# American Institute of Mathematical Sciences

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
##### References:
 [1] R. Abraham, J. E. Marsden and T. S. Ratiu, "Manifolds, Tensor Analysis, and Applications," Springer-Verlag, 2nd edition New York, 1988. doi: 10.1007/978-1-4612-1029-0.  Google Scholar [2] V. I. Arnold, "Mathematical Methods of Classical Mechanics," Graduate Texts in Mathematics, 60, Springer-Verlag, 1989.  Google Scholar [3] A. Banyaga, Sur la structure du groupe des diffmorphismes qui preservent une forme symplectique, Comment. Math. Helvetici, 53 (1978), 174-227. doi: 10.1007/BF02566074.  Google Scholar [4] A. Banyaga, "The Structure of Classical Diffeomorphism Groups," Mathematics and its Applications, 400, Kluwer Academic Publishers Group, Dordrecht, 1997.  Google Scholar [5] S. Benenti, "Hamiltonian Structures and Generating Families," Universitext v223. Springer, 2011. Google Scholar [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., Aix-en-Provence/Salamanca, 1979), Lect. Notes in Math, Springer, Berlin, 836 (1980), 9-21.  Google Scholar [7] M. Chaperon, On generating families, 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] M. Crampin and F. A. E. Pirani, "Applicable Differential Geometry," Cambridge University Press, Cambridge, 1986.  Google Scholar [9] M. De Le and P. R. Rodrigues, "Methods of Differential Geometry in Analytical Mechanics," North-Holland Mathematics Studies, 158, North-Holland, Amsterdam, 1989.  Google Scholar [10] P. Dombrowski, On the geometry of the tangent bundle, J. Reine Angew. Math., 210 (1962), 73-88.  Google Scholar [11] O. Esen and H. Gümral, Lifts, jets and reduced dynamics, 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] J. Gancarzewicz, Liftings of functions and vector fields to natural bundles, 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] F. Gay-Balmaz and C. Vizman, Dual pairs in fluid dynamics, Ann. Glob. Anal. Geom., 41 (2012), 1-24. doi: 10.1007/s10455-011-9267-z.  Google Scholar [17] J. Gibbons, Collisionless Boltzmann equations and integrable moment equations, Physica D, 3 (1981), 503-511. doi: 10.1016/0167-2789(81)90036-1.  Google Scholar [18] V. Guillemin and S. Sternberg, The moment map and collective motion, Ann. of Phys., 127 (1980), 220-253. doi: 10.1016/0003-4916(80)90155-4.  Google Scholar [19] V. Guillemin and S. Sternberg, "Symplectic Techniques in Physics," Cambridge University Press (Cambridge), 1984.  Google Scholar [20] H. Gümral, Geometry of plasma dynamics I: Group of canonical diffeomorphisms, J. Math. Phys., 51 (2010), 23 pp. 083501.  Google Scholar [21] R. S. Ismagilov, M. Losik and P. W. Michor, A 2-cocycle on a group of symplectomorphisms, Moscow Math. J., 6 (2006), 307-315.  Google Scholar [22] A. N. Kaufman and R. L. Dewar, Canonical derivation of the Vlasov-Coulomb noncanonical Poisson structure, Cont. Math. AMS, 28 (1984), 51-54. doi: 10.1090/conm/028/751974.  Google Scholar [23] P. Kobak, Natural liftings of vector fields to tangent bundles of bundles of 1-forms, Math. Boch., 116 (1991), 319-326.  Google Scholar [24] S. Kobayashi and K. Nomizu, "Foundations of Differential Geometry Vol.1," Interscience Tract, No. 15, 1963.  Google Scholar [25] I. Kolar, P. W. Michor and J. Slovak, "Natural Operations in Differential Geometry," Springer-Verlag, Berlin, Heidelberg, 1993.  