Euler’s optimal profile problem (original) (raw)
Regularity properties of solutions to the basic problem in the calculus of variations
Transactions of the American Mathematical Society, 1985
This paper concerns the basic problem in the calculus of variations: minimize a functional J J defined by \[ J ( x ) = ∫ a b L ( t , x ( t ) , x ˙ ( t ) ) d t J(x) = \int _a^b {L(t,x(t),\dot x(t))\;dt} \] over a class of arcs x x whose values at a a and b b have been specified. Existence theory provides rather weak conditions under which the problem has a solution in the class of absolutely continuous arcs, conditions which must be strengthened in order that the standard necessary conditions apply. The question arises: What necessary conditions hold merely under hypotheses of existence theory, say the classical Tonelli conditions? It is shown that, given a solution x x , there exists a relatively open subset Ω \Omega of [ a , b ] [a,b] , of full measure, on which x x is locally Lipschitz and satisfies a form of the Euler-Lagrange equation. The main theorem, of which this is a corollary, can also be used in conjunction with various classes of additional hypotheses to deduce the globa...
Foundations of the Calculus of Variations and Optimal Control
International Series in Operations Research & Management Science, 2010
In this chapter, we treat time as a continuum and derive optimality conditions for the extremization of certain functionals. We consider both variational calculus problems that are not expressed as optimal control problems and optimal control problems themselves. In this chapter, we relie on the classical notion of the variation of a functional. This classical perspective is the fastest way to obtain useful results that allow simple example problems to be solved that bolster one's understanding of continuous-time dynamic optimization.
Constrained variational calculus: the second variation (part I
2010
, the problem of minimality for constrained calculus of variations is analysed among the class of differentiable curves. A fully covariant representation of the second variation of the action functional, based on a suitable gauge transformation of the Lagrangian, is explicitly worked out. Both necessary and sufficient conditions for minimality are proved, and are then reinterpreted in terms of Jacobi fields.
1 Constrained Variational Calculus: The Second Variation
2016
Within the geometrical framework developed in [1], the problem of minimality for constrained calculus of variations is analysed among the class of differentiable curves. A fully covariant representation of the second variation of the action functional, based on a suitable gauge transformation of the Lagrangian, is explicitly worked out. Both necessary and sufficient conditions for minimality are proved, and are then reinterpreted in terms of Jacobi fields.
The extended Euler-Lagrange condition for
1997
This paper provides necessary conditions of optimality for a general variational problem for which the dynamic constraint is a dierential inclusion with a possibly nonconvex right side. They take the form of an Euler-Lagrange inclusion involving convexication in only one co-ordinate, supplemented by the transversality and Weierstrass conditions. It is also shown that for time-invariant, free time problems, the adjoint arc can be chosen so that the Hamiltonian func-tion is constant along the minimizing state arc. The methods used here, based on simple "finite dimensional" nonsmooth calculus, Clarke decoupling, and a rudimentary version of the maximum principle, oer an alternative, and somewhat simpler, derivation of such results to those used by Ioe and Rockafellar in concurrent research.
A transmission problem in the calculus of variations
Calculus of Variations and Partial Differential Equations, 1994
We study h61der regularity of minimizers of the functional fs? [Dul p(x)dx, where p(x) takes only two values and jumps across a Lipschitz surface. No restriction on the two values is imposed.
Calculus of Variations and the Euler Lagrange Equations
A foray into variational calculus and functionals. The mantelpiece of the subject, the Euler-Lagrange equation, is derived and applied to several canonical examples, namely Hamilton's principle. Hamilton's principle, expressed as the principle of least Action, is also derived, whose importance and power is demonstrated on examples in classical mechanics, and discussed in the context of general relativity and quantum field theory.
Preface to “Optimization, Convex and Variational Analysis”
Set-Valued and Variational Analysis, 2021
This collection of works in the honor of professor Terry Rockafellar is a follow-up of the "Workshop on Optimization and Variational Analysis", dedicated to Terry's 85th birthday. The meeting, jointly organized by the CMM Center for Mathematical Modeling of the University of Chile (Chile) and the University of Perpignan (France), was held in Santiago on January 20-21, 2020. That workshop was one of the last meetings we could attend physically, before Coronavirus changed our lives in so many ways. Globetrotting has become virtual since then. Suddenly, the beauty of the world found itself flattened to a screen. Fortunately, some things have not changed: our admiration and appreciation for Terry's unique career has remained intact, as has the momentum to duly celebrate his birthday, through the edition of this special volume. We are very grateful to the authors and referees for their valuable contributions and careful work. The two volumes that make up the special issue in Terry's honor sample the tremendous breadth of subjects where Terry has made fruitful contributions. This special issue is a modest gift for someone who has gifted us with seminal textbooks, whose content has marked generations of researchers, influencing the way of doing mathematics when it involves "variations", regarding not only theory or analysis in optimization but also applications.
