An Introduction To The Calculus Of Variations Fox Pdf 14
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I am currently working on problems that require familiarity with calculus of variations. I am fairly new to this field. Please suggest a good introductory book for the same that could help me pick up the concepts quickly.
I know this post is old, but if anyone else is looking for a good, concise and intuitive introduction to the calculus of variations, the chapter 'calculus of variations' in Peter Olver's as yet unpublished 'Applied Mathematics' (well, the first 10 chapters are published as 'Applied Linear Algebra') is very readable.
Charles MacCluer wrote a book on the subject in 2008 for students with a minimal background (basically calculus and some differential equations), Calculus of Variations: Mechanics, Control and Other Applications. I haven't seen the whole book,but what I have seen is excellent and very readable. MacCluer says in the introduction his goal was to write a book on the subject that doesn't replace the old classics, but updates and supplements them with a lot of real-world applications and without heavy prerequisites. From what I've seen, he's succeeded and best of all, the book's available from Dover in a cheap paperback. This might be just what you're looking for.
I want to study the calculus of variations. I understand this to be a "more advanced version" of calculus, in the sense that we maximize functionals (functions of functions), by choosing a particular function, rather than maximizing a function, by choosing a particular variable.
My background: I have mostly knowledge of applied mathematics (multivariable calculus for physics and economics, linear algebra, differential equations, PDE's), and some rudimentary knowledge of pure math (analysis 101, algebra 101, mathematical logic).
Consider, Dido's iso-perimetric problem (colloquially said to be the oldest calculus of variation problem) which can be viewed as an optimal control problem, in the sense that what you get to control is the 'shape' of the curve, and your objective is to maximize the area.
Similarly, another classic problem in calculus of variation is the Brachistochrone Problem which got much attention from the likes of Newton, Bernoullis, Leibniz etc. Again in this case, we can consider the control to be the shape of the curve, and the objective to minimize time.
The field of optimal control only really took off in the 1960's due to Bellman and Pontryagin who introduced dynamic programmingand the maximum principle respectively. Of these the latter approach is specifically a great generalization of ideas from calculus of variations.
Very simplistically, in calculus of variations, we take a function from a space of functionals, 'perturb it a bit' (that is take its variation) and then derive conditions that function and the variations would satisfy if the function were optimal to begin with. So generally this gives necessary conditions (and indeed the maximum principle is a necessary condition where as the HJB equation is necessary and sufficient).
The great leap from calculus of variations to optimal control was a broad generalization of the kinds of variations we can consider. And so we say that calculus of variations is a special case of optimal control theory.
As mentioned in the comments, Dr. Liberzon's book is an excellent introductory resource that combines both calculus of variations and optimal control in a very concise and readable form. There is a couple of chapters introducing calculus of variations and then moving into optimal control theory. So yes, studying calculus of variations first is recommended, but it needn't be a very deep study to get to optimal control. If you have a background in real and functional analysis, that should be sufficient for the Liberzon text.
Like most people above, I am not really sure what you are doing with this information. However, after you have looked at the continuous case, you might consider looking at the discrete calculus of variations. [1] (listed below) has a very nice chapter (chapter 8) on the discrete calculus of variations.
The calculus of variations may be said to begin with Newton's minimal resistance problem in 1687, followed by the brachistochrone curve problem raised by Johann Bernoulli (1696).[2] It immediately occupied the attention of Jakob Bernoulli and the Marquis de l'Hôpital, but Leonhard Euler first elaborated the subject, beginning in 1733. Lagrange was influenced by Euler's work to contribute significantly to the theory. After Euler saw the 1755 work of the 19-year-old Lagrange, Euler dropped his own partly geometric approach in favor of Lagrange's purely analytic approach and renamed the subject the calculus of variations in his 1756 lecture Elementa Calculi Variationum.[3][4][b]
In the 20th century David Hilbert, Oskar Bolza, Gilbert Ames Bliss, Emmy Noether, Leonida Tonelli, Henri Lebesgue and Jacques Hadamard among others made significant contributions.[5] Marston Morse applied calculus of variations in what is now called Morse theory.[6] Lev Pontryagin, Ralph Rockafellar and F. H. Clarke developed new mathematical tools for the calculus of variations in optimal control theory.[6] The dynamic programming of Richard Bellman is an alternative to the calculus of variations.[7][8][9][c]
Calculus of variations is concerned with variations of functionals, which are small changes in the functional's value due to small changes in the function that is its argument. The first variation[l] is defined as the linear part of the change in the functional, and the second variation[m] is defined as the quadratic part.[22]
Calculus of variations (COV) sits in the gap between calculus and optimization. In regular calculus, we learn to findmax or min of functions. COV is not about finding optimal values for a given function but about finding a whole functionthat optimize a given objective. This is a higher-level goal and is what makes the subject both challenging and useful.