Google Scholar [26] Y. Kosmann-Schwarzbach, Vector fields and generalized vector fields on fibered manifolds, 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] A. J. Ledger and K. Yano, The tangent bundle of a locally symmetric space, J. London Math. Soc., 40 (1963), 487-492. doi: 10.1112/jlms/s1-40.1.487.  Google Scholar [28] P. Libermann and C. M. Marle, "Symplectic Geometry and Analytical Mechanics," D. Reidel Publishing Company, Kluwer Academic Publishers Group, 1987.  Google Scholar [29] J. E. Marsden, A correspondence principle for momentum operators, Can. Math. Bull., 10 (1967), 247-250. doi: 10.4153/CMB-1967-023-x.  Google Scholar [30] J. E. Marsden and T. S. Ratiu, "Introduction to Mechanics and Symmetry," Texts in Applied Mathematics, 17, Springer-Verlag, New York, 1994.  Google Scholar [31] J. E. Marsden and A. Weinstein, The Hamiltonian structure of the Maxwell-Vlasov equations, Physica D, 4 (1982), 394-406. doi: 10.1016/0167-2789(82)90043-4.  Google Scholar [32] J. E. Marsden, A group theoretic approach to the equations of plasma physics, Canad. Math. Bull., 25 (1982), 129-142. doi: 10.4153/CMB-1982-019-9.  Google Scholar [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, Atti della Academia della Scienze di Torino, 117 (1983), 289-340.  Google Scholar [34] J. E. Marsden, P. J. Morrison and A. Weinstein, Hamiltonian structure of the BBGKY hierarchy equations, Comtemp. Math., AMS, 28 (1984), 115-124.  Google Scholar [35] D. McDuff and D. Salamon, "Introduction to Symplectic Topology," Second edition. Oxford Mathematical Monographs. The Clarendon Press, Oxford University Press, 1998.  Google Scholar [36] P. Michor, Manifolds of smooth maps, Cahiers Top. Geo. Diff., 19 (1978), 47-78.  Google Scholar [37] P. J. Morrison, The Maxwell-Vlasov equations as a continuous Hamiltonian system, Phys. Lett. A, 80 (1980), 383-386. doi: 10.1016/0375-9601(80)90776-8.  Google Scholar [38] P. J. Morrison and J. M. Greene, Noncanonical Hamiltonian density formulation of hydrodynamics and magnetohydrodynamics, Phys. Rev. Lett., 45 (1980), 790-794. doi: 10.1103/PhysRevLett.45.790.  Google Scholar [39] P. J. Morrison, "Hamiltonian Field Description of One-Dimensional Poisson-Vlasov Equations," PPPL-1788, 1981. Google Scholar [40] P. J. Morrison, Poisson brackets for fluids and plasmas, in Mathematical Methods in Hydrodynamics and Integrability in Dynamical Systems, (La Jolla Institute, 1981) AIP Conf. Proc., (eds. M. Taber and Y. Treve), (AIP, New York), 88 (1982), 13-46. Google Scholar [41] P. J. Olver, "Applications of Lie Groups to Differential Equations," Graduate Texts in Mathematics, 107. Springer-Verlag, New York, 1986.  Google Scholar [42] Z. Pogoda, Horizontal lifts and foliations, Suppl. ai Rendiconti del Circolo Mat. di Palermo, 21 (1989), 279-283.  Google Scholar [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, (ed. V. Kac), Math. Sci. Res. Inst. Publ., Springer, New York, 4 (1985), 1-69.  Google Scholar [44] T. S. Ratiu and R. Schmid, The differentiable structure of three remarkable diffeomorphism groups, Mathematische Zeitschrift, 177 (1981), 81-100. doi: 10.1007/BF01214340.  Google Scholar [45] S. Sasaki, On the differential geometry of tangent bundles of Riemannian manifolds, Tohoku Math. J. (2), 10 (1958), 338-354. doi: 10.2748/tmj/1178244668.  