Physics
Calculus of Variations
The calculus of variations is a mathematical field that deals with finding the path, curve, or function for which a certain integral expression has an extremum. In physics, it is used to find the path taken by a particle that minimizes the action, which is a fundamental principle in classical mechanics. This mathematical tool is essential for understanding the behavior of physical systems.
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10 Key excerpts on "Calculus of Variations"
eBook - PDFMathematics for Physics
A Guided Tour for Graduate Students
- Michael Stone, Paul Goldbart(Authors)
- 2009(Publication Date)
- Cambridge University Press(Publisher)
1 Calculus of Variations We begin our tour of useful mathematics with what is called the Calculus of Variations. Many physics problems can be formulated in the language of this calculus, and once they are there are useful tools to hand. In the text and associated exercises we will meet some of the equations whose solution will occupy us for much of our journey. 1.1 What is it good for? The classical problems that motivated the creators of the Calculus of Variations include: (i) Dido’s problem: In Virgil’s Aeneid , Queen Dido of Carthage must find the largest area that can be enclosed by a curve (a strip of bull’s hide) of fixed length. (ii) Plateau’s problem: Find the surface of minimum area for a given set of bounding curves. A soap film on a wire frame will adopt this minimal-area configuration. (iii) Johann Bernoulli’s brachistochrone: A bead slides down a curve with fixed ends. Assuming that the total energy 1 2 mv 2 + V (x) is constant, find the curve that gives the most rapid descent. (iv) Catenary: Find the form of a hanging heavy chain of fixed length by minimizing its potential energy. These problems all involve finding maxima or minima, and hence equating some sort of derivative to zero. In the next section we define this derivative, and show how to compute it. 1.2 Functionals In variational problems we are provided with an expression J [y] that “eats” whole func- tions y(x) and returns a single number. Such objects are called functionals to distinguish them from ordinary functions. An ordinary function is a map f : R → R. A functional J is a map J : C ∞ (R) → R where C ∞ (R) is the space of smooth (having derivatives of all orders) functions. To find the function y(x) that maximizes or minimizes a given functional J [y] we need to define, and evaluate, its functional derivative. 1
- Liviu I Nicolaescu(Author)
- 2007(Publication Date)
- World Scientific(Publisher)
Chapter 5 Elements of the Calculus of Variations This is a very exciting subject lying at the frontier between mathematics and physics. The limited space we will devote to this subject will hardly do it jus-tice, and we will barely touch its physical significance. We recommend to anyone looking for an intellectual feast the Chapter 16 in vol.2 of “The Feynmann Lectures on Physics” [35], which in our opinion is the most eloquent argument for the raison d’ˆ etre of the Calculus of Variations. 5.1 The least action principle 5.1.1 The 1-dimensional Euler-Lagrange equations From a very “dry” point of view, the fundamental problem of the calculus of vari-ations can be easily formulated as follows. Consider a smooth manifold M , and let L : R × TM → R by a smooth function called the lagrangian . Fix two points p 0 ,p 1 ∈ M . The action of a piecewise smooth path γ : [0 , 1] → M connecting these points is the real number S ( γ ) = S L ( γ ) defined by S ( γ ) = S L ( γ ) := integraldisplay 1 0 L ( t, ˙ γ ( t ) ,γ ( t )) dt. In the Calculus of Variations one is interested in those paths as above with minimal action. Example 5.1.1. Given a smooth function U : R 3 → R called the potential , we can form the lagrangian L ( ˙ q,q ) : R 3 × R 3 ∼ = T R 3 → R , given by L = Q − U = kinetic energy − potential energy = 1 2 m | ˙ q | 2 − U ( q ) . The scalar m is called the mass . The action of a path (trajectory) γ : [0 , 1] → R 3 is a quantity called the Newtonian action . Note that, as a physical quantity, the Newtonian action is measured in the same units as the energy. ⊓⊔ 193 194 LECTURES ON THE GEOMETRY OF MANIFOLDS Example 5.1.2. To any Riemann manifold ( M,g ) one can naturally associate two lagrangians L 1 , L 2 : TM → R defined by L 1 ( v,q ) = g q ( v,v ) 1 / 2 ( v ∈ T q M ) , and L 2 ( v,q ) = 1 2 g q ( v,v ) . We see that the action defined by L 1 coincides with the length of a path. The action defined by L 2 is called the energy of a path.
