In 1696, Johann Bernoulli solved the brachistochrone problem by an ingenious method of combining Fermat’s principle of minimum time, Snell’s law of refraction and “finite element” discretization. This appears to be the first application of a “direct method.” By taking the limits of these “broken-line solutions,” Bernoulli arrived at an equation for the cycloid. About fifty years later (1744), E...

For a given PDE system (or an exterior differential system) possessing a Lie group of internal symmetries the orbit reduction procedure is introduced. It is proved that the solutions of the reduced exterior differential system are in one-to-one correspondence with the moduli space of regular solutions of the prolongation of the original system. The isomorphism between the local characteristic c...

In this paper, we present a theory of elastic images deformation and matching. Our theory is based on the physical theory of elastic deformation of a continuum, and is very general: it applies to matching n-dimensional images with n-dimensional templates. We derive the variational formulation of the optimal deformation problem, and the Euler-Lagrange equations to solve it. Also, we address the ...

Let S ⊂ R be a bounded C domain and let g denote the flat metric in R. We prove that there exist minimizers of the Willmore functional restricted to a class of isometric immersions of the Riemannian surface (S, g) into R. We derive the Euler-Lagrange equations satisfied by such constrained minimizers. Our main motivation comes from nonlinear elasticity, where this constrained Willmore functiona...

We prove that any distribution q satisfying the equation ∇q = div f for some tensor f = (f i j), f i j ∈ h (U) (1 ≤ r < ∞) -the local Hardy space, q is in h, and is locally represented by the sum of singular integrals of f i j with Calderón-Zygmund kernel. As a consequence, we prove the existence and the local representation of the hydrostatic pressure p (modulo constant) associated with incomp...

The study of fractional variational problems with derivatives in the sense of Caputo is a recent subject, the main results being Agrawal’s necessary optimality conditions of Euler-Lagrange and respective transversality conditions. Using Agrawal’s Euler-Lagrange equation and the Lagrange multiplier technique, we obtain here a Noether-like theorem for fractional optimal control problems in the se...

This paper reviews literature, current concepts and approaches in computational anatomy (CA). The model of CA is a Grenander deformable template, an orbit generated from a template under groups of diffeomorphisms. The metric space of all anatomical images is constructed from the geodesic connecting one anatomical structure to another in the orbit. The variational problems specifying these metri...

In this paper, we derive an explicit group-invariant formula for the EulerLagrange equations associated with an invariant variational problem. The method relies on a group-invariant version of the variational bicomplex induced by a general equivariant moving frame construction, and is of independent interest. Mathematics Subject Classification (2000): Primary: 53A55, 58J70, 58E40 Secondary: 35A...

This paper presents the necessary optimality conditions of Euler–Lagrange type for variational problems with natural boundary conditions and problems with holonomic constraints where the fuzzy fractional derivative is described in the combined Caputo sense. The new results are illustrated by computing the extremals of two fuzzy variational problems. AMS subject classifications: 65D10, 92C45

The purpose of this paper is to present the basic mathematical modeling of microcopters, which could be used to develop proper methods for stabilization and trajectory control. The microcopter taken into account consists of six rotors, with three pairs of counter-rotating fixedpitch blades. The microcopter is controlled by adjusting the angular velocities of the rotors which are spun by electri...