Google Scholar [46] D. J. Saunders, "The Geometry of Jet Bundles," London Math. Soc., Lecture Notes Series, 142, Cambridge Univ. Press, 1989.  Google Scholar [47] C. Scovel and A. Weinstein, Finite-dimensional Lie-Poisson approximations to Vlasov-Poisson equations, Comm. Pure Appl. Math., 47 (1994), 683-709. doi: 10.1002/cpa.3160470505.  Google Scholar [48] J. Sniatycki and W. M. Tulczyjew, Generating forms of Lagrangian submanifolds, Indiana Univ. Math. J., 22 (1972), 267-275. doi: 10.1512/iumj.1972.22.22021.  Google Scholar [49] T. Swift, A note on the space of lagrangian submanifolds of a symplectic 4-manifold, Journal of Geometry and Physics, 35 (2000), 183-192. doi: 10.1016/S0393-0440(00)00008-5.  Google Scholar [50] S. Sternberg, "Lectures on Differential Geometry," Prentice-Hall, Inc., Englewood Cliffs, N. J., 1964  Google Scholar [51] S. Tanno, An almost complex structure of the tangent bundle of an almost contact manifold, Tohoku Math. J. (2), 17 (1965), 7-15. doi: 10.2748/tmj/1178243616.  Google Scholar [52] P. Tondeur, Structure presque klienne naturelle sur la fibrdes vecteurs covariants d'une vari riemannienne, C. R. Acad. Sci. Paris, 254 (1962), 407-408.  Google Scholar [53] W. M. Tulczyjew, The Legendre transformation, Ann. Inst. Henri PoincarSec. A: Phys. Thr., XXVII (1977), 101-114.  Google Scholar [54] W. M. Tulczyjew, A symplectic formulation of relativistic particle dynamics, Acta Physica Polonica B, 8 (1977), 431-447.  Google Scholar [55] W. M. Tulczyjew, A symplectic formulation of particle dynamics, Differential Geometric Methods in Mathematical Physics, Lect. Notes in Math., 570 (1977), 457-463. doi: 10.1007/BFb0087795.  Google Scholar [56] W. M. Tulczyjew, A symplectic framework of linear field theories, Ann. Mat. Pura Appl. (4), 130 (1981), 177-195.  Google Scholar [57] W. M. Tulczyjew, The Euler-Lagrange resolution, 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] W. M. Tulczyjew, Hamiltonian systems, Lagrangian systems and the Legendre transformation, Istituto Nazionale di Alta Matematica, Symposia Mathematica, 14 (1974), 247-258.  Google Scholar [59] W. M. Tulczyjew and P. Urbanski, A slow and careful Legendre transformation for singular Lagrangians, Acta Phys. Polon. B, 30 (1999), 2909-2978.  Google Scholar [60] K. Yano and E. T. Davies, On the tangent bundle of Finsler and Riemannian manifolds, Rend. Circ. Mat. Palermo (2), 12 (1963), 211-228. doi: 10.1007/BF02843966.  Google Scholar [61] K. Yano and S. Kobayashi, Prolongations of tensor fields and connections to tangent bundles I. General Theory, J. Math. Soc. Japan, 18 (1966), 194-210. doi: 10.2969/jmsj/01820194.  Google Scholar [62] K. Yano and A. J. Ledger, Linear connections on tangent bundles, J. London Math. Soc., 39 (1964), 495-500. doi: 10.1112/jlms/s1-39.1.495.  Google Scholar [63] K. Yano and E. M. Patterson, Vertical and complete lifts from a manifold to its cotangent bundle, J. Math. Soc. Japan, 19 (1967), 91-113. doi: 10.2969/jmsj/01910091.  Google Scholar [64] I. Vaisman, Locally conformal symplectic manifolds, Int. J. Math. and Math. Sci., 8 (1985), 521-536. doi: 10.1155/S0161171285000564.  Google Scholar [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), 610-620.  