1 Variational Calculus Overview
1–1
Chapter 1: VARIATIONAL CALCULUS OVERVIEW
TABLE OF CONTENTS Page
§1.1.
Why Variational Calculus May be Good For You
1–3
§1.2.
Functionals versus Functions §1.2.1. Functions . . . . . . . . . . . . . . . . . . . §1.2.2. Functionals . . . . . . . . . . . . . . . . . §1.2.3. Basic 1D Functional . . . . . . . . . . . . . . . §1.2.4. Admissible Functions . . . . . . . . . . . . . . §1.2.5. Variation and Extrema of a Function . . . . . . . . . . §1.2.6. Extrema of Functionals . . . . . . . . . . . . . §1.2.7. Extrema of Functionals . . . . . . . . . . . . . .
1–4 1–5 1–5 1–5 1–7 1–8 1–9 1–9
§1.3.
Variational Problem §1.3.1. Admissible Functions §1.3.2. Admissible Variations §1.3.3. Extremals . . . .
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§1.1
WHY VARIATIONAL CALCULUS MAY BE GOOD FOR YOU
§1.1. Why Variational Calculus May be Good For You A quick motivational spiel extracted from the web-posted Encyclopœdia Britannica article follows. (Edited into better English.) Variational Calculus is the branch of mathematics concerned with the problem of finding a function for which the value of a certain integral is either the largest or the smallest possible. That integral is technically known as a functional. Many problems of this kind are easy to state, but their solutions often entail dif ficult procedures of differential calculus. In turn these typically involve ordinary differential equations (ODE) as well as partial differential equations (PDE)
The isoperimetric problem — that of finding, among all plane figures of a given perimeter, the one enclosing the greatest area — was known to Greek mathematicians of the 2nd century BC. The term has been extended in the modern era to mean any problem in variational calculus in which a function is to be made a maximum or a minimum, subject to an auxiliary condition called the isoperimetric constraint , although it may have nothing to do with perimeters. For example, the problem of finding a solid of given volume that has the least surface area is an isoperimetric problem, the given volume being the isoperimetric, or auxiliary, condition. The first problem of that nature stated using the just-invented calculus and studied by Newton ca. 1684, came from the field of aerodynamics: find the shape of a solid having a given volume that will encounter minimum resistance as it travels through the atmosphere at constant velocity. Modern interest in variational calculus began in 1696 when Johann Bernoulli proposed a brachistochrone (”least-time”) problem as a challenge to his peers. Suppose that a thin wire in the shape of a plane curve joins two points at different elevations. A bead is placed on the wire at the higher point and allowed to slide under gravity, starting from rest and assuming no friction. The question was: what should be the shape of the curve so that the bead will reach the lower point in the least time? The problem was solved independently by the proposer, his brother Jakob, Leibniz, L’Hˆopital, and Newton. Their basic idea was to set up an integral for the traversal time in terms of the unknown curve, and then vary the curve so that a minimum time is obtained. This technique, typical of variational calculus, leads to an ODE whose solution is a curve called the cycloid . It is possible to formulate various scientific laws in terms of general principles involving variational calculus. These are called variational principles and are usually expressed by stating that some integral is a maximum or a minimum. One example is Maupertuis’ principle of least action (ca. 1744), which sought to explain all processes of nature as driven by a demand that some property be economized or minimized. In particular, minimizing an integral, called the action, led several mathematicians (notably Euler and Lagrange in the 18th century and Hamilton in the 19th century) to a teleological explanation of Newton’s laws of motion. Nevertheless, a general appreciation of the action principles came only with its use in the 1940s as a foundation for quantum electrodynamics. Applications of variational principles also occur in solid and structural mechanics, fluid dynamics (aeroelasticity, hydroelasticity, poroelasticity, etc.) electromagnetics, thermodynamics, and guidance and control, as well as other areas in science and engineering. Some terminology pertaining to variational calculus is compiled in Table 1.1 for the convenience of the reader. It is not necessary to soak all those defintions right away — that will naturally happen as this and following chapters are traversed.
