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Calculus of variations

From Wikipedia, the free encyclopedia
(Redirected fromVariational calculus)
Differential calculus on function spaces
"Variational method" redirects here. For the use as an approximation method in quantum mechanics, seeVariational method (quantum mechanics).
Part of a series of articles about
Calculus
∫abf′(t)dt=f(b)−f(a){\displaystyle \int _{a}^{b}f'(t)\,dt=f(b)-f(a)}

Thecalculus of variations (orvariational calculus) is a field ofmathematical analysis that uses variations, which are small changes infunctionsandfunctionals, to find maxima and minima of functionals:mappings from a set offunctions to thereal numbers.[a] Functionals are often expressed asdefinite integrals involving functions and theirderivatives. Functions that maximize or minimize functionals may be found using theEuler–Lagrange equation of the calculus of variations.

A simple example of such a problem is to find the curve of shortest length connecting two points. If there are no constraints, the solution is astraight line between the points. However, if the curve is constrained to lie on a surface in space, then the solution is less obvious, and possibly many solutions may exist. Such solutions are known asgeodesics. A related problem is posed byFermat's principle: light follows the path of shortestoptical length connecting two points, which depends upon the material of the medium. One corresponding concept inmechanics is theprinciple of least/stationary action.

Many important problems involve functions of several variables. Solutions ofboundary value problems for theLaplace equation satisfy theDirichlet's principle.Plateau's problem requires finding a surface of minimal area that spans a given contour in space: a solution can often be found by dipping a frame in soapy water. Although such experiments are relatively easy to perform, their mathematical formulation is far from simple: there may be more than one locally minimizing surface, and they may have non-trivialtopology.

History

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The calculus of variations began with the work ofIsaac Newton, such as withNewton's minimal resistance problem, which he formulated and solved in 1685, and later published in hisPrincipia in 1687,[2] which was the first problem in the field to be formulated and correctly solved,[2] and was also one of the most difficult problems tackled by variational methods prior to the twentieth century.[3][4][5] This problem was followed by thebrachistochrone curve problem raised byJohann Bernoulli (1696),[6] which was similar to one raised byGalileo Galilei in 1638, but he did not solve the problem explicitly nor did he use the methods based on calculus.[3] Bernoulli solved the problem using the principle of least time in the process, but not calculus of variations. In 1697 Newton solved the problem using variational techniques, and as a result, he pioneered the field with his work on the two problems.[4] The problem would immediately occupy the attention ofJacob Bernoulli and theMarquis de l'Hôpital, butLeonhard Euler first elaborated the subject, beginning in 1733.Joseph-Louis Lagrange was influenced by Euler's work to contribute greatly 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 thecalculus of variations in his 1756 lectureElementa Calculi Variationum.[7][8][b]

Adrien-Marie Legendre (1786) laid down a method, not entirely satisfactory, for the discrimination of maxima and minima.Isaac Newton andGottfried Leibniz also gave some early attention to the subject.[9] To this discriminationVincenzo Brunacci (1810),Carl Friedrich Gauss (1829),Siméon Poisson (1831),Mikhail Ostrogradsky (1834), andCarl Jacobi (1837) have been among the contributors. An important general work is that ofPierre Frédéric Sarrus (1842) which was condensed and improved byAugustin-Louis Cauchy (1844). Other valuable treatises and memoirs have been written byStrauch[which?] (1849),John Hewitt Jellett (1850),Otto Hesse (1857),Alfred Clebsch (1858), and Lewis Buffett Carll (1885), but perhaps the most important work of the century is that ofKarl Weierstrass. His celebrated course on the theory is epoch-making, and it may be asserted that he was the first to place it on a firm and unquestionable foundation. The20th and the23rdHilbert problem published in 1900 encouraged further development.[9]

In the 20th centuryDavid Hilbert,Oskar Bolza,Gilbert Ames Bliss,Emmy Noether,Leonida Tonelli,Henri Lebesgue andJacques Hadamard among others made significant contributions.[9]Marston Morse applied calculus of variations in what is now calledMorse theory.[10]Lev Pontryagin,Ralph Rockafellar and F. H. Clarke developed new mathematical tools for the calculus of variations inoptimal control theory.[10] Thedynamic programming ofRichard Bellman is an alternative to the calculus of variations.[11][12][13][c]

