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1995 | 32 | 1 | 245-260
Tytuł artykułu

Systems of rays in the presence of distribution of hyperplanes

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Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
Horizontal systems of rays arise in the study of integral curves of Hamiltonian systems $v_H$ on T*X, which are tangent to a given distribution V of hyperplanes on X. We investigate the local properties of systems of rays for general pairs (H,V) as well as for Hamiltonians H such that the corresponding Hamiltonian vector fields $v_H$ are horizontal with respect to V. As an example we explicitly calculate the space of horizontal geodesics and the corresponding systems of rays for the canonical distribution on the Heisenberg group. Local stability of systems of horizontal rays based on the standard singularity theory of Lagrangian submanifolds is also considered.
Rocznik
Tom
32
Numer
1
Strony
245-260
Opis fizyczny
Daty
wydano
1995
Twórcy
autor
  • Institute of Mathematics, Warsaw University of Technology, Pl. Politechniki 1, 00-661 Warszawa, Poland
Bibliografia
  • [1] V. I. Arnold, S. M. Gusein-Zade and A. N. Varchenko, Singularities of Differentiable Maps, Vol. 1, Birkhäuser, Boston, 1985.
  • [2] V. I. Arnold, Lagrangian submanifolds with singularities, asymptotic rays and the open swallowtail, Funktsional. Anal. i Prilozhen. 15 (4) (1981), 1-14 (in Russian).
  • [3] S. Bloch, The dilogarithm and extensions of Lie algebras, in: Lecture Notes in Math. 854, Springer 1981, 1-23.
  • [4] Yu. V. Chekanov, Caustics in geometrical optics, Functional. Anal. Appl. 30 (1986), 223-226.
  • [5] J. Guckenheimer, Caustics and nondegenerate Hamiltonians, Topology 13 (1974), 127-133.
  • [6] V. Guillemin and S. Sternberg, Symplectic Techniques in Physics, Cambridge Univ. Press, Cambridge, 1984.
  • [7] S. Janeczko, Generalized Luneburg canonical varieties and vector fields on quasicaustics, J. Math. Phys. 31 (1991), 997-1009.
  • [8] S. Janeczko and R. Montgomery, On systems of gliding rays in sub-Riemannian geometry, to appear.
  • [9] J. E. Marsden, Lectures on Mechanics, London Math. Soc. Lecture Note Ser. 174, Cambridge Univ. Press 1992.
  • [10] J. Martinet, Singularities of Smooth Functions and Maps, Cambridge Univ. Press, Cambridge, 1982.
  • [11] J. Martinet, Sur les singularités des formes différentielles, Ann. Inst. Fourier (Grenoble) 20 (1970), 95-178.
  • [12] J. Mitchell, On Carnot-Carathéodory metrics, J. Differential Geometry 21 (1985), 35-45.
  • [13] R. S. Strichartz, Sub-Riemannian geometry, ibid. 24 (1986), 221-263.
  • [14] R. S. Strichartz, Corrections to 'Sub-Riemannian Geometry' ibid. 30 (1989), 595-596.
  • [15] A. Weinstein, Lectures on Symplectic Manifolds, CBMS Regional Conf. Ser. in Math. 29, Amer. Math. Soc., 1977.
  • [16] A. M. Vershik and V. Ya. Gershkovich, Non-holonomic Riemannian manifolds, in: Dynamical Systems 7, Mathematical Encyclopaedia, vol. 16, 1987 (in Russian).
Typ dokumentu
Bibliografia
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bwmeta1.element.bwnjournal-article-bcpv32z1p245bwm
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