Lie group structure of diffeomorphism groups and invertible Fourier integral operators,, Infinite dimensional Groups with Applications, 4 (1985), 1.

[44]

T. S. Ratiu and R. Schmid, The differentiable structure of three remarkable diffeomorphism groups,, Mathematische Zeitschrift, 177 (1981), 81. doi: 10.1007/BF01214340.

[45]

S. Sasaki, On the differential geometry of tangent bundles of Riemannian manifolds,, Tohoku Math. J. (2), 10 (1958), 338. doi: 10.2748/tmj/1178244668.

[46]

D. J. Saunders, "The Geometry of Jet Bundles,", London Math. Soc., 142 (1989).

[47]

C. Scovel and A. Weinstein, Finite-dimensional Lie-Poisson approximations to Vlasov-Poisson equations,, Comm. Pure Appl. Math., 47 (1994), 683. doi: 10.1002/cpa.3160470505.

[48]

J. Sniatycki and W. M. Tulczyjew, Generating forms of Lagrangian submanifolds,, Indiana Univ. Math. J., 22 (1972), 267. doi: 10.1512/iumj.1972.22.22021.

[49]

T. Swift, A note on the space of lagrangian submanifolds of a symplectic 4-manifold,, Journal of Geometry and Physics, 35 (2000), 183. doi: 10.1016/S0393-0440(00)00008-5.

[50]

S. Sternberg, "Lectures on Differential Geometry,", Prentice-Hall, (1964).

[51]

S. Tanno, An almost complex structure of the tangent bundle of an almost contact manifold,, Tohoku Math. J. (2), 17 (1965), 7. doi: 10.2748/tmj/1178243616.

[52]

P. Tondeur, Structure presque klienne naturelle sur la fibrdes vecteurs covariants d'une vari riemannienne,, C. R. Acad. Sci. Paris, 254 (1962), 407.

[53]

W. M. Tulczyjew, The Legendre transformation,, Ann. Inst. Henri PoincarSec. A: Phys. Thr., XXVII (1977), 101.

[54]

W. M. Tulczyjew, A symplectic formulation of relativistic particle dynamics,, Acta Physica Polonica B, 8 (1977), 431.

[55]

W. M. Tulczyjew, A symplectic formulation of particle dynamics,, Differential Geometric Methods in Mathematical Physics, 570 (1977), 457. doi: 10.1007/BFb0087795.

[56]

W. M. Tulczyjew, A symplectic framework of linear field theories,, Ann. Mat. Pura Appl. (4), 130 (1981), 177.

[57]

W. M. Tulczyjew, The Euler-Lagrange resolution,, Differential geometrical methods in mathematical physics (Proc. Conf., (1979), 22.

[58]

W. M. Tulczyjew, Hamiltonian systems, Lagrangian systems and the Legendre transformation,, Istituto Nazionale di Alta Matematica, 14 (1974), 247.

[59]

W. M. Tulczyjew and P. Urbanski, A slow and careful Legendre transformation for singular Lagrangians,, Acta Phys. Polon. B, 30 (1999), 2909.

[60]

K. Yano and E. T. Davies, On the tangent bundle of Finsler and Riemannian manifolds,, Rend. Circ. Mat. Palermo (2), 12 (1963), 211. doi: 10.1007/BF02843966.

[61]

K. Yano and S. Kobayashi, Prolongations of tensor fields and connections to tangent bundles I. General Theory,, J. Math. Soc. Japan, 18 (1966), 194. doi: 10.2969/jmsj/01820194.

[62]

K. Yano and A. J. Ledger, Linear connections on tangent bundles,, J. London Math. Soc., 39 (1964), 495. doi: 10.1112/jlms/s1-39.1.495.

[63]

K. Yano and E. M. Patterson, Vertical and complete lifts from a manifold to its cotangent bundle,, J. Math. Soc. Japan, 19 (1967), 91. doi: 10.2969/jmsj/01910091.

[64]

I. Vaisman, Locally conformal symplectic manifolds,, Int. J. Math. and Math. Sci., 8 (1985), 521. doi: 10.1155/S0161171285000564.

[65]

L. Van Hove, Sur le proble des relations entre les transformations unitaires de la manique quantique et les transformations canoniques de la manique classique,, Acad. Roy. Belgique. Bull. Cl. Sci. (5), 37 (1951), 610.

