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Toward Hamiltonian Dynamics with Continuous Hamiltonians

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Toward Hamiltonian Dynamics with Continuous Hamiltonians Research by KIAS Visitors Research by KIAS Visitors Newsletter Vol 3 Vol Newsletter The KIAS Research by KIAS Visitors Research Toward Hamiltonian Dynamics (2008), 217-229, [arXiv:grqc/0608118]. strongly coupled quantum field theory and with continuous Hamiltonians [9] S. Hollands, A. Ishibashi and R. M. Wald, A higher condensed matter which rely on the fascinating dimensional stationary rotating black hole must be subject of black holes in higher dimensions. axisymmetric, Commun. Math. Phys. 271, 699 (2007) [10] R. Emparan, T. Harmark, V. Niarchos and N. A. Obers, Blackfolds, Phys. Rev. Lett. 102, 191301 (2009) Augustin Banyaga [1] R. Penrose, Gravitational Collapse and Space- [arXiv:0902.0427 [hep-th]]. KIAS Visitor (Pennsylvania State University) Time Singularities, Phys.Rev.Lett. 14, 57 (1965). [11] R. Emparan, T. Harmark, V. Niarchos, N. A. [2] Roy P. Kerr, Gravitational Field of a Spinning Mass Obers and M. J. Rodriguez, The Phase Structure of as an example of Algebraically Special Metrics, Higher-Dimensional Black Rings and Black Holes, Phys.Rev.Lett. 11, 237 (1963). JHEP 0710, 110 (2007) [arXiv:0708.2181 [hep-th]]. [3] J. Michell, Letter by Rev. John Michell to Herny [12] M. Caldarelli, R. Emparan and M. J. Rodriguez, Abstract. In this expository note, we define continuous Hamiltonian dynamical systems Cavendish, Phil.Trans. Royal Soc. 75, 35 (1784). Black Rings in (Anti)-deSitter space, JHEP 0811, 011 Pierre S. Laplace, Exposition du systeme du Monde, (2008) [arXiv:0806.1954 [hep-th]]. and formulate some basic notions in their study. (1796). [13] H. Elvang and P. Figueras, Black Saturn, JHEP [4] Karl Schwarzschild, On the gravitational field 0705, 050 (2007) [arXiv:hep-th/0701035]. 1. Introduction of a mass point according to Einstein’s theory, [14] H. Iguchi and T. Mishima, Black di-ring and Symplectic geometry, a foundation of classical mechanics, belongs to the C∞ category: the objects are Sitzungsber. Preuss. Akad. Wiss. Berlin (Math. infinite nonuniqueness, Phys. Rev. D75, 064018 Phys.) 1916, 189 (1916) [arXiv:physics/9905030]. symplectic manifolds (smooth manifolds M equipped with closed non-degenerate 2-forms ω), and the morphisms (2007) [arXiv:hep-th/0701043]. J. Evslin and C. f f * 5 [5] R. Emparan and H. S. Reall, A rotating black ring Krishnan, The black di-ring: An inverse scattering are symplectic diffeomorphisms (smooth diffeomorphisms such that ω ω). Each (time-dependent) smooth in five dimensions, Phys. Rev. Lett. 88, 101101 (2002) 5 5 t t construction, Class. Quant. Grav. 26:125018, (2009) function H : (Ht) on M defines a Hamiltonian dynamical system HΦ : (Φ H), where Φ H are the solutions of the [arXiv:hep-th/0110260]. [arXiv:0706.1231 [hep-th]]. Hamilton equation [6] A. A. Pomeransky and R. A. Sen’kov, Black ring . [15] H. Elvang and M. J. Rodriguez, Bicycling Black x = X (x), with two angular momenta, arXiv:hep-th/0612005. Rings, JHEP 0804, 045 (2008) [arXiv:0712.2425 [hep- H and X is the Hamiltonian vector field defined by i(X )ω = dH . [7] G. J. Galloway and R. Schoen, A generalization th]]. H H t of Hawking’s black hole topology theorem to higher [16] D. Astefanesei, M. J. Rodriguez, S. Theisen, If the Hamiltonian function above was just continuous, nothing of the above made sense. The