A typical problem in COV will ask us to find a function $y = f(x)$ that maximizes the area bounded by $f(x)$ whileholding its perimeter constant. Or a shape that minimizes the distance between two fixed points. A bit of pondering willconvince you that such problems cannot be solved by regular calculus alone. What should we differentiate and set tozero to find an entire function? Some piece seems missing.
There is, however, a branch of regular calculus that concerns itself with calculating whole functions: DifferentialEquations. If we could somehow get differential equations for these optimization problems, we could find the functionthat satisfies our requirements. COV provides these differential equations.
We dipped our toes in the fascinating world of calculus of variations. We stated the fundamental equation of COV, theEuler-Lagrange equation, and saw a few of its generalizations. Though we didnt delve into it, the form of theEuler-Lagrange equations has deeper significance and it makes recurring appearances in Physics and Optimization.
This clear and concise textbook provides a rigorous introduction to the calculus of variations, depending on functions of one variable and their first derivatives. It is based on a translation of a German edition of the book Variationsrechnung (Vieweg+Teubner Verlag, 2010), translated and updated by the author himself. Topics include: the Euler-Lagrange equation for one-dimensional variational problems, with and without constraints, as well as an introduction to the direct methods. The book targets students who have a solid background in calculus and linear algebra, not necessarily in functional analysis. Some advanced mathematical tools, possibly not familiar to the reader, are given along with proofs in the appendix. Numerous figures, advanced problems and proofs, examples, and exercises with solutions accompany the book, making it suitable for self-study.
This textbook offers a concise yet rigorous introduction to calculus of variations and optimal control theory, and is a self-contained resource for graduate students in engineering, applied mathematics, and related subjects. Designed specifically for a one-semester course, the book begins with calculus of variations, preparing the ground for optimal control. It then gives a complete proof of the maximum principle and covers key topics such as the Hamilton-Jacobi-Bellman theory of dynamic programming and linear-quadratic optimal control.
Calculus of Variations and Optimal Control Theory also traces the historical development of the subject and features numerous exercises, notes and references at the end of each chapter, and suggestions for further study.
MATH 8370 - Calculus of Variations and Optimal Control3 Credits (3 Contact Hours)Fundamental theory of the calculus of variations; variable end points; the parametric problem; the isoperimetric problem; constraint inequalities; introduction to the theory of optimal control; connections with the calculus of variations; geometric concepts. Students are expected to have completed an undergraduate-level course in advanced calculus I before enrolling in this course.
Course Objective To provide a gentle introduction to the direct methods in Calculus of Variations concerning minimizations problems, excluding minmax methods. The focus will be on illustrating the main methods using important prototype examples and not on proving the most general or the sharpest results.
Solving the Amazon interview question is not straightforward. The question asks \u201Chow far apart are the poles?,\u201D but this is not what it\u2019s really asking. It is really asking you to find an expression for the shape of a rope hanging between two fixed points, and the rest of the question follows easily. We can formulate this as an optimization problem using calculus of variations, and then solve it.
Calculus of variations is concerned with maximizing (or minimizing) a functional, which is a function of a function. If that\u2019s not immediately clear, don\u2019t worry, we\u2019ll go through an example.