Advanced

On the geometric approach to the motion of inertial mechanical systems

Constantin, Adrian LU and Kolev, B (2002) In Journal of Physics A: Mathematical and General1975-01-01+01:002007-01-01+01:00 35(32). p.51-79
Abstract
According to the principle of least action, the spatially periodic motions of one-dimensional mechanical systems with no external forces are described in the Lagrangian formalism by geodesics on a manifold-configuration space, the group D of smooth orientation-preserving diffeomorphisms of the circle. The periodic inviscid Burgers equation is the geodesic equation on D with the L-2 right-invariant metric. However, the exponential map for this right-invariant metric is not a C-1 local diffeomorphism and the geometric structure is therefore deficient. On the other hand, the geodesic equation on D for the H-1 right-invariant metric is also a re-expression of a model in mathematical physics. We show that in this case the exponential map is a... (More)
According to the principle of least action, the spatially periodic motions of one-dimensional mechanical systems with no external forces are described in the Lagrangian formalism by geodesics on a manifold-configuration space, the group D of smooth orientation-preserving diffeomorphisms of the circle. The periodic inviscid Burgers equation is the geodesic equation on D with the L-2 right-invariant metric. However, the exponential map for this right-invariant metric is not a C-1 local diffeomorphism and the geometric structure is therefore deficient. On the other hand, the geodesic equation on D for the H-1 right-invariant metric is also a re-expression of a model in mathematical physics. We show that in this case the exponential map is a C-1 local diffeomorphism and that if two diffeomorphisms are sufficiently close on D, they can be joined by a unique length-minimizing geodesic-a state of the system is transformed to another nearby state by going through a uniquely determined flow that minimizes the energy. We also analyse for both metrics the breakdown of the geodesic flow. (Less)
Please use this url to cite or link to this publication:
author
organization
publishing date
type
Contribution to journal
publication status
published
subject
in
Journal of Physics A: Mathematical and General1975-01-01+01:002007-01-01+01:00
volume
35
issue
32
pages
51 - 79
publisher
IOP Publishing
external identifiers
  • wos:000178047800003
  • scopus:0042279206
ISSN
0305-4470
DOI
10.1088/0305-4470/35/32/201
language
English
LU publication?
yes
id
b4102394-54f6-4bc5-ba09-d7fbb1521cd5 (old id 328289)
date added to LUP
2007-08-14 16:41:27
date last changed
2017-12-10 04:30:51
@article{b4102394-54f6-4bc5-ba09-d7fbb1521cd5,
  abstract     = {According to the principle of least action, the spatially periodic motions of one-dimensional mechanical systems with no external forces are described in the Lagrangian formalism by geodesics on a manifold-configuration space, the group D of smooth orientation-preserving diffeomorphisms of the circle. The periodic inviscid Burgers equation is the geodesic equation on D with the L-2 right-invariant metric. However, the exponential map for this right-invariant metric is not a C-1 local diffeomorphism and the geometric structure is therefore deficient. On the other hand, the geodesic equation on D for the H-1 right-invariant metric is also a re-expression of a model in mathematical physics. We show that in this case the exponential map is a C-1 local diffeomorphism and that if two diffeomorphisms are sufficiently close on D, they can be joined by a unique length-minimizing geodesic-a state of the system is transformed to another nearby state by going through a uniquely determined flow that minimizes the energy. We also analyse for both metrics the breakdown of the geodesic flow.},
  author       = {Constantin, Adrian and Kolev, B},
  issn         = {0305-4470},
  language     = {eng},
  number       = {32},
  pages        = {51--79},
  publisher    = {IOP Publishing},
  series       = {Journal of Physics A: Mathematical and General1975-01-01+01:002007-01-01+01:00},
  title        = {On the geometric approach to the motion of inertial mechanical systems},
  url          = {http://dx.doi.org/10.1088/0305-4470/35/32/201},
  volume       = {35},
  year         = {2002},
}