Differentiability properties of the minimal average action |
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Authors: | W M Senn |
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Institution: | (1) Mathemathisches Institut der Universität Bern, Sidlerstrasse 5, CH-3012 Bern, Switzerland |
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Abstract: | Given aZ
n+1-periodic variational principle onR
n+1 we look for solutionsu:R
n R minimizing the variational integral with respect to compactly supported variations. To every vector R
n we consider a subset of solutions which have an average slope when averaging overR
n. The minimal average action A() is defined by the average value of the variational integral given by a solution with average slope . Our main result is:A is differentiable at if and only if the set is totally ordered (in the natural sense). In case that is not totally ordered,A is differentiable at in some direction R
n{0} if and only if is orthogonal to the subspace defined by the rational dependency of . Assuming that the ith component of is rational with denominator si N in lowest terms, we show: The difference of right- and left-sided derivative in the ith standard unit direction is bounded by const ·
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Keywords: | 58C20 46G05 26B25 |
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