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.