On the geometric approach to the motion of inertial mechanical systems

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 of smooth orientation-preserving diffeomorphisms of the circle. The periodic inviscid Burgers equation is the geodesic equation on that group, with the L2 right-invariant metric. However, the exponential map for this right-invariant metric is not a local diffeomorphism and the geometric structure is therefore deficient. On the other hand, the geodesic equation on for the H1 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 local diffeomorphism and that if two diffeomorphisms are sufficiently close, 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.
Type de document :
Article dans une revue
Journal of Physics A: Mathematical and Theoretical, IOP Publishing, 2002, 35, pp.R51-R79. 〈10.1088/0305-4470/35/32/201〉
Liste complète des métadonnées

https://hal.archives-ouvertes.fr/hal-00003269
Contributeur : Boris Kolev <>
Soumis le : samedi 13 novembre 2004 - 15:49:39
Dernière modification le : mercredi 10 octobre 2018 - 01:25:44

Lien texte intégral

Identifiants

Collections

Citation

Boris Kolev, Adrian Constantin. On the geometric approach to the motion of inertial mechanical systems. Journal of Physics A: Mathematical and Theoretical, IOP Publishing, 2002, 35, pp.R51-R79. 〈10.1088/0305-4470/35/32/201〉. 〈hal-00003269〉

Partager

Métriques

Consultations de la notice

270