Homogeneous variational problems and Lagrangian sections

D.J. Saunders

Communications in Mathematics (2016)

  • Volume: 24, Issue: 2, page 115-123
  • ISSN: 1804-1388

Abstract

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We define a canonical line bundle over the slit tangent bundle of a manifold, and define a Lagrangian section to be a homogeneous section of this line bundle. When a regularity condition is satisfied the Lagrangian section gives rise to local Finsler functions. For each such section we demonstrate how to construct a canonically parametrized family of geodesics, such that the geodesics of the local Finsler functions are reparametrizations.

How to cite

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Saunders, D.J.. "Homogeneous variational problems and Lagrangian sections." Communications in Mathematics 24.2 (2016): 115-123. <http://eudml.org/doc/287911>.

@article{Saunders2016,
abstract = {We define a canonical line bundle over the slit tangent bundle of a manifold, and define a Lagrangian section to be a homogeneous section of this line bundle. When a regularity condition is satisfied the Lagrangian section gives rise to local Finsler functions. For each such section we demonstrate how to construct a canonically parametrized family of geodesics, such that the geodesics of the local Finsler functions are reparametrizations.},
author = {Saunders, D.J.},
journal = {Communications in Mathematics},
keywords = {Finsler geometry; line bundle; geodesics; Finsler geometry; line bundle; geodesics},
language = {eng},
number = {2},
pages = {115-123},
publisher = {University of Ostrava},
title = {Homogeneous variational problems and Lagrangian sections},
url = {http://eudml.org/doc/287911},
volume = {24},
year = {2016},
}

TY - JOUR
AU - Saunders, D.J.
TI - Homogeneous variational problems and Lagrangian sections
JO - Communications in Mathematics
PY - 2016
PB - University of Ostrava
VL - 24
IS - 2
SP - 115
EP - 123
AB - We define a canonical line bundle over the slit tangent bundle of a manifold, and define a Lagrangian section to be a homogeneous section of this line bundle. When a regularity condition is satisfied the Lagrangian section gives rise to local Finsler functions. For each such section we demonstrate how to construct a canonically parametrized family of geodesics, such that the geodesics of the local Finsler functions are reparametrizations.
LA - eng
KW - Finsler geometry; line bundle; geodesics; Finsler geometry; line bundle; geodesics
UR - http://eudml.org/doc/287911
ER -

References

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  1. Chern, S.-S., Finsler geometry is just Riemannian geometry without the quadratic restriction, Not. A.M.S., 43, 9, 1996, 959-963, (1996) Zbl1044.53512MR1400859
  2. Crampin, M., Some remarks on the Finslerian version of Hilbert's Fourth Problem, Houston J. Math., 37, 2, 2011, 369-391, (2011) Zbl1228.53085MR2794554
  3. Crampin, M., Mestdag, T., Saunders, D.J., 10.1016/j.difgeo.2012.07.004, Diff. Geom. Appl., 30, 6, 2012, 604-621, (2012) Zbl1257.53105MR2996856DOI10.1016/j.difgeo.2012.07.004
  4. Crampin, M., Saunders, D.J., 10.1016/j.geomphys.2006.03.007, J. Geom. Phys., 57, 2, 2007, 691-727, (2007) Zbl1114.53014MR2271212DOI10.1016/j.geomphys.2006.03.007
  5. Hebda, J., Roberts, C., Examples of Thomas--Whitehead projective connections, Diff. Geom. Appl., 8, 1998, 87-104, (1998) Zbl0897.53009MR1601526
  6. Massa, E., Pagani, E., Lorenzoni, P., 10.1080/00411450008205861, Transport Theory and Statistical Physics, 29, 1--2, 2000, 69-91, (2000) Zbl0968.70014MR1774182DOI10.1080/00411450008205861
  7. Roberts, C., 10.1016/0926-2245(95)92848-Y, Diff. Geom. Appl., 5, 1995, 237-255, (1995) Zbl0833.53023MR1353058DOI10.1016/0926-2245(95)92848-Y
  8. Thomas, T.Y., 10.1007/BF01283864, Math. Zeit., 25, 1926, 723-733, (1926) MR1544836DOI10.1007/BF01283864

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