I want to discuss the problem from differential geometry of describing those plane curves C which minimize the integral
$$\int\limits_C {(\alpha k^2 + \beta )ds.}$$
Here α and β are constants, kis the curvature of C, ds the arc length and, to make the fewest boundary conditions, we mean minimizing for infinitesimal variations of C on a compact set not containing the endpoints of C. Alternately, one may minimize
$$\int\limits_C {k^2 ds}$$
over variations of C which preserve the total length.


Manifold Etate 


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© Springer-Verlag New York, Inc. 1994

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  • David Mumford

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