goal of this dimensions, Commun. Math. Phys. 266, 571 (2006) Thermodynamic instability of doubly spinning black expository note is to show that there is a way to give a meaning to Hamiltonian dynamics with continuous [arXiv:grqc/0509107]. objects, JHEP 1008, 046 (2010) [arXiv:1003.2421 Hamiltonians, and formulate some basic notions on the first steps of what we want to call continuous [8] G. J. Galloway, Rigidity of outer horizons and the [hep-th]]. topology of black holes, Commun. Anal. Geom. 16 Hamiltonian dynamical systems, or a continuous version of mechanics. The motions in that world will be called hameomorphisms [4], a nickname for “Hamiltonian homeomorphisms.” Time-dependent mechanics is modeled by contact geometry, which is also a C∞ object. One also can define quantomorphisms with continuous contact Hamiltonian of the prequantization of a symplectic manifold with Maria J. RodrigueziscurrentlyapostdoctoralresearcherattheMaxPlanckInstitutefor GravitationalPhysics-AlbertEinsteinInstitute(AEI)inGermanyconductingtheoreticalresearch integral periods [1]. In the continuous framework, we evoke the conservation of the energy, first integrals, onblackholesandquantumgravity.ShereceivedherbachelordegreeinphysicsfromUniversidad Hamiltonian actions and momentum maps. NacionaldeLaPlatainArgentina,andherMasterDiplomaofAdvancedStudiesaswellasthe Ph.D.fromtheUniversidaddeBarcelona,Spain. 2. Hamiltonian homeomorphisms Contactinformation:MaxPlanckInstituteforGravitationalPhysics(AlbertEinsteinInstitute),Am Let (M, ω) be a symplectic manifold. To each smooth family of smooth functions H 5 (H : M→ ) corresponds a Mühlenberg1,D-14476Golm,Germany.E-mail:[email protected] t 2000 Mathematics Subject Classification. 53D05; 53D35. Key words and phrases. Hamiltonian homeomorphisms, Hamiltonian topology, symplectic topology, strong symplectic homeomorphisms, C 0 symplectic topology. 34 Korea Institute for Advanced Study | 35 Research by KIAS Visitors Newsletter Vol 3 Vol Newsletter The KIAS 3. Some results in continuous Hamiltonian dynamical systems by KIAS Visitors Research family of vector fieldsX H, called Hamiltonian vector fields, with HamiltonianH , uniquely determined g by the equation As in mechanics, we call a continuous function f a “first integral” of a continuous Hamiltonian system ( t, ht) 5 if f(gt (x)) = f(x) for all t. In [5], Oh has already observed that h is a first integral of the dynamical system g( t, h) i(XH) ω dHt, 5 when h is autonomous. where i(.) is the operation of interior product by a vector field: i(XH) ω (x) ω (XH , x) for all vector fields x. We denote by ΦH the family of diffeomorphisms obtained by integrating the differential 3.1. Conservation of energy. The energy h is conserved along the trajectory of the system (gt). Therefore gt equation induces a homeomorphism of each level set of h. We have the following result. x = XH(x) t 1 2 when H has compact support. The family of maps ΦH:= ΦH is called a Hamiltonian isotopy, Proposition 6. Let ΦHn , ΦHn be two sequences defining two continuous Hamiltonian systems (g1, h1), (g2, h2). 