Google Scholar [66] C. Vizman, Some remarks on the quantomorphism group, Proc. of the Third International Workshop on Diff. Geom., Sibiu, Romania, Gen. Math., 5 (1997), 393-399.  Google Scholar [67] C. Vizman, Abelian extensions via prequantization, Annals of Global Analysis and Geometry, 39 (2011), 361-386.  Google Scholar [68] A. Weinstein, "Lectures on Symplectic Manifolds," Exp. lec. from the CBMS, Regional Conference Series in Mathematics, No. 29. A.M.S., Providence, 1977.  Google Scholar [69] A. Weinstein, Symplectic manifolds and their Lagrangian submanifolds, Advan. in Math., 6 (1971), 329-346. doi: 10.1016/0001-8708(71)90020-X.  Google Scholar [70] A. Weinstein, Lagrangian submanifolds and Hamiltonian systems, Annals of Math., 2 (1973), 377-410. doi: 10.2307/1970911.  Google Scholar [71] A. Weinstein and P. J. Morrison, Comments on: The Maxwell-Vlasov equation as a continuous Hamiltonian system, Phys. Lett. A., 86 (1981), 235-236. doi: 10.1016/0375-9601(81)90496-5.  Google Scholar

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##### References:
 [1] R. Abraham, J. E. Marsden and T. S. Ratiu, "Manifolds, Tensor Analysis, and Applications," Springer-Verlag, 2nd edition New York, 1988. doi: 10.1007/978-1-4612-1029-0.  Google Scholar [2] V. I. Arnold, "Mathematical Methods of Classical Mechanics," Graduate Texts in Mathematics, 60, Springer-Verlag, 1989.  Google Scholar [3] A. Banyaga, Sur la structure du groupe des diffmorphismes qui preservent une forme symplectique, Comment. Math. Helvetici, 53 (1978), 174-227. doi: 10.1007/BF02566074.  Google Scholar [4] A. Banyaga, "The Structure of Classical Diffeomorphism Groups," Mathematics and its Applications, 400, Kluwer Academic Publishers Group, Dordrecht, 1997.  Google Scholar [5] S. Benenti, "Hamiltonian Structures and Generating Families," Universitext v223. Springer, 2011. Google Scholar [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., Aix-en-Provence/Salamanca, 1979), Lect. Notes in Math, Springer, Berlin, 836 (1980), 9-21.  Google Scholar [7] M. Chaperon, On generating families, 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] M. Crampin and F. A. E. Pirani, "Applicable Differential Geometry," Cambridge University Press, Cambridge, 1986.  Google Scholar [9] M. De Le and P. R. Rodrigues, "Methods of Differential Geometry in Analytical Mechanics," North-Holland Mathematics Studies, 158, North-Holland, Amsterdam, 1989.  Google Scholar [10] P. Dombrowski, On the geometry of the tangent bundle, J. Reine Angew. Math., 210 (1962), 73-88.  Google Scholar [11] O. Esen and H. Gümral, Lifts, jets and reduced dynamics, 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] J. Gancarzewicz, Liftings of functions and vector fields to natural bundles, 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] F. Gay-Balmaz and C. Vizman, Dual pairs in fluid dynamics, Ann. Glob. Anal. Geom., 41 (2012), 1-24. doi: 10.1007/s10455-011-9267-z.  Google Scholar [17] J. Gibbons, Collisionless Boltzmann equations and integrable moment equations, Physica D, 3 (1981), 503-511. doi: 10.1016/0167-2789(81)90036-1.  Google Scholar [18] V. Guillemin and S. Sternberg, The moment map and collective motion, Ann. of Phys., 127 (1980), 220-253. doi: 10.1016/0003-4916(80)90155-4.  Google Scholar [19] V. Guillemin and S. Sternberg, "Symplectic Techniques in Physics," Cambridge University Press (Cambridge), 1984.  