Adrian Constantin and Boris Kolev 2002 J. Phys. A: Math. Gen. 35 R51 doi:10.1088/0305-4470/35/32/201
On the geometric approach to the motion of inertial mechanical systems
REVIEW ARTICLEAdrian Constantin1 and Boris Kolev2
Show affiliationsTOPICAL REVIEW
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 with the L2 right-invariant metric. However, the exponential map for this right-invariant metric is not a C1 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 C1 local diffeomorphism and that if two diffeomorphisms are sufficiently close on , 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.
- PACS
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45.10.Na Geometrical and tensorial methods
- MSC
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70Hxx Hamiltonian and Lagrangian mechanics (See also 37Jxx)
- Subjects
- Dates
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Issue 32 (16 August 2002)
Received 31 May 2002, in final form 19 June 2002
Published 2 August 2002
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On the geometric approach to the motion of inertial mechanical systems
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