On central fields in the calculus of variations

This article develops sufficient conditions of local optimality for the scalar and vectorial cases of the calculus of variations. The results are established through the construction of stationary fields which keep invariant what we define as the generalized Hilbert integral.Comment: 12 pages, typos correcte

A variational formulation for relativistic mechanics based on Riemannian geometry and its application to the quantum mechanics context

This article develops a variational formulation of relativistic nature applicable to the quantum mechanics context. The main results are obtained through basic concepts on Riemannian geometry. Standards definitions such as vector fields and connection have a fundamental role in the main action establishment. In the last section, as a result of an approximation for the main formulation, we obtain the relativistic Klein-Gordon equation.Comment: 16 pages, new results based on the Weinberg approach for relativistic mechanics, one more section adde

A primal dual variational formulation and a multi-duality principle for a non-linear model of plates

This article develops a new primal dual formulation for the Kirchhoff-Love non-linear plate model. At first we establish a duality principle which includes sufficient conditions of global optimality through the dual formulation. At this point we highlight this first duality principle is specially suitable for the case in which the membrane stress tensor is negative definite. In a second step, from such a general principle, we develop a primal dual variational formulation which also includes the corresponding sufficient conditions for global optimality. The results are based on standard tools of convex analysis and on a well known Toland result for D.C. optimization. Finally, in the last section, we present a multi-duality principle and qualitative relations between the critical points of the primal and dual formulations. We formally prove there is no duality gap between such primal and dual formulations in a local extremal context.Comment: 21 pages, some more corrections implemente

On the generalized method of lines and its proximal explicit and hyper-finite difference approaches

This article firstly develops a proximal explicit approach for the generalized method of lines. In such a method, the domain of the PDE in question is discretized in lines and the equation solution is written on these lines as functions of the boundary conditions and domain shape. The main objective of introducing a proximal formulation is to minimize the solution error as a typical parameter $\varepsilon>0$ is too small. In a second step we present another procedure to minimize this same error, namely, the hyper-finite differences approach. In this last method the domain is divided in sub-domains on which the solution is obtained through the generalized method of lines allowing the parameter $\varepsilon>0$ to be very small without increasing the solution error. The solutions for the sub-domains are connected through the boundary conditions and the solution of the partial differential equation in question on the node lines which separate the sub-domains. In the last sections of each text part we present the concerning softwares and perform numerical examples.Comment: 18 pages, some typos corrected, a new result adde

On General Duality Principles for Non-Convex Variational Optimization with Applications to the Ginzburg-Landau System in Superconductivity

This article develops duality principles applicable to non-convex models in the calculus of variations. The results here developed are applied to Ginzburg-Landau type equations. For the first and second duality principles, through an optimality criterion developed for the dual formulations, we qualitatively classify the critical points of the primal and dual functionals in question. We formally prove there is no duality gap between the primal and dual formulations in a local extremal context. Finally, in the last sections, we present a global existence result, a duality principle and respective optimality conditions for the complex Ginzburg-Landau system in superconductivity in the presence of a magnetic field and concerning magnetic potential.Comment: 32 pages, typos corrected, other results adde

On the numerical solution of non-linear first order ordinary differential equation systems

In this article, firstly we develop a method for a type of difference equations, applicable to solve approximately a class of first order ordinary differential equation systems. In a second step, we apply the results obtained to solve a non-linear two point boundary value problem relating a flight mechanics model. We highlight the algorithm obtained seems to be robust and of easy computational implementation.Comment: 9 page

On duality principles for non-convex variational models applied to a Ginzburg-Landau type equation

This article develops a duality principle applicable to a large class of variational problems. Firstly, we apply the results to a Ginzburg-Landau type model. In a second step, we develop another duality principle and related primal dual variational formulation and such an approach includes optimality conditions which guarantee zero duality gap between the primal and dual formulations. We emphasize in both cases the dual variational formulations obtained have large regions of concavity about the critical points in question.Comment: 14 pages, more typos correcte

Duality suitable for a class of non-convex optimization problems

In this article we develop a duality principle suitable for a large class of problems in optimization. The main result is obtained through basic tools of convex analysis and duality theory. We establish a correct relation between the critical points of the primal and dual formulations and formally prove there is no duality gap between such formulations, in a local extremal context.Comment: 7 pages, a minor error and some typos correcte

A general variational formulation for relativistic mechanics based on fundamentals of differential geometry

The first part of this article develops a variational formulation for relativistic mechanics. The results are established through standard tools of variational analysis and differential geometry. The novelty here is that the main motion manifold has a $n+1$ dimensional range. It is worth emphasizing in a first approximation we have neglected the self-interaction energy part. In its second part, this article develops some formalism concerning the causal structure in a general space-time manifold. Finally, the last article section presents a result concerning the existence of a generalized solution for the world sheet manifold variational formulation.Comment: 25 pages, minor corrections implemented, new sections adde

A note on optimization in Rn\mathbb{R}^n

In this article, we develop an algorithm suitable for constrained optimization in $\mathbb{R}^n$. The results are developed through standard tools of n-dimensional real analysis and basic concepts of optimization. Indeed, the well known Banach fixed point theorem has a fundamental role in the main result establishment.Comment: 10 page