eBook - PDF- Lawrence Turyn(Author)
- 2013(Publication Date)
- CRC Press(Publisher)
14 Calculus of Variations 14.1 Minimization Problems The Calculus of Variations, as a systematic way of modeling and solving physical problems, is historically relevant but was also reinvigorated in the twentieth century in the subjects of both control theory and finite element methods. Very natural contexts for Calculus of Variations include engineering mechanics and elec-tromagnetism where we use knowledge of the energy in the system. For example, it makes sense that if we deform a solid object then its new equilibrium shape should minimize its potential energy. There are many physical problems that are modeled by a Calculus of Variations problem of finding an “admissible” function y ( x ) so as to ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ Minimize J [ y ] b a F ( x , y ( x ) , y ( x ) ) dx Subject to y ( a ) = y a y ( b ) = y b ⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭ (14.1) or subject to conditions on or including y ( a ) , y ( b ) . Also, the integral may involve higher order derivatives of y . Often the problem has a natural interpretation in terms of energy in a system. In Chapter 13 we studied minimization of a function that depends on several inde-pendent variables, that is, unknowns to be solved for. In Chapter 14 we will minimize an integral which depends on a function which is the unknown to be solved for. That is inherently a more mind boggling problem, but results from Chapter 13 will still be relevant. A function is admissible if it is continuous and piecewise continuously differentiable on the interval [ a , b ] . If higher order derivatives are in the integrand then the class of admissible functions may be further restricted to involve higher order differentiability. A functional is a mapping from a vector space to scalar values. For example, in (14.1) the mapping y → b a F ( x , y ( x ) , y ( x ) ) dx is a functional. 1081
eBook - PDF- Nivaldo A. Lemos(Author)
- 2018(Publication Date)
- Cambridge University Press(Publisher)
2 Hamilton’s Variational Principle Suit the action to the word, the word to the action. William Shakespeare, Hamlet, Act 3, Scene 2 The most concise of all formulations of classical dynamics has the form of a variational principle. According to ideas typical of the eighteenth century, of which Maupertuis was one of the pioneers, among all alternatives at its disposal nature follows the less expensive course on the basis of some criterion for comparing the myriad of possibilities. The dif- ferential principle of d’Alembert, from which Lagrange’s equations are derived, expresses the fundamental law of motion in terms of the system’s instantaneous configuration and infinitesimal deviations therefrom. It is possible to reformulate the fundamental dynamical law as an integral principle that takes into account the entire motion during a finite time interval. Hamilton’s principle reduces the laws of mechanics to a statement that, among all imaginable paths, the actual path followed by the system is the one for which a certain quantity whose value depends on the totality of the motion is a minimum or, more generally, stationary. The precise formulation of Hamilton’s principle or the principle of least action, which is advantageous in several respects, requires a brief foray into an important branch of mathematics known as the Calculus of Variations. 2.1 Rudiments of the Calculus of Variations The Calculus of Variations occupies itself with the problem of determining extrema, that is, maxima and minima, of functionals defined by integrals. A functional is a real-valued application whose domain is a set of functions. More explicitly, a functional associates a real number with each function of a certain class for which the functional is defined. Example 2.1 Let (x 1 , y 1 ) and (x 2 , y 2 ) be two distinct points in a plane, with x 2 > x 1 .
eBook - PDFAdvanced Engineering Analysis: The Calculus Of Variations And Functional Analysis With Applications In Mechanics
The Calculus of Variations and Functional Analysis with Applications in Mechanics
- Leonid P Lebedev, Michael J Cloud, Victor A Eremeyev(Authors)
- 2012(Publication Date)
- World Scientific(Publisher)
Chapter 1 Basic Calculus of Variations 1.1 Introduction Optimization is a universal goal. Students would like to learn more, receive better grades, and have more free time; professors (at least some of them) would like to give better lectures, see students learn more, receive higher pay, and have more free time. These are the optimization problems of real life. In mathematics, optimization makes sense only when formulated in terms of a function f ( x ) or other expression. One then seeks the mini-mum value of the expression. (It suffices to discuss minimization because maximizing f is equivalent to minimizing − f .) This book treats the minimization of functionals . The notion of func-tional generalizes that of function. Although the process of generalization does yield results of greater generality, as a rule the results are not sharper in particular cases. So to understand what can be expected from the calcu-lus of variations, we should review the minimization of ordinary functions. All quantities will be assumed sufficiently differentiable for the purpose at hand. Let us recall some terminology for the one-variable case y = f ( x ). Definition 1.1. The function f ( x ) has a local minimum at a point x 0 if there is a neighborhood ( x 0 − d, x 0 + d ) in which f ( x ) ≥ f ( x 0 ). We call x 0 the global minimum of f ( x ) on [ a, b ] if f ( x ) ≥ f ( x 0 ) holds for all x ∈ [ a, b ]. The necessary condition for a differentiable function f ( x ) to have a local minimum at x 0 is f ( x 0 ) = 0 . (1.1) A simple and convenient sufficient condition is f ( x 0 ) > 0 . (1.2) 1 2 Advanced Engineering Analysis Unfortunately, no available criterion for a local minimum is both sufficient and necessary. So the approach is to solve (1.1) for possible points of local minimum of f ( x ) and then test these using an available sufficient condition. The global minimum on [ a, b ] can be attained at a point of local mini-mum.