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Chapter 1: VARIATIONAL CALCULUS OVERVIEW
Table 1.1. Variational Calculus Terminology De finition ∗
Term Variational calculus
The branch of mathematics concerned with the manipulation of functionals to solve variational problems. Functional A scalar function of a function. Usually defined by an integral over a domain. The function argument of a functional. Nominally considered as one function Input function to keep definitions compact, but can be a vector of functions, cf. Notes below. Extremal An input function that renders a functional stationary with respect to function variations taken about that function. The functional is said to be extremalized. Extremizer An extremal that makes a functional a maximum or minimum. Variational problem The determination of input functions that extremalize a functional. Variational principle The linkage of functional extremalization to a scientific law. The integral that defines the functional, in the usual case noted above. De fining integral Region (interval in 1D, surface in 2D, volume in 3D, etc.) over which the Domain input function is defined. Usually the same as the defining integral domain. Boundary conditions The set of conditions that the input function must satisfy on the domain boundary. Often abbreviated to BC, as in the next two definitions. Essential BC The set of all BC that the input function must satisfy a priori to be admissible. The set of all BC that are not essential BC. Natural BC ∗∗ The set of values that the functional may take. Functional range ∗∗ The set of values that the input function may take. Function range Independent variable The input function argument(s). Often same as the domain position coordinates. Dependent variable Another name for input function, esp. when viewed as functional argument. Primary variable Name used for the input function when connected to a physical quantity. Admissible function An input function acceptable to a functional. Acceptance hinges on two conditions: (i) belonging to a function vector space that meets appropriate smoothness conditions, and (ii) satisfying essential BC a priori. The derivative(s) of the input function that appear in the functional. Input derivatives † Function increment A finite change of the input function. The incremented function must be admissible. Generally taken while keeping all independent variables frozen. ‡ The infinitesimal limit of a function increment. Function variation † The finite change in the functional resulting from a function increment. Functional increment ‡ Functional variation The infinitesimal limit of a functional increment. First variation The first-order term (in a Taylor series sense) of a functional variation. Second variation The second-order term (in a Taylor series sense) of a functional variation. Weak (strong) increment A function increment that forces (does not force) input derivatives to become infinitesimal when the function increment becomes infinitesimal. Weak (strong) variation The infinitesimal limit of a weak (strong) increment. Weak (strong) extremal An extremal with respect to weak (strong) variations of the input function Weak (strong) extremizer An extremizer with respect to weak (strong) variations of the input function Weak (strong) minimum A functional minimum associated with a weak (strong) extremizer. Weak (strong) maximum A functional maximum associated with a weak (strong) extremizer. Notes
∗ All definitions assume that the functional has nominally only one input function. Multiple input functions may be accommodated by thinking of the argument as a vector of functions.
∗∗ The term range, without qualifier, implies functional range.
† The term increment , without qualifier, implies functional increment. ‡ The term variation, without qualifier, implies functional variation.
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§1.2
FUNCTIONALS VERSUS FUNCTIONS
§1.2. Functionals versus Functions As Table 1.1 notes, Variational Calculus deals with functionals.1 Briefly, a functional is a function of a function. The difference between a functional and an ordinary function can be appreciated in the diagrams of Figure 1.1. §1.2.1. Functions
=
Figure 1.1(a) block diagrams an ordinary function y f ( x ), in which both x and y are scalars. Argument x is the independent variable. For each x that belong to a given set, called the function domain, the function operator f ( x ) produces another number: the function value. Function values span the function range. The picture has an “input-output machine” flavor; this metaphor illustrates that for each input x the “machine” (the operator box) returns a corresponding output y . The concept of function can be generalized to vectors and tensors. The general definition is: a single-valued function is a relation between a set of inputs and a set of outputs with the property that each input is related to exactly one output. Example: y sin( x ) for real x . Since about 1950, the term “function” without qualifier, generally means single-valued .
=
If an input can produce multiple outputs, the function is said to be multivalued . For example n y x for complex x returns two values, and y x (with n a positive integer) returns n values.
= √
= √
§1.2.2. Functionals A functional is a more complicated input-output machine. It receives a function and produces a number. A simple one-dimensional case is diagrammed in Figure 1.1(b). The input to the functional operator J is the function y ( x ) of Figure 1.1(a). This is called the primary dependent variable, or primary variable for short. The functional value output is a scalar. Notations used for this case2 are (1.1) J J ( x , y ), and J F [ y ].
=
=
The first form explicitly shows the two inputs: x and y ( x ), in function style. The second form uses square brackets and only shows the function y , the dependence y f ( x ) being implied from context.