[66]

C. Vizman, Some remarks on the quantomorphism group,, Proc. of the Third International Workshop on Diff. Geom., 5 (1997), 393.

[67]

C. Vizman, Abelian extensions via prequantization,, Annals of Global Analysis and Geometry, 39 (2011), 361.

[68]

A. Weinstein, "Lectures on Symplectic Manifolds,", Exp. lec. from the CBMS, (1977).

[69]

A. Weinstein, Symplectic manifolds and their Lagrangian submanifolds,, Advan. in Math., 6 (1971), 329. doi: 10.1016/0001-8708(71)90020-X.

[70]

A. Weinstein, Lagrangian submanifolds and Hamiltonian systems,, Annals of Math., 2 (1973), 377. doi: 10.2307/1970911.

[71]

A. Weinstein and P. J. Morrison, Comments on: The Maxwell-Vlasov equation as a continuous Hamiltonian system,, Phys. Lett. A., 86 (1981), 235. doi: 10.1016/0375-9601(81)90496-5.

show all references

References:
[1]

R. Abraham, J. E. Marsden and T. S. Ratiu, "Manifolds, Tensor Analysis, and Applications,", Springer-Verlag, (1988). doi: 10.1007/978-1-4612-1029-0.

[2]

V. I. Arnold, "Mathematical Methods of Classical Mechanics,", Graduate Texts in Mathematics, 60 (1989).

[3]

A. Banyaga, Sur la structure du groupe des diffmorphismes qui preservent une forme symplectique,, Comment. Math. Helvetici, 53 (1978), 174. doi: 10.1007/BF02566074.

[4]

A. Banyaga, "The Structure of Classical Diffeomorphism Groups,", Mathematics and its Applications, 400 (1997).

[5]

S. Benenti, "Hamiltonian Structures and Generating Families,", Universitext v223. Springer, (2011).

[6]

S. Benenti and W. M. Tulczyjew, The geometrical meaning and globalization of the Hamilton-Jacobi method,, Diff. Geom. Meth. in Math. Phys. (Proc. Conf., 836 (1980), 9.

[7]

M. Chaperon, On generating families,, The Floer Memorial Volume, 133 (1995), 283.

[8]

M. Crampin and F. A. E. Pirani, "Applicable Differential Geometry,", Cambridge University Press, (1986).

[9]

M. De Le and P. R. Rodrigues, "Methods of Differential Geometry in Analytical Mechanics,", North-Holland Mathematics Studies, 158 (1989).

[10]

P. Dombrowski, On the geometry of the tangent bundle,, J. Reine Angew. Math., 210 (1962), 73.

[11]

O. Esen and H. Gümral, Lifts, jets and reduced dynamics,, Int. J. of Geom. Meth. in Modern Phys., 8 (2011), 331. doi: 10.1142/S0219887811005166.

[12]

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

[13]

J. Gancarzewicz, Liftings of functions and vector fields to natural bundles,, Proc. of the Conference (CSSR-GDR-Poland) on Diff. Geom. and its Appl., (1981), 89.

[14]

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

[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, ().

[16]

F. Gay-Balmaz and C. Vizman, Dual pairs in fluid dynamics,, Ann. Glob. Anal. Geom., 41 (2012), 1. doi: 10.1007/s10455-011-9267-z.

[17]

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

[18]

V. Guillemin and S. Sternberg, The moment map and collective motion,, Ann. of Phys., 127 (1980), 220. doi: 10.1016/0003-4916(80)90155-4.

[19]

V. Guillemin and S. Sternberg, "Symplectic Techniques in Physics,", Cambridge University Press (Cambridge), (1984).

[20]

H. Gümral, Geometry of plasma dynamics I: Group of canonical diffeomorphisms,, J. Math. Phys., 51 (2010).

[21]

R. S. Ismagilov, M. Losik and P. W. Michor, A 2-cocycle on a group of symplectomorphisms,, Moscow Math. J., 6 (2006), 307.

[22]

A. N. Kaufman and R. L. Dewar, Canonical derivation of the Vlasov-Coulomb noncanonical Poisson structure,, Cont. Math. AMS, 28 (1984), 51. doi: 10.1090/conm/028/751974.

