Elastica

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

$$F^\lambda(\gamma)=\int_\gamma (\kappa^2+\lambda),$$

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

$$0=2\kappa''(s)+\kappa^3(s)+2\kappa(s)G(s)-\lambda \kappa(s),$$

where $\kappa$ is the signed curvature of $\gamma$, $G(s)$ is the Gaussian Curvature of the oriented Riemannian surface $M$ along $\gamma$, $\kappa''$ is the second derivative of $\kappa$ with respect to $s$, and $\lambda$ 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 $\int(k^2/s)\,ds$.'' Amer. J. Math. 108, 525-570, 1986.

Langer, J. and Singer, D. A. ``Knotted Elastic Curves in $R^3$.'' 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.