=
Functionals that appear in practice often depend on function derivatives, such as the one shown in Figure 1.1(c). Here F depends on y ( x ) d y /d x . Notations used for this case are
=
= J ( x , y, y),
J
and
= J [ y].
J
(1.2)
The first one explicitly shows the three inputs: x , y ( x ) and y ( x ). The second one, with square brackets, shows only the function y f ( x ); dependence on x and y ( x ) being implied from context.
=
Functionals may also depend on higher derivatives such as y ( x ) d 2 y /d x 2 . Other important extensions include passing to multiple dimensions, and satisfaction of algebraic and differential constraint conditions. Such extensions will be illustrated in ensuing chapters.
=
1
Why is this branch not named “functional calculus”? Historical reasons. See Notes and Bibliography at end of Chap ter.
2
The compact xzsquare bracket notation follows thr textbook of Gelfand and Fomin [289]. That notation is only used by several journals, notably those of the Americal Physics Society (APS). See http://publish.aps.org.
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Chapter 1: VARIATIONAL CALCULUS OVERVIEW
Input: argument x (independent variable)
Function operator
Output: function value y (dependent variable)
x
f
y=y( x)=f ( x)
(a)
FUNCTIONS Input 1: argument x (independent variable)
(b)
x
Functional operator
Output: functional value J (a scalar)
y=f ( x)
J
J [ y]= J ( x,y)
Input 2: function y=y(x) (primary dependent variable)
Functional operator
Output: functional value J (a scalar)
y=f ( x)
J
f
Input 1: argument x (independent variable)
(c)
Input 2: function y=y(x) (primary dependent variable)
f
x
Input 3: derivative of primary dependent variable
J [ y]= J ( x,y,y' )
y'=dy / dx FUNCTIONALS
Figure 1.1.
Block diagrams that illustrate keydifferencesbetween functions and functionals in one dimension. (a) An ordinary function y y ( x ) f ( x) of the independent variable x ; (b) a functional J [ y ] J ( x , y) of the function y( x); (c) a functional J [ y ] J ( x, y, y ) of the function y( x ) and its derivative y d y/d x .
=
=
= =
=
§1.2.3. Basic 1D Functional To keep things simple, in this Chapter we focus on one specific type of functional, called the basic one-dimensional functional. It has the form (1.2), under additional restrictions: b
J [ y ]
=
F x , y ( x ), y ( x ) ,
a
x
= [a, b], a ≤ b,
y (a )
=
∈
= yˆ , a
y (b )
= yˆ . b
(1.3)
≤
In words: the function y f ( x ) is defined over the segment x [ a , b], a b , of the real line. Given x , y is assumed real and unique; that is, y ( x ) is single-valued. Furthermore y ( x ) possesses the appropriate smoothness so that y ( x ), as well as the integral in (1.3), exist. That function must satisfy the stated end conditions at x a and x b, at which ya y (a ) and yb y (b) are given.3 Those boundary conditions will be called essential.
=
=
ˆ =
ˆ =
The basicfunctional (1.3) is historically interesting since it belongs to a class of problems that prompted the birth and development of variational calculus in the XVIII Century. But it also serves as a good expository tool. Some specific examples follow.
3
The overhat over a symbol will be often used to denote prescribed values.
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§1.2
y
y
(a) y=y( x)
A y(a)=y^a
FUNCTIONALS VERSUS FUNCTIONS
y
(b) Arclength L
A B
Area A
Figure 1.2.
B b
x
Constant gravity g Straight line Cycloid
y(b)=y^ x=b
x=a
A
(c)
B
Parabola
x=b
x=a
x
x=a
x=b
x
Canonical one-dimensional functionals used in the examples of ?. (a) area under curve; (b) curve arclength, (c) brachistochrone.
Example 1.1. Area Under Curve. The area subtented by y( x ) and the real axis is given by b
A
=
(1.4)
y( x) d x.
a
See Figure 1.2(a). This has the form illustrated in Figure 1.1(b). Example 1.2. Curve Length. The length of the curve y ( x ) between x
conditions is given by
∈ [a, b] and satisfying the given end
b
L
=
+
1
a
+ ( y )
2
dx .