Extrema

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The calculus of variations is concerned with the maxima or minima (collectively calledextrema) of functionals. A functional mapsfunctions toscalars, so functionals have been described as "functions of functions." Functionals have extrema with respect to the elementsy{\displaystyle y} of a givenfunction space defined over a givendomain. A functionalJ[y]{\displaystyle J[y]} is said to have an extremum at the functionf{\displaystyle f} ifΔJ=J[y]−J[f]{\displaystyle \Delta J=J[y]-J[f]} has the samesign for ally{\displaystyle y} in an arbitrarily small neighborhood off.{\displaystyle f.}[d] The functionf{\displaystyle f} is called anextremal function or extremal.[e] The extremumJ[f]{\displaystyle J[f]} is called a local maximum ifΔJ≤0{\displaystyle \Delta J\leq 0} everywhere in an arbitrarily small neighborhood off,{\displaystyle f,} and a local minimum ifΔJ≥0{\displaystyle \Delta J\geq 0} there. For a function space of continuous functions, extrema of corresponding functionals are calledstrong extrema orweak extrema, depending on whether the first derivatives of the continuous functions are respectively all continuous or not.[15]

Examples where calculus of variations can be applied- finding minimal surfaces, finding geodesics, deriving Snell's law of refraction, getting an equation to solve the double pendulum problem numerically

Both strong and weak extrema of functionals are for a space of continuous functions but strong extrema have the additional requirement that the first derivatives of the functions in the space be continuous. Thus a strong extremum is also a weak extremum, but theconverse may not hold. Finding strong extrema is more difficult than finding weak extrema.[16] An example of anecessary condition that is used for finding weak extrema is theEuler–Lagrange equation.[17][f]

Euler–Lagrange equation

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Main article:Euler–Lagrange equation

Finding the extrema of functionals is similar to finding the maxima and minima of functions. The maxima and minima of a function may be located by finding the points where its derivative vanishes (i.e., is equal to zero). The extrema of functionals may be obtained by finding functions for which thefunctional derivative is equal to zero. This leads to solving the associatedEuler–Lagrange equation.[g]

Consider the functional

J[y]=∫x1x2L(x,y(x),y′(x))dx,{\displaystyle J[y]=\int _{x_{1}}^{x_{2}}L\left(x,y(x),y'(x)\right)\,dx,}

where

If the functionalJ[y]{\displaystyle J[y]} attains alocal minimum atf,{\displaystyle f,} andη(x){\displaystyle \eta (x)} is an arbitrary function that has at least one derivative and vanishes at the endpointsx1{\displaystyle x_{1}} andx2,{\displaystyle x_{2},} then for any numberε{\displaystyle \varepsilon } close to 0,

J[f]≤J[f+εη].{\displaystyle J[f]\leq J[f+\varepsilon \eta ]\,.}

The termεη{\displaystyle \varepsilon \eta } is called thevariation of the functionf{\displaystyle f} and is denoted byδf.{\displaystyle \delta f.}[1][h]

Substitutingf+εη{\displaystyle f+\varepsilon \eta } fory{\displaystyle y} in the functionalJ[y],{\displaystyle J[y],} the result is a function ofε,{\displaystyle \varepsilon ,}

Φ(ε)=J[f+εη].{\displaystyle \Phi (\varepsilon )=J[f+\varepsilon \eta ]\,.}

Since the functionalJ[y]{\displaystyle J[y]} has a minimum fory=f{\displaystyle y=f} the functionΦ(ε){\displaystyle \Phi (\varepsilon )} has a minimum atε=0{\displaystyle \varepsilon =0} and thus,[i]

Φ′(0)≡dΦdε|ε=0=∫x1x2dLdε|ε=0dx=0.{\displaystyle \Phi '(0)\equiv \left.{\frac {d\Phi }{d\varepsilon }}\right|_{\varepsilon =0}=\int _{x_{1}}^{x_{2}}\left.{\frac {dL}{d\varepsilon }}\right|_{\varepsilon =0}dx=0\,.}

