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On the Poisson Algebra of a Singular Map

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On the Poisson Algebra of a Singular Map Journal of Geometry and Physics 86 (2014) 194–202 Contents lists available at ScienceDirect Journal of Geometry and Physics journal homepage: www.elsevier.com/locate/jgp On the Poisson algebra of a singular map T. Fukuda a, S. Janeczko b,c,∗,1 a Department of Mathematics, College of Humanities and Sciences, Nihon University, Sakurajousui 3-25-40, Setagaya-ku, 156-8550 Tokyo, Japan b Warsaw University of Technology, Faculty of Mathematics and Information Science, Plac Politechniki 1, 00-661 Warszawa, Poland c Institute of Mathematics, Polish Academy of Sciences, ul. Sniadeckich 8, 00-956 Warsaw, Poland article info a b s t r a c t Article history: We construct the Poisson algebra associated to a singular mapping into symplectic space Received 30 April 2014 and show that this is an algebra of smooth functions generating solvable implicit Hamilto- Accepted 31 July 2014 nian systems. Available online 7 August 2014 ' 2014 Elsevier B.V. All rights reserved. MSC: primary 53D05 secondary 58K05 57R42 58A10 Keywords: Symplectic manifold Hamiltonian system Solvability Singularities 1. Introduction Let .M; !/ be a smooth, symplectic 2n-dimensional manifold and let .TM; !/P be its tangent bundle endowed with a symplectic structure !P . This structure is defined by the Liouville form θ using the canonical flat morphism β V TM 3 v 7! !.v; ·/ 2 T ∗M, between tangent and cotangent bundles of the symplectic manifold .M; !/; !P D β∗dθ. N 2n Let F V R ⊃ .U; 0/ ! .M; !/ be a smooth map-germ. With this mapping we associate all smooth, isotropic map-germs F V .U; 0/ ! .TM; !/P which are vector fields along FN such that FN D π ◦ F; π V TM ! M, and F ∗!P D 0. This is a generalization of standard Hamiltonian systems to include the implicit case (see [1,2]). For each isotropic F we have ∗ ∗ ∗ d(β ◦ F/ θ D F !P D 0. Thus (β ◦ F/ θ is a germ of a closed 1-form, so there exists a smooth function-germ h V .U; 0/ ! R which we call the generating function for F, such that (β ◦ F/∗θ D −dh. N We consider the space RFN of generating functions of isotropic map-germs F along a fixed singular map-germ F. Among the h 2 RFN appear such ones that the vector field Xh on U defined by the formula N ∗ F !.Xh; ξ/ D −ξ.h/ for each vector field ξ on U; is smooth. In this case the generating functions are called Hamiltonians and we denote by HFN the space of Hamiltonians. In this paper we investigate the spaces RFN and HFN . We show (Theorem 3.2) that the space HFN of Hamiltonians endowed with N ∗ N the brackets fk; hgFN∗! VD F !.Xk; Xh/ forms a Poisson algebra associated to F, which is not extendable to RFN . ∗ Corresponding author at: Institute of Mathematics, Polish Academy of Sciences, ul. Sniadeckich 8, 00-956 Warsaw, Poland. Tel.: +48 222346003. E-mail addresses: [email protected] (T. Fukuda), [email protected] (S. Janeczko). 1 S. Janeczko was partially supported by NCN grant no. DEC-2013/11/B/ST1/03080. http://dx.doi.org/10.1016/j.geomphys.2014.07.037 0393-0440/' 2014 Elsevier B.V. All rights reserved. T. Fukuda, S. Janeczko / Journal of Geometry and Physics 86 (2014) 194–202 195 For each isotropic F associated to FN we consider N VD F.U/ ⊂ .TM; !/P as an implicit differential system, possibly with singularities of the projection π ◦ F. We investigate the smooth solvability of such systems. The point .p; pP/ 2 N is called V − ! D 0 D P D 0 2 solvable if there exists a smooth curve γp . ϵ; ϵ/ M such that γp.0/ p; γp.0/ p and κp.t/ (γp.t/; γp.t// N for all t 2 .−ϵ; ϵ/; ϵ > 0. If κ is smooth in the neighborhood of p we call it smoothly solvable. N is called smoothly solvable if it consists only of smoothly solvable points. We show (Theorem 5.1) that F is smoothly solvable if and only if its generating function belongs to the Poisson algebra associated to FN. For each symplectically invariant singularity of FN we have its Poisson algebra which is the fundamental object of singularity theory endowed with an additional structure. In Section2 isotropic mappings into symplectic tangent bundle to the symplectic manifold are investigated through their generating functions. The Hamiltonian functions are distinguished and their Poisson algebra structure is introduced in Section3. Solvability criteria for isotropic mappings are studied in Section4 and the main theorem of the paper saying that the only solvable Hamiltonian systems are those generated by Hamiltonian generating functions is proved in Section5. An investigation of Poisson algebras associated to singular mappings into a symplectic manifold and their representative examples are presented in Section6. 