Google Scholar [20] H. Gümral, Geometry of plasma dynamics I: Group of canonical diffeomorphisms, J. Math. Phys., 51 (2010), 23 pp. 083501.  Google Scholar [21] R. S. Ismagilov, M. Losik and P. W. Michor, A 2-cocycle on a group of symplectomorphisms, Moscow Math. J., 6 (2006), 307-315.  Google Scholar [22] A. N. Kaufman and R. L. Dewar, Canonical derivation of the Vlasov-Coulomb noncanonical Poisson structure, Cont. Math. AMS, 28 (1984), 51-54. doi: 10.1090/conm/028/751974.  Google Scholar [23] P. Kobak, Natural liftings of vector fields to tangent bundles of bundles of 1-forms, Math. Boch., 116 (1991), 319-326.  Google Scholar [24] S. Kobayashi and K. Nomizu, "Foundations of Differential Geometry Vol.1," Interscience Tract, No. 15, 1963.  Google Scholar [25] I. Kolar, P. W. Michor and J. Slovak, "Natural Operations in Differential Geometry," Springer-Verlag, Berlin, Heidelberg, 1993.  Google Scholar [26] Y. Kosmann-Schwarzbach, Vector fields and generalized vector fields on fibered manifolds, 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] A. J. Ledger and K. Yano, The tangent bundle of a locally symmetric space, J. London Math. Soc., 40 (1963), 487-492. doi: 10.1112/jlms/s1-40.1.487.  Google Scholar [28] P. Libermann and C. M. Marle, "Symplectic Geometry and Analytical Mechanics," D. Reidel Publishing Company, Kluwer Academic Publishers Group, 1987.  Google Scholar [29] J. E. Marsden, A correspondence principle for momentum operators, Can. Math. Bull., 10 (1967), 247-250. doi: 10.4153/CMB-1967-023-x.  Google Scholar [30] J. E. Marsden and T. S. Ratiu, "Introduction to Mechanics and Symmetry," Texts in Applied Mathematics, 17, Springer-Verlag, New York, 1994.  Google Scholar [31] J. E. Marsden and A. Weinstein, The Hamiltonian structure of the Maxwell-Vlasov equations, Physica D, 4 (1982), 394-406. doi: 10.1016/0167-2789(82)90043-4.  Google Scholar [32] J. E. Marsden, A group theoretic approach to the equations of plasma physics, Canad. Math. Bull., 25 (1982), 129-142. doi: 10.4153/CMB-1982-019-9.  Google Scholar [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, Atti della Academia della Scienze di Torino, 117 (1983), 289-340.  Google Scholar [34] J. E. Marsden, P. J. Morrison and A. Weinstein, Hamiltonian structure of the BBGKY hierarchy equations, Comtemp. Math., AMS, 28 (1984), 115-124.  Google Scholar [35] D. McDuff and D. Salamon, "Introduction to Symplectic Topology," Second edition. Oxford Mathematical Monographs. The Clarendon Press, Oxford University Press, 1998.  Google Scholar [36] P. Michor, Manifolds of smooth maps, Cahiers Top. Geo. Diff., 19 (1978), 47-78.  Google Scholar [37] P. J. Morrison, The Maxwell-Vlasov equations as a continuous Hamiltonian system, Phys. Lett. A, 80 (1980), 383-386. doi: 10.1016/0375-9601(80)90776-8.  Google Scholar [38] P. J. Morrison and J. M. Greene, Noncanonical Hamiltonian density formulation of hydrodynamics and magnetohydrodynamics, Phys. Rev. Lett., 45 (1980), 790-794. doi: 10.1103/PhysRevLett.45.790.  