Taking thetotal derivative ofL[x,y,y′],{\displaystyle L\left[x,y,y'\right],} wherey=f+εη{\displaystyle y=f+\varepsilon \eta } andy′=f′+εη′{\displaystyle y'=f'+\varepsilon \eta '} are considered as functions ofε{\displaystyle \varepsilon } rather thanx,{\displaystyle x,} yields

dLdε=∂L∂ydydε+∂L∂y′dy′dε{\displaystyle {\frac {dL}{d\varepsilon }}={\frac {\partial L}{\partial y}}{\frac {dy}{d\varepsilon }}+{\frac {\partial L}{\partial y'}}{\frac {dy'}{d\varepsilon }}}

and becausedydε=η{\displaystyle {\frac {dy}{d\varepsilon }}=\eta } anddy′dε=η′,{\displaystyle {\frac {dy'}{d\varepsilon }}=\eta ',}

dLdε=∂L∂yη+∂L∂y′η′.{\displaystyle {\frac {dL}{d\varepsilon }}={\frac {\partial L}{\partial y}}\eta +{\frac {\partial L}{\partial y'}}\eta '.}

Therefore,

∫x1x2dLdε|ε=0dx=∫x1x2(∂L∂fη+∂L∂f′η′)dx=∫x1x2∂L∂fηdx+∂L∂f′η|x1x2−∫x1x2ηddx∂L∂f′dx=∫x1x2(∂L∂fη−ηddx∂L∂f′)dx{\displaystyle {\begin{aligned}\int _{x_{1}}^{x_{2}}\left.{\frac {dL}{d\varepsilon }}\right|_{\varepsilon =0}dx&=\int _{x_{1}}^{x_{2}}\left({\frac {\partial L}{\partial f}}\eta +{\frac {\partial L}{\partial f'}}\eta '\right)\,dx\\&=\int _{x_{1}}^{x_{2}}{\frac {\partial L}{\partial f}}\eta \,dx+\left.{\frac {\partial L}{\partial f'}}\eta \right|_{x_{1}}^{x_{2}}-\int _{x_{1}}^{x_{2}}\eta {\frac {d}{dx}}{\frac {\partial L}{\partial f'}}\,dx\\&=\int _{x_{1}}^{x_{2}}\left({\frac {\partial L}{\partial f}}\eta -\eta {\frac {d}{dx}}{\frac {\partial L}{\partial f'}}\right)\,dx\\\end{aligned}}}

whereL[x,y,y′]→L[x,f,f′]{\displaystyle L\left[x,y,y'\right]\to L\left[x,f,f'\right]} whenε=0{\displaystyle \varepsilon =0} and we have usedintegration by parts on the second term. The second term on the second line vanishes becauseη=0{\displaystyle \eta =0} atx1{\displaystyle x_{1}} andx2{\displaystyle x_{2}} by definition. Also, as previously mentioned the left side of the equation is zero so that

∫x1x2η(x)(∂L∂f−ddx∂L∂f′)dx=0.{\displaystyle \int _{x_{1}}^{x_{2}}\eta (x)\left({\frac {\partial L}{\partial f}}-{\frac {d}{dx}}{\frac {\partial L}{\partial f'}}\right)\,dx=0\,.}

According to thefundamental lemma of calculus of variations, the fact that this equation holds for any choice ofη{\displaystyle \eta } implies that the part of the integrand in parentheses is zero, i.e.

∂L∂f−ddx∂L∂f′=0{\displaystyle {\frac {\partial L}{\partial f}}-{\frac {d}{dx}}{\frac {\partial L}{\partial f'}}=0}

which is called theEuler–Lagrange equation. The left hand side of this equation is called thefunctional derivative ofJ[f]{\displaystyle J[f]} and is denotedδJ{\displaystyle \delta J} orδf(x).{\displaystyle \delta f(x).}

In general this gives a second-orderordinary differential equation which can be solved to obtain the extremal functionf(x).{\displaystyle f(x).} The Euler–Lagrange equation is anecessary, but notsufficient, condition for an extremumJ[f].{\displaystyle J[f].} A sufficient condition for a minimum is given in the sectionVariations and sufficient condition for a minimum.

Example

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In order to illustrate this process, consider the problem of finding the extremal functiony=f(x),{\displaystyle y=f(x),} which is the shortest curve that connects two points(x1,y1){\displaystyle \left(x_{1},y_{1}\right)} and(x2,y2).{\displaystyle \left(x_{2},y_{2}\right).} Thearc length of the curve is given by

A[y]=∫x1x21+[y′(x)]2dx,{\displaystyle A[y]=\int _{x_{1}}^{x_{2}}{\sqrt {1+[y'(x)]^{2}}}\,dx\,,}

with

y′(x)=dydx,  y1=f(x1),  y2=f(x2).{\displaystyle y'(x)={\frac {dy}{dx}}\,,\ \ y_{1}=f(x_{1})\,,\ \ y_{2}=f(x_{2})\,.}

Note that assumingy is a function ofx loses generality; ideally both should be a function of some other parameter. This approach is good solely for instructive purposes.

