The elastica formed by bent rods and considered in physics can be generalized to curves
in a Riemannian Manifold which are a Critical Point for

where is the Geodesic Curvature of , is a Real Number, and is closed or satisfies some specified boundary condition. The curvature of an elastica must satisfy

where is the signed curvature of , is the Gaussian Curvature of the oriented Riemannian surface along , is the second derivative of with respect to , and is a constant.

**References**

Barros, M. and Garay, O. J. ``Free Elastic Parallels in a Surface of Revolution.'' *Amer. Math. Monthly*
**103**, 149-156, 1996.

Bryant, R. and Griffiths, P. ``Reduction for Constrained Variational Problems and
.''
*Amer. J. Math.* **108**, 525-570, 1986.

Langer, J. and Singer, D. A. ``Knotted Elastic Curves in .'' *J. London Math. Soc.* **30**, 512-520,
1984.

Langer, J. and Singer, D. A. ``The Total Squared of Closed Curves.'' *J. Diff. Geom.* **20**, 1-22, 1984.

© 1996-9

1999-05-25