Variational Principles

  • Emmanuele DiBenedettoEmail author
Part of the Cornerstones book series (COR)


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 \(\lambda \in \mathbb{R}\).


Stationary Point Wave Front Isotropic Medium Refraction Index Geometrical Optic 
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Copyright information

© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  1. 1.Department of MathematicsVanderbilt UniversityNashvilleUSA

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