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.
@article{hal-00003269,
author = {Kolev, Boris and Constantin, Adrian},
title = {On the geometric approach to the motion of inertial mechanical systems},
journal = {HAL},
volume = {2002},
number = {0},
year = {2002},
language = {en},
url = {http://dml.mathdoc.fr/item/hal-00003269}
}
Kolev, Boris; Constantin, Adrian. On the geometric approach to the motion of inertial mechanical systems. HAL, Tome 2002 (2002) no. 0, . http://gdmltest.u-ga.fr/item/hal-00003269/