(1.5)
in which the sign of the square root is taken. See Figure 1.2(b). (If this L is uniquely defined by this integral the curve is said to be recti fiable.) This functional has the form illustrated in Figure 1.1(c), except that there is no explicit dependence on y ; only on its slope. Example 1.3. Brachistochrone. This is illustrated in Figure 1.2(c). A point-mass body is released at A at zero time. It is constrained to move under the influence of constant gravity g acting along y and without friction, y( x) that ends at B, which is lower than A. (The mass is pictured as a rolling disk in the along a curve y figure, but it is actually a sliding point — no rotation is involved.) The curve y ( x ) is called the trajectory or path. The traversal time taken by mass to go from A to B is given by the functional
−
=
+ b
T A B
=
1
( y )2
2g y
a
d x.
(1.6)
in whch g is the acceleration of gravity. This functional fits the form illustrated in Figure 1.1(c). The derivation of this functional is the matter of an Exercise. An important feature is to find the trajectory that minimizes T ; this is called the brachistochrone problem. We shall later show that the solution of this fastest descent time is a cycloid, colored red in 1.2(c). Example 1.4. Action Integral. In analytical dynamics (including both Newtonian and quantum mechanics) a functional like (1.3), with the integral taken over a time interval t [t a , t b ] is called the action. Its integrand F is known as the Lagrangian, which is often denoted by L or L .
∈
§1.2.4. Admissible Functions Key question: what sort of function can be fed into a functional? The question is similar to: what kind of argument can a function have? but more complicated, because we are talking about a 1–7
Chapter 1: VARIATIONAL CALCULUS OVERVIEW
(a)
(b) 1
y
B
B
y
2
1 2
3
A y(a)=y^a
A
y(b)=y^b
4
y(a)=y^a
5
y(b)=y^b
3 4 5
x
x=b
x=a
x=b
x=a
x
Figure 1.3. Visualization of function admissibity concept as regards (1.3): (a) sample admissible functions y( x ) over that satisfy: C 1 continuity(i.e., unique tangentat eachpoint), and essential BC,and single valuedness; (b) sample inadmissible functions that violate one or more of the foregoing conditions: curves 1 and 3 have corners, curve 2 is discontinuous, curve 4 is multivalued, and curve 5 violates the right essential BC.
function rather than just a number. A function that is permissible as input to a functional is called admissible. The set of such functions is the admissible class with respect to the functional under consideration. The following general guidelines may be offered. Function Smoothness. Admissible functions are usually chosen to have the minimal smoothness for which the integration over the problem domain makes sense. For example, if the functional has the form (1.2) the presence of y means that it is reasonable to ask that y ( x ) have integrable derivatives. For this to happen, it is suf ficient that y ( x ) be piecewise continuous. Requiring y ( x ) C 1 over the domain would be overkill but safe.
∈
=ˆ
=ˆ
End Conditions. Prescribed end values, such as y (a ) ya , and y (b) yb in (1.3), must be satisfied a priori. As previously noted, this kind of specification is called an essential boundary condition. Single Valuedness. This requirement is optional. It is often stipulated a priori to simplify the formulation and analysis. But it must be abandoned for certain problems. For example, those that involved closed curves or spirals.4 Real or Complex Values. If the functional is intended to model a physical problem, the choice should be obvious. Else it is a fielder’s choice re generality.
Figure 1.3 illustrates admissible function classes for the basicfunctional (1.3). Three requirements are laid down: (1) y ( x ) is real and C 1 [a , b], whence the function must be continuous and possess continuous first derivatives; (2) satisfy essential BC at both ends; and (3) be single valued. All curves drawn in Figure 1.3(a) are admissible in this sense, whereas those in Figure 1.3(b) are not; see figure legend as to why.
∈
§1.2.5. Variation and Extrema of a Function The concept of variation is fundamental since it gives variational calculus its name. It is an extension of the well known concept of differential in standard calculus. This is briefly reviewed below. Consider an ordinary 1D function y ( x ), where y and x are real and y ( x ) possesses unique derivative x . The function y ( x ) at each x in the problem domain x [ a , b]. Change a given x bytto x
∈
4
+
Sometimes single valuedness may be often restored through a parametric representation in non-Cartesian coordinates. This is often the case with isoperimetric problems.