- K. F. Riley, M. P. Hobson(Authors)
- 2011(Publication Date)
- Cambridge University Press(Publisher)
12 Calculus of Variations How to find stationary values of functions of a single variable f ( x ), of several variables f ( x , y , . . . ) and of constrained variables, where x , y , . . . are subject to the n constraints g i ( x , y , . . . ) = 0, i = 1 , 2 , . . . , n will be known to the reader and is summarized in Sections A.3 and A.7 of Appendix A. In all those cases the forms of the functions f and g i were known, and the problem was one of finding the appropriate values of the variables x , y , etc. We now turn to a different kind of problem in which we are interested in bringing about a particular condition for a given expression (usually maximizing or minimizing it) by varying the functions on which the expression depends. For instance, we might want to know in what shape a fixed length of rope should be arranged so as to enclose the largest possible area, or in what shape it will hang when suspended under gravity from two fixed points. In each case we are concerned with a general maximization or minimization criterion by which the function y ( x ) that satisfies the given problem may be found. The Calculus of Variations provides a method for finding the function y ( x ). The problem must first be expressed in a mathematical form, and the form most commonly applicable to such problems is an integral . In each of the above questions, the quantity that has to be maximized or minimized by an appropriate choice of the function y ( x ) may be expressed as an integral involving y ( x ) and the variables describing the geometry of the situation. In our example of the rope hanging from two fixed points, we need to find the shape function y ( x ) that minimizes the gravitational potential energy of the rope. Each elementary piece of the rope has a gravitational potential energy proportional both to its vertical height above an arbitrary zero level and to the length of the piece.
eBook - PDF- Harold Jeffreys, Bertha Jeffreys(Authors)
- 1999(Publication Date)
- Cambridge University Press(Publisher)
Chapter 10 Calculus of Variations When change itself can give no more 'Tis easy to be true. SIB CHABLES SEDLEY, Reasons for Constancy 10-01. Condition for an integral to be stationary. Suppose that we have an integral of the form *-£/(£•*')*• < j > where/is a given function; x is to be a function of t } but we have not yet specified what function. The problem of the Calculus of Variations is to decide what function x must be in order that S may be stationary for small variations of x. In its simplest form we can consider the determination of the shortest distance between two points. Using Cartesian coordinates and assuming that y is a differentiable function of x t with an integrable derivative, the distance along an arbitrary path is If the ends are specified, so that y(x x ) = y v y(x 0 ) = y 01 two given quantities, we know that s is made a minimum by taking y-yo = s-*o ( 3 ) 2/i-2/o * i - * o " This makes the path the straight line connecting (x 0 , y 0 ) and (a^, y t ). If we make y any other function of x we are choosing a different path, and its length will necessarily be greater than that of the straight line if the termini are kept the same. The charac- teristic feature of the Calculus of Variations, in contrast to ordinary problems of maxima and minima, is the occurrence of the unknown function or its derivative under the integral sign. To evaluate the integral (1) we must have the value of x for every value of t in the range; to make it stationary we have therefore, effectively, to deter- mine an infinite number of values of x. The meaning of * stationary* therefore needs definition. We write dxjdt = p and regard/as a function of the three variables x, p, t and suppose that within a certain region of these variables the second partial derivatives of/ with regard to x, p, t are continuous as functions of #, p, t. We outline the argument for the case when only functions x(t) with continuous second derivatives are regarded as admissible.
eBook - PDFMath Unlimited
Essays in Mathematics
- R. Sujatha, H. N. Ramaswamy, C. S. Yogananda(Authors)
- 2011(Publication Date)
- CRC Press(Publisher)
Part IV Mathematics and Physics This page intentionally left blank This page intentionally left blank C hapter 17 Variational Calculus in Mathematics and Physics Rajesh Gopakumar Harish-Chandra Research Institute, Allahabad 211019, India. e-mail: [email protected] 1 Introduction The search for unifying principles has been an integral part of the devel-opment of physical laws of nature. One of the most fruitful principles of this kind has been the so-called variational principle which has appeared in a number of di ff erent guises, most famously as the Principle of Least Action . In this article we will discuss some aspects of this principle, its applications in physics and some of the mathematics underlying it. Mathematically, the variational principle arises in questions related to extremisation. The extremisation which is involved here is more general than the extremisation of functions. However, to explain the more gen-eral class of questions, let us start by considering the latter case which is simpler. In many circumstances involving maximisation or minimisation, the question can be reduced to one that can be addressed by di ff erential calculus. For instance, consider the following simple question. Among all rectangles of fixed perimeter 4 L , which is the one of maximum area? This is easy to answer since we can take the sides of any such rectangle to be of length x and 2 L − x . The area of such a rectangle is then A ( x ) = x (2 L − x ). 284 Math Unlimited The one with the maximum area is the one for which the function A ( x ) is extremised i.e. dA ( x ) dx = 0 ⇒ 2( L − x ) = 0 ⇒ x = L . (1) This is easily checked to be a maximum i.e. d 2 A dx 2 | x = L < 0 Thus the rectangle which maximises the area is the square with all sides of equal length L . There are many such questions which can be reduced to extremisation of functions. One quickly realises that one needs to consider functions of many variables and extremise with respect to all the variables.
