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On geometric variation theory. (English) Zbl 0948.58008

We construct here a framework for a geometric variation of one-dimensional geometric figures regarding them as sets of ordered points. In this framework, we can make full use of cut and paste technique to find a way to go down to the geometric smallest figure, including the topological change of the parameter space. Therefore we can discuss practical problems like switching of current flows and the minimal networks not only multiple closed geodesics.

MSC:

58E10 Variational problems in applications to the theory of geodesics (problems in one independent variable)
53C22 Geodesics in global differential geometry
Full Text: DOI

References:

[1] Geometric integration theory (1956)
[2] Lectures on closed geodesics (1978) · Zbl 0397.58018
[3] Variationsrechnung im Grossen (1938)
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