The Euler–Lagrange equation will now be used to find the extremal functionf(x){\displaystyle f(x)} that minimizes the functionalA[y].{\displaystyle A[y].}

∂L∂f−ddx∂L∂f′=0{\displaystyle {\frac {\partial L}{\partial f}}-{\frac {d}{dx}}{\frac {\partial L}{\partial f'}}=0}

with

L=1+[f′(x)]2.{\displaystyle L={\sqrt {1+[f'(x)]^{2}}}\,.}

Sincef{\displaystyle f} does not appear explicitly inL,{\displaystyle L,} the first term in the Euler–Lagrange equation vanishes for allf(x){\displaystyle f(x)} and thus,

ddx∂L∂f′=0.{\displaystyle {\frac {d}{dx}}{\frac {\partial L}{\partial f'}}=0\,.}

Substituting forL{\displaystyle L} and taking the derivative,

ddx f′(x)1+[f′(x)]2 =0.{\displaystyle {\frac {d}{dx}}\ {\frac {f'(x)}{\sqrt {1+[f'(x)]^{2}}}}\ =0\,.}

Thus

f′(x)1+[f′(x)]2=c,{\displaystyle {\frac {f'(x)}{\sqrt {1+[f'(x)]^{2}}}}=c\,,}

for some constantc{\displaystyle c}. Then

[f′(x)]21+[f′(x)]2=c2,{\displaystyle {\frac {[f'(x)]^{2}}{1+[f'(x)]^{2}}}=c^{2}\,,}

where

0≤c2<1.{\displaystyle 0\leq c^{2}<1.}

Solving, we get

[f′(x)]2=c21−c2{\displaystyle [f'(x)]^{2}={\frac {c^{2}}{1-c^{2}}}}

which implies that

f′(x)=m{\displaystyle f'(x)=m}

is a constant and therefore that the shortest curve that connects two points(x1,y1){\displaystyle \left(x_{1},y_{1}\right)} and(x2,y2){\displaystyle \left(x_{2},y_{2}\right)} is

f(x)=mx+bwith  m=y2−y1x2−x1andb=x2y1−x1y2x2−x1{\displaystyle f(x)=mx+b\qquad {\text{with}}\ \ m={\frac {y_{2}-y_{1}}{x_{2}-x_{1}}}\quad {\text{and}}\quad b={\frac {x_{2}y_{1}-x_{1}y_{2}}{x_{2}-x_{1}}}}

and we have thus found the extremal functionf(x){\displaystyle f(x)} that minimizes the functionalA[y]{\displaystyle A[y]} so thatA[f]{\displaystyle A[f]} is a minimum. The equation for a straight line isy=mx+b.{\displaystyle y=mx+b.} In other words, the shortest distance between two points is a straight line.[j]

Beltrami's identity

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In physics problems it may be the case that∂L∂x=0,{\displaystyle {\frac {\partial L}{\partial x}}=0,} meaning the integrand is a function off(x){\displaystyle f(x)} andf′(x){\displaystyle f'(x)} butx{\displaystyle x} does not appear separately. In that case, the Euler–Lagrange equation can be simplified to theBeltrami identity[20]

L−f′∂L∂f′=C,{\displaystyle L-f'{\frac {\partial L}{\partial f'}}=C\,,}

whereC{\displaystyle C} is a constant. The left hand side is theLegendre transformation ofL{\displaystyle L} with respect tof′(x).{\displaystyle f'(x).}

The intuition behind this result is that, if the variablex{\displaystyle x} is actually time, then the statement∂L∂x=0{\displaystyle {\frac {\partial L}{\partial x}}=0} implies that the Lagrangian is time-independent. ByNoether's theorem, there is an associated conserved quantity. In this case, this quantity is the Hamiltonian, the Legendre transform of the Lagrangian, which (often) coincides with the energy of the system. This is (minus) the constant in Beltrami's identity.

