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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.
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