What are the conditions for the union of two Elastica curves to be an Elastica curve as well? An Elastica curve is defined as one that minimises the bending energy, i.e. the curvature squared. Suppose I have two Elastica curves, and I decide to join them end to end. Then what are the conditions at the joint for the final curve (the union of the two curves) to be also an Elastica?
I presume one necessary condition is smoothness, i.e. continuity of the first derivative. Otherwise there will be infinite bending energy at the joint. Do we also need continuity of the second derivative? If yes, could you explain why?
Is there a theorem regarding this? Could someone give me a reference?
P.S. I presume that what I am describing above is actually spline curves.

UPDATE
The two responses below answer most of my query, one in the the applied context of splines/Bézier curves, and the other in the abstract context of the Elastica equation itself.
I have one remaining doubt, which I ask in a separate question: Can an Elastica curve have a discontinuity in the curvature?

It really depends what you mean by “elastica curves”.
In most applications, true elastica curves (mimimizing bending energy) are not used; some approximation is used instead.
Suppose you approximated each elastica curve by a Bézier curve of degree three). Then two such curves could be joined to form a single Bézier curve iff all derivatives up to order three match at the junction.
Another way to approximate an elastica curve is by using a (cubic) spline curve, i.e. a piece wise cubic polynomial. Joining together two cubic splines always gives you a cubic spline.