Euler–Poisson equation

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IfS{\displaystyle S} depends on higher-derivatives ofy(x){\displaystyle y(x)}, that is, if

S=∫abf(x,y(x),y′(x),…,y(n)(x))dx,{\displaystyle S=\int _{a}^{b}f(x,y(x),y'(x),\dots ,y^{(n)}(x))dx,}

theny{\displaystyle y} must satisfy the Euler–Poisson equation,[21]

∂f∂y−ddx(∂f∂y′)+⋯+(−1)ndndxn[∂f∂y(n)]=0.{\displaystyle {\frac {\partial f}{\partial y}}-{\frac {d}{dx}}\left({\frac {\partial f}{\partial y'}}\right)+\dots +(-1)^{n}{\frac {d^{n}}{dx^{n}}}\left[{\frac {\partial f}{\partial y^{(n)}}}\right]=0.}

Du Bois-Reymond's theorem

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The discussion thus far has assumed that extremal functions possess two continuous derivatives, although the existence of the integralJ{\displaystyle J} requires only first derivatives of trial functions. The condition that the first variation vanishes at an extremal may be regarded as aweak form of the Euler–Lagrange equation. The theorem of Du Bois-Reymond asserts that this weak form implies the strong form. IfL{\displaystyle L} has continuous first and second derivatives with respect to all of its arguments, and if

∂2L∂f′2≠0,{\displaystyle {\frac {\partial ^{2}L}{\partial f'^{2}}}\neq 0,}

thenf{\displaystyle f} has two continuous derivatives, and it satisfies the Euler–Lagrange equation.

Lavrentiev phenomenon

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Hilbert was the first to give good conditions for the Euler–Lagrange equations to give a stationary solution. Within a convex area and a positive thrice differentiable Lagrangian the solutions are composed of a countable collection of sections that either go along the boundary or satisfy the Euler–Lagrange equations in the interior.

HoweverLavrentiev in 1926 showed that there are circumstances where there is no optimum solution but one can be approached arbitrarily closely by increasing numbers of sections. The Lavrentiev Phenomenon identifies a difference in the infimum of a minimization problem across different classes of admissible functions. For instance the following problem, presented by Manià in 1934:[22]

L[x]=∫01(x3−t)2x′6,{\displaystyle L[x]=\int _{0}^{1}(x^{3}-t)^{2}x'^{6},}

A={x∈W1,1(0,1):x(0)=0, x(1)=1}.{\displaystyle {A}=\{x\in W^{1,1}(0,1):x(0)=0,\ x(1)=1\}.}

Clearly,x(t)=t13{\displaystyle x(t)=t^{\frac {1}{3}}}minimizes the functional, but we find any functionx∈W1,∞{\displaystyle x\in W^{1,\infty }} gives a value bounded away from the infimum.

Examples (in one-dimension) are traditionally manifested acrossW1,1{\displaystyle W^{1,1}} andW1,∞,{\displaystyle W^{1,\infty },} but Ball and Mizel[23] procured the first functional that displayed Lavrentiev's Phenomenon acrossW1,p{\displaystyle W^{1,p}} andW1,q{\displaystyle W^{1,q}} for1≤p<q<∞.{\displaystyle 1\leq p<q<\infty .} There are several results that gives criteria under which the phenomenon does not occur - for instance 'standard growth', a Lagrangian with no dependence on the second variable, or an approximating sequence satisfying Cesari's Condition (D) - but results are often particular, and applicable to a small class of functionals.

Connected with the Lavrentiev Phenomenon is the repulsion property: any functional displaying Lavrentiev's Phenomenon will display the weak repulsion property.[24]

Functions of several variables

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For example, ifφ(x,y){\displaystyle \varphi (x,y)} denotes the displacement of a membrane above the domainD{\displaystyle D} in thex,y{\displaystyle x,y} plane, then its potential energy is proportional to its surface area:

U[φ]=∬D1+∇φ⋅∇φdxdy.{\displaystyle U[\varphi ]=\iint _{D}{\sqrt {1+\nabla \varphi \cdot \nabla \varphi }}\,dx\,dy.}

Plateau's problem consists of finding a function that minimizes the surface area while assuming prescribed values on the boundary ofD{\displaystyle D}; the solutions are calledminimal surfaces. The Euler–Lagrange equation for this problem is nonlinear:

φxx(1+φy2)+φyy(1+φx2)−2φxφyφxy=0.{\displaystyle \varphi _{xx}(1+\varphi _{y}^{2})+\varphi _{yy}(1+\varphi _{x}^{2})-2\varphi _{x}\varphi _{y}\varphi _{xy}=0.}

See Courant (1950) for details.