The variational principles of mechanics are firmly rooted in the soil of that great century of Liberalism which starts with Descartes and ends with the French Revolution and which has witnessed the lives of Leibniz, Spinoza, Goethe, and Johann Sebastian Bach. It is the only period of cosmic thinking in the entire history of Europe since the time of the Greeks. 1
Lecture Notes on Calculus of Variations
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.
En Route for the Calculus of Variations
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...
Geometric constrained variational calculus. III: The second variation (Part II)
International Journal of Geometric Methods in Modern Physics, 2016
A geometric setup for constrained variational calculus is presented. The analysis deals with the study of the extremals of an action functional defined on piecewise differentiable curves, subject to differentiable, non-holonomic constraints. Special attention is paid to the tensorial aspects of the theory. As far as the kinematical foundations are concerned, a fully covariant scheme is developed through the introduction of the concept of infinitesimal control. The standard classification of the extremals into normal and abnormal ones is discussed, pointing out the existence of an algebraic algorithm assigning to each admissible curve a corresponding abnormality index, related to the corank of a suitable linear map. Attention is then shifted to the study of the first variation of the action functional. The analysis includes a revisitation of Pontryagin's equations and of the Lagrange multipliers method, as well as a reformulation of Pontryagin's algorithm in Hamiltonian terms. The analysis is completed by a general result, concerning the existence of finite deformations with fixed endpoints.
Variational Analysis, Optimization and Applications: Preface
Journal of Optimization Theory and Applications, 2013
This Special Issue of JOTA is mainly based on selected invited papers presented at the Tenth International Seminar in Optimization and Related Areas (ISORA), which was held in Lima (Perú) on October 3-7, 2011. Perú, the old Inca's country, had not long traditions in mathematical research and in mathematical conferences before the first ISORA meeting organized in 1993. It was dedicated to the memory of Eugen Blum, a Swiss mathematician who spent the last twenty years of his life in Perú. His area of research was Mathematical Optimization to which he made fundamental contributions. His seminal paper with Werner Oettli entitled "From optimization and variational inequalities to equilibrium problems", published in 1994, has initiated equilibrium problems in optimization theory, and then has been quoted by many researchers. According to MathSciNet, it was cited 404 times up to July 2013. Eugen Blum started the research on optimization theory in Lima. His students and collaborators have later continued these lines of research and have succeeded to build a strong research team that includes a number of very promising young mathematicians.
2015
Variational calculus studied methods for finding maximum and minimum values of functional. It has its inception in 1696 year by Johan Bernoulli with its glorious problem: to find a curve, connecting two points A and B , which does not lie in a vertical, so that heavy point descending on this curve from position A to reach position in for at least time. In functional analysis variational calculus takes the same space, as well as theory of maxima and minimum intensity in the classic analysis . We will prove a theorem for functional where prove that necessary condition for extreme of functional is the variation of functional is equal to zero. We describe the solution of the equation of Euler with example of application, such as the Bernoulli's problem for the shortest time.
Natural boundary conditions in the calculus of variations
Mathematical Methods in the …, 2010
We prove necessary optimality conditions for problems of the calculus of variations on time scales with a Lagrangian depending on the free end-point. 1. Introduction. The calculus on time scales was introduced by Bernd Aulbach and Stefan Hilger in 1988 [7]. The new theory unify and extends the traditional areas of continuous and discrete analysis and the various dialects of q-calculus [14] into a single theory [13, 24], and is finding numerous applications in such areas as engineering, biology, economics, finance, and physics [1]. The present work is dedicated to the study of problems of calculus of variations on a generic time scale T. As particular cases, one gets the classical calculus of variations [17] by choosing T = R; the discrete-time calculus of variations [23] by choosing T = Z; and the q-calculus of variations [8] by choosing T