[23]

P. Kobak, Natural liftings of vector fields to tangent bundles of bundles of 1-forms,, Math. Boch., 116 (1991), 319.

[24]

S. Kobayashi and K. Nomizu, "Foundations of Differential Geometry Vol.1,", Interscience Tract, (1963).

[25]

I. Kolar, P. W. Michor and J. Slovak, "Natural Operations in Differential Geometry,", Springer-Verlag, (1993).

[26]

Y. Kosmann-Schwarzbach, Vector fields and generalized vector fields on fibered manifolds,, Geometry and differential geometry (Proc. Conf. Univ. Haifa, 792 (1980), 307.

[27]

A. J. Ledger and K. Yano, The tangent bundle of a locally symmetric space,, J. London Math. Soc., 40 (1963), 487. doi: 10.1112/jlms/s1-40.1.487.

[28]

P. Libermann and C. M. Marle, "Symplectic Geometry and Analytical Mechanics,", D. Reidel Publishing Company, (1987).

[29]

J. E. Marsden, A correspondence principle for momentum operators,, Can. Math. Bull., 10 (1967), 247. doi: 10.4153/CMB-1967-023-x.

[30]

J. E. Marsden and T. S. Ratiu, "Introduction to Mechanics and Symmetry,", Texts in Applied Mathematics, 17 (1994).

[31]

J. E. Marsden and A. Weinstein, The Hamiltonian structure of the Maxwell-Vlasov equations,, Physica D, 4 (1982), 394. doi: 10.1016/0167-2789(82)90043-4.

[32]

J. E. Marsden, A group theoretic approach to the equations of plasma physics,, Canad. Math. Bull., 25 (1982), 129. doi: 10.4153/CMB-1982-019-9.

[33]

J. E. Marsden, A. Weinstein, T. S. Ratiu, R. Schmid and R. G. Spencer, Hamiltonian systems with symmetry, coadjoint orbits and plasma physics,, Proc. IUTAM-ISIMM Symposium on Modern Developments in Analytical Mechanics, 117 (1983), 289.

[34]

J. E. Marsden, P. J. Morrison and A. Weinstein, Hamiltonian structure of the BBGKY hierarchy equations,, Comtemp. Math., 28 (1984), 115.

[35]

D. McDuff and D. Salamon, "Introduction to Symplectic Topology,", Second edition. Oxford Mathematical Monographs. The Clarendon Press, (1998).

[36]

P. Michor, Manifolds of smooth maps,, Cahiers Top. Geo. Diff., 19 (1978), 47.

[37]

P. J. Morrison, The Maxwell-Vlasov equations as a continuous Hamiltonian system,, Phys. Lett. A, 80 (1980), 383. doi: 10.1016/0375-9601(80)90776-8.

[38]

P. J. Morrison and J. M. Greene, Noncanonical Hamiltonian density formulation of hydrodynamics and magnetohydrodynamics,, Phys. Rev. Lett., 45 (1980), 790. doi: 10.1103/PhysRevLett.45.790.

[39]

P. J. Morrison, "Hamiltonian Field Description of One-Dimensional Poisson-Vlasov Equations,", PPPL-1788, (1981).

[40]

P. J. Morrison, Poisson brackets for fluids and plasmas, in Mathematical Methods in Hydrodynamics and Integrability in Dynamical Systems,, (La Jolla Institute, 88 (1982), 13.

[41]

P. J. Olver, "Applications of Lie Groups to Differential Equations,", Graduate Texts in Mathematics, 107 (1986).

[42]

Z. Pogoda, Horizontal lifts and foliations,, Suppl. ai Rendiconti del Circolo Mat. di Palermo, 21 (1989), 279.

[43]

T. S. Ratiu, R. Schmid and M. R. Adams, The Lie group structure of diffeomorphism groups and invertible Fourier integral operators,, Infinite dimensional Groups with Applications, 4 (1985), 1.

[44]

T. S. Ratiu and R. Schmid, The differentiable structure of three remarkable diffeomorphism groups,, Mathematische Zeitschrift, 177 (1981), 81. doi: 10.1007/BF01214340.

[45]

S. Sasaki, On the differential geometry of tangent bundles of Riemannian manifolds,, Tohoku Math. J. (2), 10 (1958), 338. doi: 10.2748/tmj/1178244668.

[46]

D. J. Saunders, "The Geometry of Jet Bundles,", London Math. Soc., 142 (1989).

[47]

C. Scovel and A. Weinstein, Finite-dimensional Lie-Poisson approximations to Vlasov-Poisson equations,, Comm. Pure Appl. Math., 47 (1994), 683. doi: 10.1002/cpa.3160470505.