Dirichlet's principle

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It is often sufficient to consider only small displacements of the membrane, whose energy difference from no displacement is approximated by

V[φ]=12∬D∇φ⋅∇φdxdy.{\displaystyle V[\varphi ]={\frac {1}{2}}\iint _{D}\nabla \varphi \cdot \nabla \varphi \,dx\,dy.}

The functionalV{\displaystyle V} is to be minimized among all trial functionsφ{\displaystyle \varphi } that assume prescribed values on the boundary ofD{\displaystyle D}. Ifu{\displaystyle u} is the minimizing function andv{\displaystyle v} is an arbitrary smooth function that vanishes on the boundary ofD{\displaystyle D}, then the first variation ofV[u+εv]{\displaystyle V[u+\varepsilon v]} must vanish:

ddεV[u+εv]|ε=0=∬D∇u⋅∇vdxdy=0.{\displaystyle \left.{\frac {d}{d\varepsilon }}V[u+\varepsilon v]\right|_{\varepsilon =0}=\iint _{D}\nabla u\cdot \nabla v\,dx\,dy=0.}

Provided thatu{\displaystyle u} has two derivatives, we may apply the divergence theorem to obtain

∬D∇⋅(v∇u)dxdy=∬D∇u⋅∇v+v∇⋅∇udxdy=∫Cv∂u∂nds,{\displaystyle \iint _{D}\nabla \cdot (v\nabla u)\,dx\,dy=\iint _{D}\nabla u\cdot \nabla v+v\nabla \cdot \nabla u\,dx\,dy=\int _{C}v{\frac {\partial u}{\partial n}}\,ds,}

whereC{\displaystyle C} is the boundary ofD,{\displaystyle D,}s{\displaystyle s} is arclength alongC{\displaystyle C} and∂u/∂n{\displaystyle \partial u/\partial n} is the normal derivative ofu{\displaystyle u} onC.{\displaystyle C.} Sincev{\displaystyle v} vanishes onC{\displaystyle C} and the first variation vanishes, the result is

∬Dv∇⋅∇udxdy=0{\displaystyle \iint _{D}v\nabla \cdot \nabla u\,dx\,dy=0}

for all smooth functionsv{\displaystyle v} that vanish on the boundary ofD{\displaystyle D}. The proof for the case of one dimensional integrals may be adapted to this case to show that

∇⋅∇u=0{\displaystyle \nabla \cdot \nabla u=0}inD.{\displaystyle D.}

The difficulty with this reasoning is the assumption that the minimizing functionu{\displaystyle u} must have two derivatives. Riemann argued that the existence of a smooth minimizing function was assured by the connection with the physical problem: membranes do indeed assume configurations with minimal potential energy. Riemann named this idea theDirichlet principle in honor of his teacherPeter Gustav Lejeune Dirichlet. However Weierstrass gave an example of a variational problem with no solution: minimize

W[φ]=∫−11(xφ′)2dx{\displaystyle W[\varphi ]=\int _{-1}^{1}(x\varphi ')^{2}\,dx}

among all functionsφ{\displaystyle \varphi } that satisfyφ(−1)=−1{\displaystyle \varphi (-1)=-1} andφ(1)=1.{\displaystyle \varphi (1)=1.}W{\displaystyle W} can be made arbitrarily small by choosing piecewise linear functions that make a transition between −1 and 1 in a small neighborhood of the origin. However, there is no function that makesW=0.{\displaystyle W=0.}[k] Eventually it was shown that Dirichlet's principle is valid, but it requires a sophisticated application of the regularity theory forelliptic partial differential equations; see Jost and Li–Jost (1998).

