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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:
 37K65 Hamiltonian systems on groups of diffeomorphisms and on manifolds of mappings and metrics 37K10 Completely integrable systems, integrability tests, bi-Hamiltonian structures, hierarchies 58D05 Groups of diffeomorphisms and homeomorphisms as manifolds