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Calculus of Variations
Joel G. Broida
University of Colorado, Boulder
Copyright
c?2009 by Joel G. Broida. All rights reserved.
October 1, 2009
0
Calculus of Variations
1 Functional Derivatives
The fundamental equation of the calculus of variations is the Euler-Lagrange equation d dt? ∂f∂x? -∂f∂x= 0. There are several ways to derive this result, and we will cover three of the most common approaches. Our first method I think gives the most intuitive treatment, and this will then serve as the model for the othermethods that follow. To begin with, recall that a (real-valued)functiononRnis a mapping f:U?Rn→R. In other words,ftakes apointin some subsetUofRn and gives back a number, i.e., a point inR. In particular, the domain offis a subset ofRn. We write this mapping asf(x). In contrast to this, afunctionalFis a "function" whose domain is the space ofcurvesinRn, and hence it depends on theentire curve, not just a single point. Very loosely speaking, we will take acurveto be a differentiable mappingy:U?Rn→Rm. So a curve is just a function defined on some interval, and a functional is a "function of a function." For example, lety(x) be a real valued curve defined on the interval [x1,x2]?
R. Then we can define a functionalF[y] by
F[y] :=?
x2 x
1[y(x)]2dx?R.
(The notationF[y] is the standard way to denote a functional.) So a functional is a mapping from the space of curves into the real numbers. We now want to define the derivative of such a functional. There are several ways to go about this, and we will take the most intuitive approach that is by analogy with the usual notion of derivative. So, letf(t) be a function of a single real variable, and recall the definition of the derivativef?(t): f ?(t) =df dt(t) = limh→0f(t+h)-f(t)h.(1) This is equivalent to saying thatfis differentiable attif there exists some numberL(called thederivativeoffatt) and a function?with the property that lim h→0?(h) h= 0 such that f(t+h) =f(t) +Lh+?(h).(2) 1 Before proving the equivalence of these formulations, let me make two re- marks. First, we say that such a function?(h) isO(h2) (orderh2). And second, note that the numberLis just a linear map fromRtoR. (In this case, L:R→Ris defined byL(h) =Lhforh?R.) In fact, it is this formulation of the derivative that is used to generalize differentiation tofunctions fromRnto R m, in which case the linear mapLbecomes the Jacobian matrix (∂yi/∂xj). Let us now show that equations (1) and (2) are equivalent. Note that if we start from (1) anddefinethe function?by ?(h) =? f(t+h)-f(t)-f?(t)hforh?= 0
0 forh= 0
then f(t+h) =f(t) +Lh+?(h) whereL=f?(t) and (by equation (1)) lim?(h)/h= 0. Conversely, if we start from equation (2), then f(t+h)-f(t) h=L+?(h)h and taking the limit ash→0 we see thatf?(t) =L. Now let us return to functionals. Letγbe a curve in the plane:
γ={(t,x) :x(t) =xfort0< t < t1}.
Let?γbe an approximation toγ, i.e.,
?γ={(t,x) :x=x(t) +h(t)} for some functionh(t). We abbreviate this by?γ=γ+h. LetFbe a functional and consider the differenceF[?γ]-F[γ] =F[γ+h]-F[γ]. t0t1x t
γeγ
We say thatFisdifferentiableif there exists a linear mapL(i.e., for fixedγ we haveL(h1+h2) =L(h1)+L(h2) andL(ch) =cL(h)) and a remainderR(h,γ) with the property thatR(h,γ) =O(h2) (i.e., for|h|< εand|h?|=|dh/dt|< ε we have|R|
F[γ+h]-F[γ] =L(h) +R(h,γ) (3) 2 The linear part of equation (3),L(h), is called thedifferentialofF. We now want to prove the following theorem. As is common, we will denote the derivative with respect totby a dot, although in this casetis not necessarily the time - it is simply the independent variable. Theorem 1.Letγbe a curve in the plane, and letf=f(x(t),x(t),t)be a differentiable function. Then the functional F[γ] =?
t1 t 0f(x(t),x(t),t)dt
is differentiable and its derivative is given by L(h) =?
t1 t 0? ∂f ∂x-ddx? ∂f∂x?? hdt+∂f∂xh????t 1 t 0(4) Proof.Sincefis a differentiable function we have (using equation (2) in the case wherefis a function of the two variablesxand x)quotesdbs_dbs2.pdfusesText_3
The Calculus of Variations - University of Minnesota