Systems of rays in the presence of distribution of hyperplanes

S. Janeczko

Banach Center Publications (1995)

  • Volume: 32, Issue: 1, page 245-260
  • ISSN: 0137-6934

Abstract

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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.

How to cite

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Janeczko, S.. "Systems of rays in the presence of distribution of hyperplanes." Banach Center Publications 32.1 (1995): 245-260. <http://eudml.org/doc/262790>.

@article{Janeczko1995,
abstract = {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.},
author = {Janeczko, S.},
journal = {Banach Center Publications},
keywords = {optical rays; Hamiltonian systems; distributions; singularities of Lagrange projections},
language = {eng},
number = {1},
pages = {245-260},
title = {Systems of rays in the presence of distribution of hyperplanes},
url = {http://eudml.org/doc/262790},
volume = {32},
year = {1995},
}

TY - JOUR
AU - Janeczko, S.
TI - Systems of rays in the presence of distribution of hyperplanes
JO - Banach Center Publications
PY - 1995
VL - 32
IS - 1
SP - 245
EP - 260
AB - 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.
LA - eng
KW - optical rays; Hamiltonian systems; distributions; singularities of Lagrange projections
UR - http://eudml.org/doc/262790
ER -

References

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  14. [14] R. S. Strichartz, Corrections to 'Sub-Riemannian Geometry' ibid. 30 (1989), 595-596. 
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  16. [16] A. M. Vershik and V. Ya. Gershkovich, Non-holonomic Riemannian manifolds, in: Dynamical Systems 7, Mathematical Encyclopaedia, vol. 16, 1987 (in Russian). 

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