Google Scholar [39] P. J. Morrison, "Hamiltonian Field Description of One-Dimensional Poisson-Vlasov Equations," PPPL-1788, 1981. Google Scholar [40] P. J. Morrison, Poisson brackets for fluids and plasmas, in Mathematical Methods in Hydrodynamics and Integrability in Dynamical Systems, (La Jolla Institute, 1981) AIP Conf. Proc., (eds. M. Taber and Y. Treve), (AIP, New York), 88 (1982), 13-46. Google Scholar [41] P. J. Olver, "Applications of Lie Groups to Differential Equations," Graduate Texts in Mathematics, 107. Springer-Verlag, New York, 1986.  Google Scholar [42] Z. Pogoda, Horizontal lifts and foliations, Suppl. ai Rendiconti del Circolo Mat. di Palermo, 21 (1989), 279-283.  Google Scholar [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, (ed. V. Kac), Math. Sci. Res. Inst. Publ., Springer, New York, 4 (1985), 1-69.  Google Scholar [44] T. S. Ratiu and R. Schmid, The differentiable structure of three remarkable diffeomorphism groups, Mathematische Zeitschrift, 177 (1981), 81-100. doi: 10.1007/BF01214340.  Google Scholar [45] S. Sasaki, On the differential geometry of tangent bundles of Riemannian manifolds, Tohoku Math. J. (2), 10 (1958), 338-354. doi: 10.2748/tmj/1178244668.  Google Scholar [46] D. J. Saunders, "The Geometry of Jet Bundles," London Math. Soc., Lecture Notes Series, 142, Cambridge Univ. Press, 1989.  Google Scholar [47] C. Scovel and A. Weinstein, Finite-dimensional Lie-Poisson approximations to Vlasov-Poisson equations, Comm. Pure Appl. Math., 47 (1994), 683-709. doi: 10.1002/cpa.3160470505.  Google Scholar [48] J. Sniatycki and W. M. Tulczyjew, Generating forms of Lagrangian submanifolds, Indiana Univ. Math. J., 22 (1972), 267-275. doi: 10.1512/iumj.1972.22.22021.  Google Scholar [49] T. Swift, A note on the space of lagrangian submanifolds of a symplectic 4-manifold, Journal of Geometry and Physics, 35 (2000), 183-192. doi: 10.1016/S0393-0440(00)00008-5.  Google Scholar [50] S. Sternberg, "Lectures on Differential Geometry," Prentice-Hall, Inc., Englewood Cliffs, N. J., 1964  Google Scholar [51] S. Tanno, An almost complex structure of the tangent bundle of an almost contact manifold, Tohoku Math. J. (2), 17 (1965), 7-15. doi: 10.2748/tmj/1178243616.  Google Scholar [52] P. Tondeur, Structure presque klienne naturelle sur la fibrdes vecteurs covariants d'une vari riemannienne, C. R. Acad. Sci. Paris, 254 (1962), 407-408.  Google Scholar [53] W. M. Tulczyjew, The Legendre transformation, Ann. Inst. Henri PoincarSec. A: Phys. Thr., XXVII (1977), 101-114.  Google Scholar [54] W. M. Tulczyjew, A symplectic formulation of relativistic particle dynamics, Acta Physica Polonica B, 8 (1977), 431-447.  Google Scholar [55] W. M. Tulczyjew, A symplectic formulation of particle dynamics, Differential Geometric Methods in Mathematical Physics, Lect. Notes in Math., 570 (1977), 457-463. doi: 10.1007/BFb0087795.  Google Scholar [56] W. M. Tulczyjew, A symplectic framework of linear field theories, Ann. Mat. Pura Appl. (4), 130 (1981), 177-195.  Google Scholar [57] W. M. Tulczyjew, The Euler-Lagrange resolution, 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] W. M. Tulczyjew, Hamiltonian systems, Lagrangian systems and the Legendre transformation, Istituto Nazionale di Alta Matematica, Symposia Mathematica, 14 (1974), 247-258.  Google Scholar [59] W. M. Tulczyjew and P. Urbanski, A slow and careful Legendre transformation for singular Lagrangians, Acta Phys. Polon. B, 30 (1999), 2909-2978.  Google Scholar [60] K. Yano and E. T. 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