eBook - PDFDifferential Equations
A first course on ODE and a brief introduction to PDE
- Shair Ahmad, Antonio Ambrosetti(Authors)
- 2019(Publication Date)
- De Gruyter(Publisher)
16 The Euler–Lagrange equations in the Calculus of Variations: an introduction This chapter is intended to serve as an elementary introduction to the Calculus of Vari-ations, one of the most classical topics in mathematical analysis. For a more complete discussion we refer, e. g., to R. Courant, Calculus of Varia-tions: With Supplementary Notes and Exercises, Courant Inst. of Math. Sci., N. Y. U., 1962 , or B. Dacorogna, Introduction to the Calculus of Variations, 2nd Edition, World Scien-tific, 2008 . Notation: in this chapter we let x denote the independent variable and y = y ( x ) the dependent variable. 16.1 Functionals Given two points in the plane A = ( a , α ) , B = ( b , β ) , the length of a smooth curve y = y ( x ) such that y ( a ) = α , y ( b ) = β is given by ℓ[ y ] = b ∫ a √ 1 + y ? 2 ( x ) dx . The map y → ℓ[ y ] is an example of a functional . In general, given a class of functions Y , a functional is a map defined on Y with values in the set of real numbers ℝ . We will be mainly concerned with functionals of the form I [ y ] = b ∫ a L ( x , y ( x ), y ? ( x ) ) dx , y ∈ Y , (I) where the class Y is given by Y = { y ∈ C 2 ([ a , b ]) : y ( a ) = α , y ( b ) = β } (Y) and the Lagrangian L = L ( x , y , p ) is a function of three variables ( x , y , p ) ∈ [ a , b ]×ℝ×ℝ , such that L ( x , y ( x ), y ? ( x )) is integrable on [ a , b ] , ∀ y ∈ Y . In the preceding arclength example, L is given by L ( p ) = √ 1 + p 2 . To keep the presentation as simple as possible, here and in the sequel we will not deal with the least possible regularity. For example, though I [ y ] would make sense for y ∈ C 1 ([ a , b ]) , we take C 2 functions to avoid technicalities in what follows. The analysis of functionals as (I) is carried out in the Calculus of Variations . This is a branch of mathematical analysis dealing with geometrical or physical problems https://doi.org/10.1515/9783110652864-016
- N.N. Puri(Author)
- 2009(Publication Date)
- CRC Press(Publisher)
Chapter 8 Calculus of Variations 8.1 Introduction The Calculus of Variations plays a very important role in the design of optimal con-trol systems that can be stated as an extremal value problem. The purpose of this chapter is to make the reader familiar with the principles of variational calculus and thus prepare him for the synthesis of optimal control systems. In the next section, we introduce preliminaries involving the calculus of maxima and minima and then derives the various results in the Calculus of Variations. 8.2 Maxima, Minima, and Stationary Points 1. Extrema of a function of a single variable Given: A scalar function V = f ( y ) of a single variable y . The extrema points of a function are defined as those where its slope vanishes: d V d y = d d y f ( y ) = 0 678 Calculus of Variations Let y = y * be one of the extrema points. Assuming all the necessary derivatives exist and are continous, we can expand f ( y ) in the Taylor series about y * : 4 V = 4 f ( y ) = f ( y * + 4 y ) -f ( y * ) = d f d y y = y * 4 y + 1 2 d 2 f d y 2 y = y * 4 y 2 + H.O.T. (8.1) Neglecting higher order terms 4 V = 4 f ( y ) ≈ d f d y y = y * 4 y + 1 2 d 2 f d y 2 y = y * 4 y 2 The first term is called the “ First Variation ” δ f or the “variation” and the second term is called the Second Variation . At the extrema point y = y * it is necessary that the first variation, δ f = d f d y 4 y , vanishes for an arbitrarily small 4 y . Thus, d f d y = 0 at y = y * Necessary condition for an extrema Change in f ( y ) in the extremal point neighborhood is approximated by the ( 4 y ) 2 term. The classification of the extremals is given by the following: d f d y y = y * = 0 and d 2 f d y 2 y = y * > 0 , then f ( y ) has a local minimum < 0 , then f(y) has a local maximum = 0 , then f ( y ) has a “saddle” point Example 8.1: Find the extremal and its classification for f ( y ) = tan -1 y -tan -1 ky , 0 < k < 1
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