[48]

J. Sniatycki and W. M. Tulczyjew, Generating forms of Lagrangian submanifolds,, Indiana Univ. Math. J., 22 (1972), 267. doi: 10.1512/iumj.1972.22.22021.

[49]

T. Swift, A note on the space of lagrangian submanifolds of a symplectic 4-manifold,, Journal of Geometry and Physics, 35 (2000), 183. doi: 10.1016/S0393-0440(00)00008-5.

[50]

S. Sternberg, "Lectures on Differential Geometry,", Prentice-Hall, (1964).

[51]

S. Tanno, An almost complex structure of the tangent bundle of an almost contact manifold,, Tohoku Math. J. (2), 17 (1965), 7. doi: 10.2748/tmj/1178243616.

[52]

P. Tondeur, Structure presque klienne naturelle sur la fibrdes vecteurs covariants d'une vari riemannienne,, C. R. Acad. Sci. Paris, 254 (1962), 407.

[53]

W. M. Tulczyjew, The Legendre transformation,, Ann. Inst. Henri PoincarSec. A: Phys. Thr., XXVII (1977), 101.

[54]

W. M. Tulczyjew, A symplectic formulation of relativistic particle dynamics,, Acta Physica Polonica B, 8 (1977), 431.

[55]

W. M. Tulczyjew, A symplectic formulation of particle dynamics,, Differential Geometric Methods in Mathematical Physics, 570 (1977), 457. doi: 10.1007/BFb0087795.

[56]

W. M. Tulczyjew, A symplectic framework of linear field theories,, Ann. Mat. Pura Appl. (4), 130 (1981), 177.

[57]

W. M. Tulczyjew, The Euler-Lagrange resolution,, Differential geometrical methods in mathematical physics (Proc. Conf., (1979), 22.

[58]

W. M. Tulczyjew, Hamiltonian systems, Lagrangian systems and the Legendre transformation,, Istituto Nazionale di Alta Matematica, 14 (1974), 247.

[59]

W. M. Tulczyjew and P. Urbanski, A slow and careful Legendre transformation for singular Lagrangians,, Acta Phys. Polon. B, 30 (1999), 2909.

[60]

K. Yano and E. T. Davies, On the tangent bundle of Finsler and Riemannian manifolds,, Rend. Circ. Mat. Palermo (2), 12 (1963), 211. doi: 10.1007/BF02843966.

[61]

K. Yano and S. Kobayashi, Prolongations of tensor fields and connections to tangent bundles I. General Theory,, J. Math. Soc. Japan, 18 (1966), 194. doi: 10.2969/jmsj/01820194.

[62]

K. Yano and A. J. Ledger, Linear connections on tangent bundles,, J. London Math. Soc., 39 (1964), 495. doi: 10.1112/jlms/s1-39.1.495.

[63]

K. Yano and E. M. Patterson, Vertical and complete lifts from a manifold to its cotangent bundle,, J. Math. Soc. Japan, 19 (1967), 91. doi: 10.2969/jmsj/01910091.

[64]

I. Vaisman, Locally conformal symplectic manifolds,, Int. J. Math. and Math. Sci., 8 (1985), 521. doi: 10.1155/S0161171285000564.

[65]

L. Van Hove, Sur le proble des relations entre les transformations unitaires de la manique quantique et les transformations canoniques de la manique classique,, Acad. Roy. Belgique. Bull. Cl. Sci. (5), 37 (1951), 610.

[66]

C. Vizman, Some remarks on the quantomorphism group,, Proc. of the Third International Workshop on Diff. Geom., 5 (1997), 393.

[67]

C. Vizman, Abelian extensions via prequantization,, Annals of Global Analysis and Geometry, 39 (2011), 361.

[68]

A. Weinstein, "Lectures on Symplectic Manifolds,", Exp. lec. from the CBMS, (1977).

[69]

A. Weinstein, Symplectic manifolds and their Lagrangian submanifolds,, Advan. in Math., 6 (1971), 329. doi: 10.1016/0001-8708(71)90020-X.

[70]

A. Weinstein, Lagrangian submanifolds and Hamiltonian systems,, Annals of Math., 2 (1973), 377. doi: 10.2307/1970911.

[71]

A. Weinstein and P. J. Morrison, Comments on: The Maxwell-Vlasov equation as a continuous Hamiltonian system,, Phys. Lett. A., 86 (1981), 235. doi: 10.1016/0375-9601(81)90496-5.

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