On the geometric approach to the motion of inertial mechanical systems
(2002) In Journal of Physics A: Mathematical and General 35(32). p.5179 Abstract
 According to the principle of least action, the spatially periodic motions of onedimensional mechanical systems with no external forces are described in the Lagrangian formalism by geodesics on a manifoldconfiguration space, the group D of smooth orientationpreserving diffeomorphisms of the circle. The periodic inviscid Burgers equation is the geodesic equation on D with the L2 rightinvariant metric. However, the exponential map for this rightinvariant metric is not a C1 local diffeomorphism and the geometric structure is therefore deficient. On the other hand, the geodesic equation on D for the H1 rightinvariant metric is also a reexpression 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 onedimensional mechanical systems with no external forces are described in the Lagrangian formalism by geodesics on a manifoldconfiguration space, the group D of smooth orientationpreserving diffeomorphisms of the circle. The periodic inviscid Burgers equation is the geodesic equation on D with the L2 rightinvariant metric. However, the exponential map for this rightinvariant metric is not a C1 local diffeomorphism and the geometric structure is therefore deficient. On the other hand, the geodesic equation on D for the H1 rightinvariant metric is also a reexpression of a model in mathematical physics. We show that in this case the exponential map is a C1 local diffeomorphism and that if two diffeomorphisms are sufficiently close on D, they can be joined by a unique lengthminimizing geodesica 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:
https://lup.lub.lu.se/record/328289
 author
 Constantin, Adrian ^{LU} and Kolev, B
 organization
 publishing date
 2002
 type
 Contribution to journal
 publication status
 published
 subject
 in
 Journal of Physics A: Mathematical and General
 volume
 35
 issue
 32
 pages
 51  79
 publisher
 IOP Publishing
 external identifiers

 wos:000178047800003
 scopus:0042279206
 ISSN
 03054470
 DOI
 10.1088/03054470/35/32/201
 language
 English
 LU publication?
 yes
 id
 b410239454f64bc5ba09d7fbb1521cd5 (old id 328289)
 date added to LUP
 20160401 16:08:57
 date last changed
 20201208 05:06:40
@article{b410239454f64bc5ba09d7fbb1521cd5, abstract = {According to the principle of least action, the spatially periodic motions of onedimensional mechanical systems with no external forces are described in the Lagrangian formalism by geodesics on a manifoldconfiguration space, the group D of smooth orientationpreserving diffeomorphisms of the circle. The periodic inviscid Burgers equation is the geodesic equation on D with the L2 rightinvariant metric. However, the exponential map for this rightinvariant metric is not a C1 local diffeomorphism and the geometric structure is therefore deficient. On the other hand, the geodesic equation on D for the H1 rightinvariant metric is also a reexpression of a model in mathematical physics. We show that in this case the exponential map is a C1 local diffeomorphism and that if two diffeomorphisms are sufficiently close on D, they can be joined by a unique lengthminimizing geodesica 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 = {03054470}, language = {eng}, number = {32}, pages = {5179}, publisher = {IOP Publishing}, series = {Journal of Physics A: Mathematical and General}, title = {On the geometric approach to the motion of inertial mechanical systems}, url = {http://dx.doi.org/10.1088/03054470/35/32/201}, doi = {10.1088/03054470/35/32/201}, volume = {35}, year = {2002}, }