Methods of Variations and Their Applications

2019

Optimal control deals with the problem of finding a control law for a given system such that a certain optimality criterion is achieved. An optimal control is an extension of the calculus of variations. It is a mathematical optimization method for deriving control policies. The calculus of variations is concerned with the extrema of functionals. The different approaches tried out in its solution may be considered, in a more or less direct way, as the starting point for new theories. While the true “mathematical” demonstration involves what we now call the calculus of variations, a theory for which Euler and then Lagrange established the foundations, the solution which Johann Bernoulli originally produced, obtained with the help analogy with the law of refraction on optics, was empirical. A similar analogy between optics and mechanics reappears when Hamilton applied the principle of least action in mechanics which Maupertuis justified in the first instance, on the basis of the laws o...

The lecture notes consist of 53 lectures and are intended for senior undergraduate students. Topics convered include the first and second variations, the Euler–Lagrange equation, the Hamilton’s canonical system, the Hamilton-Jacobi equation, brachistochrone curve, geodesics, and necessary / sufficient conditions.

Fluid Dynamics, 1975

2013

Die Deutsche Bibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data is available in the Internet at

Nonlinear Dynamics and Systems Theory

Employing the contemporary theory of functional differential equations, we propose an effective test on the existence of a minimum for a wide class of functionals in various Banach spaces.

This paper presents use of variational calculus to evolve third order functionals for continuum analysis. The governing equilibrium equation of forces of a line continuum was integrated in the open domain with respect to deflection to obtain three different valid forms of total energy functional for the continuum. They are second order (Ritz energy functional), fourth order (work error functional) and any functional hereinafter called the third order energy functional. Third order energy functional for a rectangular plate was also formulated. These third order energy functionals were subjected to direct variation (differentiating with respect to the coefficient of deflection) to obtain the weak form equilibrium of forces of continuums. Line continuum of four different boundary conditions and a plate with one edge clamped and the other three edges simply supported were used to test this new third order energy functional. In this numerical study, pure bending, buckling and free vibration analysis were performed. The results obtained indicated that the values obtained using this new method are exactly the same as the values obtained using either Ritz or work error energy functionals. Thus, one can comfortably and confidently use the third order energy functionals in continuum analysis

2016

The São Paulo Journal of Mathematical Sciences, 2011

São Paulo Journal of Mathematical Sciences, 2015

In this note we present a brief overview of variational methods to solve homogenization problems. The purpose is to give a first insight on the subject by presenting some fundamental theoretical tools, both classical and modern. We conclude by mentioning some open problems.

Lesson for the module Mathematical Physics II, Dept of Physics, University of York - Years 2005 - 2007