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Publications [#243396] of Robert Bryant
search www.ams.org.Papers Published
 with Griffiths, PA; Bryant, RL, Reduction for constrained variational problems and $\int{1\over 2}k\sp 2\,ds$,
Amer. J. Math., vol. 108 no. 3
(1986),
pp. 525570 [MR88a:58044]
(last updated on 2018/06/23)
Author's Comments: This paper gives an exposition of a way of computing
the EulerLagrange
equations and the conservation laws for them that arise
from symmetries
in geometrically defined variational problems. The main
technical advantage
of this method over the more classical Pontrjagin
Maximum Principle is
the way it avoids choosing coordinates that are not
needed, but works directly
on the invariant coframing of the group of symmetries.
Some extended examples are computed for Euler
elastica in space forms
and on surfaces of constant curvature.
Since this paper appeared, David Mumford has
shown how to get a complete
integration of the equations in the flat case by a very
clever use of thetafunctions.
It would be interesting to see if this would work also in
the case of space
elastica or for elastica in other space forms.


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