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Variational Principles

  • Emmanuele DiBenedetto
Chapter
Part of the Cornerstones book series (COR)

Abstract

Given two points q o and q 1 in \({\mathbb{R}}^{N}\)and an interval \([{t}_{o},{t}_{1}] \subset \mathbb{R}\), consider the convex set \(\mathcal{K}\)of all smooth curves parameterized with t ∈ [t o , t 1], and of extremities q o and q 1, i.e.,
$$\mathcal{K} = \left \{q \in {C}^{1}[{t}_{ o},{t}_{1}]\bigm |q({t}_{o}) = {q}_{o}, q({t}_{1}) = {q}_{1}\right \}.$$
Given \(q \in \mathcal{K}\)and a vector-valued function φ ∈ C∞o(to, t1), the curve {q + λφ} is still in \(\mathcal{K}\)for all \(\lambda \in \mathbb{R}\).

Keywords

Stationary Point Wave Front Isotropic Medium Refraction Index Geometrical Optic 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Copyright information

© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  1. 1.Department of MathematicsVanderbilt UniversityNashvilleUSA

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