1 2 i g g generated by the Hamiltonian (Ht). We will denote by HIso(M, ω) the set of all Hamiltonian Suppose that {Hn , Hn } = 0 and Hn are time-independent, then h1, h2 are both first integrals of both 1 and 2. isotopies and by Ham(M, ω) the set of all time 1 maps of Hamiltonian isotopies. The set Ham( Definition 7. A continuous Hamiltonian action of a topological group G on a symplectic manifold (M, ω) is a M, ω) is a group called the group of Hamiltonian diffeomorphisms. It plays a central role in the homomorphism u : G → Hameo(M, ω). If G is a Lie group and is its Lie algebra, a continuous Hamiltonian development of symplectic geometry and topology (Arnold conjecture, Floer homology, ...). action u of G induces an action û : →Hameo(M, ω) : x →u(exp(x)). Definition 1. A continuous Hamiltonian system is a continuous path (gt) of homeomorphisms of M with g0 = identity, such that there exists a sequence of Hamiltonian isotopies ΦHk converging 3.2. The momentum map. A continuous Hamiltonian action u on a Lie group G gives raise to a momentum k uniformly to gt, and such that the Hamiltonians H converge uniformly to some continuous map * function ht. I : M → If (x) Œ and u(exp(tx)) = g, and g is generated by a time-independent function h [5], then we set We have the following fundamental result due to Viterbo [6]. I(x)(x) = h(x). 0 Theorem 2. H H We hope to be able to define more notions of classical mechanics and prove interesting theorems in the Let Φ k ; Φ k9 be two Hamiltonian isotopies, which converge in the C topology (uniformly) to the same path gtŒ Homeo(M). If Hk and Hk9 converge uniformly to continuous continuous setting in future papers. functions H and H9, then H = H9. Hence the limiting path of homeomorphism g(t) determines References uniquely the limiting Hamiltonian h = H = H9. [1] A. Banyaga and P. W. Spaeth, The C0-contact topology and the group of contact homeomorphisms, preprint. [2] L. Buhovsky and S. Seyfaddini, Uniqueness of generating Hamiltonians for continuous Hamiltonian flows, preprint. Therefore to a continuous Hamiltonian system (gt) corresponds a unique continuous function h [3] S. Müller, The group of Hamiltonian homeomorphisms in the L∞ norm, J. Korean Math. Soc. 45 (2008)1769-1784. = (ht), we call the “generator” of (gt). It will play the role of the “energy” of the system evolving [4] Y.-G. Oh and S. Müller, The group of Hamiltonian homeomorphisms and C0-symplectic topology, J. Symp. Geometry 5 (2007) under the motion (gt). 167-225. Remark 3. Viterbo’s theorem is a deep fact. Its proof is intricate and uses a deep result by [5] Y.-G. Oh, The group of Hamiltonian homeomorphisms and continuous Hamiltonian flows, Contemporary Math. Vol 512 Gromov in “hard” symplectic topology. A more elementary (and more general) proof has recently (2010) 149-177 been given by Buhovsky and Seyfaddini [2]. [6] C. Viterbo, On the uniqueness of generating Hamiltonians for continuous limits of Hamiltonian flows, Inter. Math. Res. Notices, vol 2006, article ID 34028, 9 pages; Erratum, ibid, vol 2006, article ID 38784, 4 pages. Definition 4. g g The time 1 map 1 of a continuous Hamiltonian system ( t) is called a Hamiltonian homeomorphism “generated” by the “Hamiltonian” h = (ht). Hamiltonian homeomorphisms have Augustin Banyaga isaProfessorintheMathematicsDepartmentofthePennsylvaniaState been nicknamed hameomorphisms [4]. University.AfterobtaininghisPhDfromtheUniversityofGenevain1976,heheldregularpositions attheUniversityofGeneva,IASPrinceton,HarvardUniversity,andBostonUniversity.Professor We will denote by Hameo(M, ω) the set of all hameomorphisms of (M, ω). The following result Banyagahaspublishedover50researcharticlesandseveralmonographs.Hismainresearch interestsaresymplecticandcontactgeometryandtopology,andhecurrentlyworksonlocally is due to Müller and Oh [3,4].
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