Generalization to other boundary value problems

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A more general expression for the potential energy of a membrane is

V[φ]=∬D[12∇φ⋅∇φ+f(x,y)φ]dxdy+∫C[12σ(s)φ2+g(s)φ]ds.{\displaystyle V[\varphi ]=\iint _{D}\left[{\frac {1}{2}}\nabla \varphi \cdot \nabla \varphi +f(x,y)\varphi \right]\,dx\,dy\,+\int _{C}\left[{\frac {1}{2}}\sigma (s)\varphi ^{2}+g(s)\varphi \right]\,ds.}

This corresponds to an external force densityf(x,y){\displaystyle f(x,y)} inD,{\displaystyle D,} an external forceg(s){\displaystyle g(s)} on the boundaryC,{\displaystyle C,} and elastic forces with modulusσ(s){\displaystyle \sigma (s)}acting onC{\displaystyle C}. The function that minimizes the potential energywith no restriction on its boundary values will be denoted byu{\displaystyle u}. Provided thatf{\displaystyle f} andg{\displaystyle g} are continuous, regularity theory implies that the minimizing functionu{\displaystyle u} will have two derivatives. In taking the first variation, no boundary condition need be imposed on the incrementv{\displaystyle v}. The first variation ofV[u+εv]{\displaystyle V[u+\varepsilon v]} is given by

∬D[∇u⋅∇v+fv]dxdy+∫C[σuv+gv]ds=0.{\displaystyle \iint _{D}\left[\nabla u\cdot \nabla v+fv\right]\,dx\,dy+\int _{C}\left[\sigma uv+gv\right]\,ds=0.}

If we apply the divergence theorem, the result is

∬D[−v∇⋅∇u+vf]dxdy+∫Cv[∂u∂n+σu+g]ds=0.{\displaystyle \iint _{D}\left[-v\nabla \cdot \nabla u+vf\right]\,dx\,dy+\int _{C}v\left[{\frac {\partial u}{\partial n}}+\sigma u+g\right]\,ds=0.}

If we first setv=0{\displaystyle v=0} onC,{\displaystyle C,} the boundary integral vanishes, and we conclude as before that

−∇⋅∇u+f=0{\displaystyle -\nabla \cdot \nabla u+f=0}

inD{\displaystyle D}. Then if we allowv{\displaystyle v} to assume arbitrary boundary values, this implies thatu{\displaystyle u} must satisfy the boundary condition

∂u∂n+σu+g=0,{\displaystyle {\frac {\partial u}{\partial n}}+\sigma u+g=0,}

onC{\displaystyle C}. This boundary condition is a consequence of the minimizing property ofu{\displaystyle u}: it is not imposed beforehand. Such conditions are callednatural boundary conditions.

The preceding reasoning is not valid ifσ{\displaystyle \sigma } vanishes identically onC.{\displaystyle C.} In such a case, we could allow a trial functionφ≡c{\displaystyle \varphi \equiv c}, wherec{\displaystyle c} is a constant. For such a trial function,

V[c]=c[∬Dfdxdy+∫Cgds].{\displaystyle V[c]=c\left[\iint _{D}f\,dx\,dy+\int _{C}g\,ds\right].}

By appropriate choice ofc{\displaystyle c},V{\displaystyle V} can assume any value unless the quantity inside the brackets vanishes. Therefore, the variational problem is meaningless unless

∬Dfdxdy+∫Cgds=0.{\displaystyle \iint _{D}f\,dx\,dy+\int _{C}g\,ds=0.}

This condition implies that net external forces on the system are in equilibrium. If these forces are in equilibrium, then the variational problem has a solution, but it is not unique, since an arbitrary constant may be added. Further details and examples are in Courant and Hilbert (1953).

Eigenvalue problems

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Both one-dimensional and multi-dimensionaleigenvalue problems can be formulated as variational problems.

Sturm–Liouville problems

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See also:Sturm–Liouville theory

The Sturm–Liouvilleeigenvalue problem involves a general quadratic form

Q[y]=∫x1x2[p(x)y′(x)2+q(x)y(x)2]dx,{\displaystyle Q[y]=\int _{x_{1}}^{x_{2}}\left[p(x)y'(x)^{2}+q(x)y(x)^{2}\right]\,dx,}

wherey{\displaystyle y} is restricted to functions that satisfy the boundary conditions

y(x1)=0,y(x2)=0.{\displaystyle y(x_{1})=0,\quad y(x_{2})=0.}

LetR{\displaystyle R} be a normalization integral

R[y]=∫x1x2r(x)y(x)2dx.{\displaystyle R[y]=\int _{x_{1}}^{x_{2}}r(x)y(x)^{2}\,dx.}

The functionsp(x){\displaystyle p(x)} andr(x){\displaystyle r(x)} are required to be everywhere positive and bounded away from zero. The primary variational problem is to minimize the ratioQ/R{\displaystyle Q/R} among ally{\displaystyle y} satisfying the endpoint conditions, which is equivalent to minimizingQ[y]{\displaystyle Q[y]} under the constraint thatR[y]{\displaystyle R[y]} is constant. It is shown below that the Euler–Lagrange equation for the minimizingu{\displaystyle u} is

