Cálculo variacional e aplicações

This paper consists of a brief review and introduction to the main concepts of Classic Variational Calculus. Starting from thedefinitions of the concepts of first and second variation of a functional, we present a mathematically rigorous treatment for theVariational Calculus, establishing necessary and sufficient conditions for obtaining extrema. In this context, the notion of conjugatepoints is introduced, which is fundamental for the classification of weak extrema. Some simple and enlightening examples are dealtwith throughout the paper. Strong extrema are characterized and sufficient conditions for their occurrence are given. The paperconcludes with a brief application to Lagrange mechanics, showing the existence of actions whose stationary points are saddlepoints instead of minima.

THE FIRST AND SECOND VARIATION OF THE TOTAL ENERGY OF CLOSED DUPLEX DNA IN PLANAR CASE

DNA's shape mostly lies on its total energy F. Its corresponding equilibrium shape equations can be obtained by classical variation method: letting the first energy variation δ(1)F = 0. Here, we not only provide the first variation δ(1)F but also give the second variation δ(2)F in planar case. Moreover, the general shape equations of DNA are abstained and a mistake in Zhang et al., [Phys. Rev. E70, 051902 (2004)] is pointed out.

Harmonic G-structures

For closed and connected subgroups G of SO(n), we study the energy functional on the space of G-structures of a (compact) Riemannian manifold (M, 〈⋅, ⋅〉), where G-structures are considered as sections of the quotient bundle (M)/G. We deduce the corresponding first and second variation formulae and the characterising conditions for critical points by means of tools closely related to the study of G-structures. In this direction, we show the rôle in the energy functional played by the intrinsic torsion of the G-structure. Moreover, we analyse the particular case G=U(n) for 2n-dimensional manifolds. This leads to the study of harmonic almost Hermitian manifolds and harmonic maps from M into (M)/U(n).

GEOMETRY OF UNITARIES IN A FINITE ALGEBRA: VARIATION FORMULAS AND CONVEXITY

Given a C*-algebra [Formula: see text] with trace τ, we compute the first and second variation formulas for the p-energy functional Fp of the unitary group [Formula: see text] of [Formula: see text], for p = 2n an even integer, namely: [Formula: see text] where [Formula: see text] is a smooth curve for t ∈ [a, b]. As an application of these formulas, we prove that if dp denotes the geodesic distance of the Finsler metric induced by the p-norm [Formula: see text] with [Formula: see text] and δ(t) is a geodesic of [Formula: see text] joining δ(0) = u0 and δ(1) = u1, then the mapping [Formula: see text] is convex.

Stability of Elastic Half-Spaces by an Energy Method

An energy stability criterion is used to study the stability of deformations of a compressible elastic half-space. A minimization problem is formulated in an unbounded domain, and the first and second variation conditions are derived for this problem. Algebraic stability conditions are derived for general compressible isotropic materials, as well as for neo-Hookean class of Hadamard materials.