Papers Published
Author's Comments:
This paper gives an exposition of a way of computing
the Euler-Lagrange
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 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.