2. Generating functions of isotropic mappings 2n D Pn ^ Let .R ; !/ be a Euclidean symplectic space endowed with ! iD1 dyi dxi in canonical Darboux coordinates .x; y/ D .x1;:::; xn; y1;:::; yn/. ∗ 2n ∗ 2n Let θ be the Liouville 1-form on the cotangent bundle T R . Then dθ is the standard symplectic structure on T R . Let 2n ∗ 2n β V T R ! T R be the canonical bundle map defined through ! by 2n ∗ 2n β V T R 3 v 7! !.v; ·/ 2 T R : 2n Then we can define (cf. [3]) the canonical symplectic structure !P on T R by n ∗ ∗ X !P D β dθ D d(β θ/ D .dyPi ^ dxi − dxPi ^ dyi/; iD1 P P 2n ∗ D Pn P − P where .x; y; x; y/ are local coordinates on T R and β θ iD1.yidxi xidyi/. 2n Throughout the paper, unless otherwise stated, all objects are germs at 0 2 R of smooth functions, mappings, forms 2n etc. or their representatives on an open neighborhood of 0 in R . 2n 2n ∗ Definition 2.1. Let F V .R ; 0/ ! T R be a smooth map-germ. We say that F is isotropic if F !P D 0. 2n 2n ∗ If a map-germ F V .R ; 0/ ! T R is isotropic, then the germ of a differential of the 1-form (β ◦ F/ θ vanishes; d(β ◦ F/∗θ D F ∗β∗dθ D F ∗!P D 0. Thus (β ◦ F/∗θ is the germ of a closed 1-form and there exists a smooth function-germ 2n h V .R ; 0/ ! R such that ∗ (β ◦ F/ θ D −dh: (2.1) We call h a generating function for F. For each smooth isotropic map-germ F, its generating function is unique up to an additive constant. 2n 2n Let U be an open neighborhood of 0 in the source space R and let F V .U; 0/ ! T R be an isotropic map-germ. Let N 2n F D π ◦ F V .U; 0/ ! T R ; 2n 2n N where π V T R ! R is the tangent bundle projection. We express F and F in the form N 2n P 2n F D .f ; g/ V .U; 0/ ! R and F D .f ; g; f ; gP/ V .U; 0/ ! T R respectively. By (and by 2n respectively) we denote the -algebra of smooth function-germs at 0 on U (and on the target space EU ER R 2n respectively). Let m (m 2n respectively) denote the maximal ideal in (in 2n respectively). R U R EU ER N N ∗ 2n In general F can be regarded as a vector field along F, i.e. a section of an induced fiber bundle F T R . To each isotropic N map-germ F along F, there exists a unique generating function-germ h 2 mU for F. Now we introduce a natural equivalence group acting on isotropic mappings through a natural lifting of diffeomorphic or symplectic equivalences of FN and GN . Before introducing it, we first recall the standard equivalence relations of smooth N 2n N 2n map-germs and of Lagrange projections. Two map-germs F V .U; 0/ ! R and G V .U; 0/ ! R are symplectomorphic if 2n 2n ∗ there exists a diffeomorphism-germ ' V .U; 0/ ! .U; 0/ and a symplectomorphism-germ Φ V .R ; 0/ ! .R ; 0/; Φ ! D N N 2n 2n !, such that G D Φ ◦ F ◦ '. Two isotropic mappings F V .U; 0/ ! T R and G V .U; 0/ ! T R are Lagrangian equivalent 2n (L-equivalent [4]) if there exist a diffeomorphism-germ ' V .U; 0/ ! .U; 0/ and a symplectomorphism-germ Ψ V .T R ; 2n ∗ F.0// ! .T R ; G.0//; Ψ !P DP!, preserving the fibering π such that G D Ψ ◦ F ◦ '. 2n 2n N Definition 2.2. Let F V .U; 0/ ! T R and G V .U; 0/ ! T R be two isotropic map-germs along smooth map-germs F V 2n N 2n .U; 0/ ! R and G V .U; 0/ ! R respectively. We say that F and G are L-symplectic equivalent if there exist a 196 T. Fukuda, S. Janeczko / Journal of Geometry and Physics 86 (2014) 194–202 2n 2n diffeomorphism-germ ' V .U; 0/ ! .U; 0/, a symplectomorphism-germ Ψ V .T R ; F.0// ! .T R ; G.0// preserving 2n 2n ∗ the fibering π and a symplectomorphism-germ Φ V .R ; 0/ ! .R ; 0/; Φ ! D ! such that π ◦ Ψ D Φ ◦ π; G D Ψ ◦ F ◦ ' and GN D Φ ◦ FN ◦ '. In this case we also say that F and G are symplectomorphic or symplectically equivalent.
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