Math @ Duke
Publications [#243396] of Robert Bryant
- with Bryant, R; Griffiths, P, Reduction for Constrained Variational Problems and � κ 2 2 ds,
American Journal of Mathematics, vol. 108 no. 3
pp. 525-525, JSTOR [MR88a:58044], [doi]
(last updated on 2019/09/24)
This paper gives an exposition of a way of computing
equations and the conservation laws for them that arise
in geometrically defined variational problems. The main
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 theta-functions.
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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