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§1.3
VARIATIONAL PROBLEM
changes by y x ) y ( x ) y ( x y ( x ) x O ( x 2 ) Make x infinitesimal so that x d x , and y d y to first order one has dy y ( x ) d x so y ( x ) d y /d x The differentials are d y and d x . If further derivatives exist we can construct the Taylor series expansion
=
→
→
+
−
=
+ | | =
+ d x ) = y ( x ) + y ( x ) d x +
1 y 2
y ( x
=
( x ) (d x )2 + . . .
(1.7)
An function extremum of y ( x ) is a location x ∗ at which y ( x ) has a minimum, maximum or inflexion point within the problem domain: x 0, (a , b). At that point, calculus tells us that y ( x ∗ ) whence d y 0 at x ∗ since d x is nonzero.
∈
=
=
Moving to functionals, the analog of x is y ( x ), and that of y ( x ) is J [ y ]. Take two admissible functions: y ( x ) and y ( x ) h ( x ). The difference h ( x ) is called a finite variation see Figure. This is the analog of the function increment y x ) y ( x ) of ordinary calculus. The y ( x correspondng functional values, in abbreviated bracket notation, are J [ y ] and J [ y y ]. The functional finite variation is J y ] J [ y J [ y ]. To find the analog of d y , parametrize parametrize the function variation as y where varies from 0 to 1; see Figure. Expand about 0 using a Taylor series analog:
+
=
= −
+
+
−
+
=
J [ y
def
+ y] =
J [ y ]
+ δ J [ y ] +
1 2 2
in which δ J [ y ]
=
2
+ . . . =
δ J
1 J [ y
δ J δ y δ y
+ u]
→
+ δ y +
1 2 2
δ 2 J 2 δ y δ y 2
+ ...
(1.8)
(1.9)
, et c
0
Here δ J is called the variation of J [ y ] with respect to y ; it is the analog of d y . Likewise, δ 2 J is the second variation, which is the analog of d y . And so on. In ordinary calculus, d y y dx . the analog is δ J J u δ y . (The symbol δ was introduced by lagrange to emphasize the analogy).
=
=
§1.2.6. Extrema of Functionals An function extremum of y ( x ) is a location x ∗ at which y ( x ) has a minimum, maximum or inflexion point within the problem domain: x (a , b). At that point, calculus tells us that y ( x ∗ ) 0, ∗ whence d y 0 at x since d x is nonzero.
∈
=
=
§1.2.7. Extrema of Functionals An function extremum of y ( x ) is a location x ∗ at which y ( x ) has a minimum, maximum or inflexion point within the problem domain: x (a , b). At that point, calculus tells us that y ( x ∗ ) 0, ∗ whence d y 0 at x since d x is nonzero.
∈
=
=
§1.3. Variational Problem The most important application of functionals is associated with the so-called variational problem. Given a functional J [ y ] with possible some end conditions, find the function(s) for which J is extremized with respect to admissible variations of the function. In summary: Find the function(s) that extremize the functional This statement requires some clarification. What are admissble functions? What is a variation? What kind of extrema are of interest? 1–9
Chapter 1: VARIATIONAL CALCULUS OVERVIEW
(a)
B
y
δ y(b)=0 δ y( x)
A y(a)=y^
a
(b)
y( x)+δ y( x)
y(b)=y^b δ y(a)=0
B
y A
y( x)
y( x) x=b
x=a Figure 1.4.
y( x)+ε δ y( x), ε=1/4,1/2,3/4,1
x
x=a
x=b
x
Concept of a function variation as difference between two admnissible functions: (a) an admissible variation; (b) -parametrized version of (a).
§1.3.1. Admissible Functions An important concept is the admissible class of functions that are permissible in the functional. This is generally chosen to be the class with minimal restrictions for which that the functional can make sense. For the functionals of Exampoles 2 and 3, the presence of y means that it is reasonable to ask that y ( x ) have integrable derivatives. In addition the function should be single valued between a and b. §1.3.2. Admissible Variations §1.3.3. Extremals The central problem of the Calculus of Variations is: which function(s) make the functional stationary with respect to variations of the function? Such functions are called extremals. This problem is solved by considering the first variation of the functional. As in ordinary calculus, a stationary point of J can be either a maximum, a minimum, or an inflexion point. Such classification conditions depend on the second variation of the functional.
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