#Best book on calculus of variations how to#

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. Here traditionally the choice of control has been to stabilize a system, drive a system from one state to another etc. Now as you observed classical control theory is concerned with transfer functions, root locus, stabilization etc. Again in this case, we can consider the control to be the shape of the curve, and the objective to minimize time. Similarly, another classic problem in calculus of variation is the Brachistochrone Problem which got much attention from the likes of Newton, Bernoullis, Leibniz etc. 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).Ĭalculus of variation is a special case of optimal control theory in a particular sense.Ĭonsider, 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.

What do you think is the best resource for studying the calculus of variations, given my background? (or optimal control theory, depending on your answer to the above). So it is not entirely clear to me what the relation is between $(1), (2), (3)$?Īlso: Is it recommendable to study calculus of variations first, and then study Optimal Control Theory? Or is it old-fashioned to study calculus of variations in isolation?

#Best book on calculus of variations how to#

However, I understand control theory to be about dynamical systems (in engineering), and how to set a variable in order to control other variables in that system in some way ( is that correct?).

I read online that the calculus of variations $(1)$ is a special case of 'Optimal Control Theory' $(2)$, which in turn is a part of 'control theory' $(3)$.

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. Students are assumed to have a knowledge of partial differentiation and differential equations.I want to study the calculus of variations. Ideal as a text, this volume offers an exceptionally clear presentation of the mathematics involved, with many illustrative examples, while numerous references cite additional source readings for those interested in pursuing a topic further. The last three chapters examine variable end points and strong variations, including an account of Weierstrass's theory of strong variations, based upon the work of Hilbert. Applied mathematics are discussed in Chapters V, VI, and VII, including studies of least action, a proof of Hamilton's principle and its use in dealing with dynamical problems in the special theory of relativity, and such methods of approximation as the Rayleigh-Ritz method, illustrated by applications to the theory of elasticity. Chapters III and IV delve into pure mathematics, exploring generalizations and isoperimetrical problems. The first two chapters deal with the first and second variation of an integral in the simplest case, illustrated by applications of the principle of least action to dynamical problems.

In this highly regarded text, aimed at advanced undergraduate and graduate students in mathematics, the author develops the calculus of variations both for its own intrinsic interest and because of its wide and powerful applications to modern mathematical physics.

An understanding of variational methods, the source of such fundamental theorems as the principle of least action and its various generalizations, is essential to the study of mathematical physics and applied mathematics.