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On the geometric approach to the motion of inertial mechanical systems. (English) Zbl 1039.37068
Summary: 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.
MSC:
37K65Hamiltonian systems on groups of diffeomorphisms and on manifolds of mappings and metrics
37K10Completely integrable systems, integrability tests, bi-Hamiltonian structures, hierarchies
58D05Groups of diffeomorphisms and homeomorphisms as manifolds