By Jurgen Jost, Xianqing Li-Jost

ISBN-10: 0521642035

ISBN-13: 9780521642033

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**Extra resources for Calculus of Variations**

**Example text**

Examples show that without such an invertibility condition, regularity need not hold. This invertibility condition det Fpp ^ 0 implies that the Euler-Lagrange equations allow the expression of u(i) in terms of u(i) and ii(t). 3 The second variation. e. 6I(u,

5) has to be independent of them, too. In order to study this more closely, let f:U-+V be another chart with c([0,T}) C f(U). Then there exists a curve 7 in U with c(t) = /(7(t)) for all t. 1). e. a bijective map between open subsets of E n whose derivative D

E. e. e. the reparameterized curve c = cor is parameterized by arc-length. 1. Let c : [0, L(c)] —• Rd be a curve parameterized on [0,L(c)]. e. we keep the interval of definition fixed, namely [0, L(c)]), the parameterization by arc-length leads to the smallest energy. 12) whereas for any other parameterization of c on the same interval, L(c) < 2E(c). 13) We now return to those curves c that are confined to lie on M, in order to discover a third invariance. 6) for its energy. 5) has to be independent of them, too.

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