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 normal
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.
."
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.