THE RESULTS obtained here have to do with the following problem. Imagine the ends of a straight length of springy wire are joined together smoothly and the wire is held in some configuration described by an immersion y of the circle into the plane or into R3. According to the Bernoulli-Euler theory of elastic rods the bending energy of the wire is proportional to the total squared curvature of y, which we will denote by F (7) = 5, k2 ds. Suppose now the wire is released and it moves so as to decrease its bending energy as efficiently as possible, i.e. following “the negative gradient of F” (so our dynamics are Aristotelian rather than Newtonian, and we are also making the physically unrealistic assumption that the wire can pass through itself freely). How does the wire evolve, and what will happen ultimately (as time goes to infinity)? Of course, one wants to know first that one can actually define such a flow on the space of immersed circles, that it exists for all time, and that one can sensibly speak of a limiting curve yrn for the trajectory through a given initial curve yo. It is shown here that this is indeed the case and that in fact the Palais-Smale condition holds for this flow. It is proved, moreover, that if y0 is a plane curve of rotation index one (e.g. if y0 is embedded) then the flow carries y0 to a circle. Our main result, however, pertains to the non-planar case, where the situation is more complicated. In a space form it is possible to integrate the equations for an elastica, i.e. for a critical point of F, and this enables one to prove, in particular, that there is a countably infinite family of (similarity classes of) closed elastic curves in R3 (see Theorem 0.1). Thus, not all wire loops in R3 will flow to a circle. On the other hand, this leaves open the possibility that ycu is a circle for almost any initial curve yO, and indeed, our concluding Theorem 3.2 states that the circle is the only stable closed elastica in R3. The proof of this theorem itself depends on the dynamical, i.e. gradient flow approach to the study of F (and avoids a detailed analysis of the Hessian of F, which is quite complicated for non-planar elastic curves). The idea is as follows. One considers a discrete group G of rotations of R3 and an associated pair of multiply covered circular elastic curves which are Gequivariantly regularly homotopic, and which are both local minima for the restriction of F to G-symmetric curves (though multiple circles are unstable with respect to general variations). An appeal to the minimax and symmetric criticality principles then enables one to conclude that there exists a non-circular elastica of “saddle type”. Comparision with the classification theorem shows that one can account in this way for all non-circular solutions, hence all are unstable. We remark that a similar critical structure occurs for “free” (length unconstrained) elastic curves in the standard two-sphere: it was shown in [S] (by an entirely different method) that all closed non-geodesic solutions in S2 are unstable and can be regarded as minimax critical points arising from symmetrical regular homotopies between certain multiple coverings of a prime geodesic (though the minimax argument in [SJ is made only heuristically). To the extent that a similar picture holds as well for manifolds of (non-constant) positive curvature one gains a new view of closed geodesics as the limits of almost all trajectories of -VF. The organization of the paper is as follows. Section 0 is a brief review of some basic facts concerning elastic curves in space forms and the classification of closed elastic curves in R3 (details can be found in [S], [6] ). Section 1 is devoted mostly to the proof ofcondition (C)for the curve straightening flow. We have included details and have attempted to keep the discussion as self-contained as possible. In Section 2 we derive a second variation formula