−(pu′)′+qu−λru=0,{\displaystyle -(pu')'+qu-\lambda ru=0,}

whereλ{\displaystyle \lambda } is the quotient

λ=Q[u]R[u].{\displaystyle \lambda ={\frac {Q[u]}{R[u]}}.}

It can be shown (see Gelfand and Fomin 1963) that the minimizingu{\displaystyle u} has two derivatives and satisfies the Euler–Lagrange equation. The associatedλ{\displaystyle \lambda } will be denoted byλ1{\displaystyle \lambda _{1}}; it is the lowest eigenvalue for this equation and boundary conditions. The associated minimizing function will be denoted byu1(x){\displaystyle u_{1}(x)}. This variational characterization of eigenvalues leads to theRayleigh–Ritz method: choose an approximatingu{\displaystyle u} as a linear combination of basis functions (for example trigonometric functions) and carry out a finite-dimensional minimization among such linear combinations. This method is often surprisingly accurate.

The next smallest eigenvalue and eigenfunction can be obtained by minimizingQ{\displaystyle Q} under the additional constraint

∫x1x2r(x)u1(x)y(x)dx=0.{\displaystyle \int _{x_{1}}^{x_{2}}r(x)u_{1}(x)y(x)\,dx=0.}

This procedure can be extended to obtain the complete sequence of eigenvalues and eigenfunctions for the problem.

The variational problem also applies to more general boundary conditions. Instead of requiring thaty{\displaystyle y} vanish at the endpoints, we may not impose any condition at the endpoints, and set

Q[y]=∫x1x2[p(x)y′(x)2+q(x)y(x)2]dx+a1y(x1)2+a2y(x2)2,{\displaystyle Q[y]=\int _{x_{1}}^{x_{2}}\left[p(x)y'(x)^{2}+q(x)y(x)^{2}\right]\,dx+a_{1}y(x_{1})^{2}+a_{2}y(x_{2})^{2},}

wherea1{\displaystyle a_{1}} anda2{\displaystyle a_{2}} are arbitrary. If we sety=u+εv{\displaystyle y=u+\varepsilon v}, the first variation for the ratioQ/R{\displaystyle Q/R} is

V1=2R[u](∫x1x2[p(x)u′(x)v′(x)+q(x)u(x)v(x)−λr(x)u(x)v(x)]dx+a1u(x1)v(x1)+a2u(x2)v(x2)),{\displaystyle V_{1}={\frac {2}{R[u]}}\left(\int _{x_{1}}^{x_{2}}\left[p(x)u'(x)v'(x)+q(x)u(x)v(x)-\lambda r(x)u(x)v(x)\right]\,dx+a_{1}u(x_{1})v(x_{1})+a_{2}u(x_{2})v(x_{2})\right),}

whereλ{\displaystyle \lambda } is given by the ratioQ[u]/R[u]{\displaystyle Q[u]/R[u]} as previously.After integration by parts,

R[u]2V1=∫x1x2v(x)[−(pu′)′+qu−λru]dx+v(x1)[−p(x1)u′(x1)+a1u(x1)]+v(x2)[p(x2)u′(x2)+a2u(x2)].{\displaystyle {\frac {R[u]}{2}}V_{1}=\int _{x_{1}}^{x_{2}}v(x)\left[-(pu')'+qu-\lambda ru\right]\,dx+v(x_{1})[-p(x_{1})u'(x_{1})+a_{1}u(x_{1})]+v(x_{2})[p(x_{2})u'(x_{2})+a_{2}u(x_{2})].}

If we first require thatv{\displaystyle v} vanish at the endpoints, the first variation will vanish for all suchv{\displaystyle v} only if

−(pu′)′+qu−λru=0forx1<x<x2.{\displaystyle -(pu')'+qu-\lambda ru=0\quad {\hbox{for}}\quad x_{1}<x<x_{2}.}

Ifu{\displaystyle u} satisfies this condition, then the first variation will vanish for arbitraryv{\displaystyle v} only if

−p(x1)u′(x1)+a1u(x1)=0,andp(x2)u′(x2)+a2u(x2)=0.{\displaystyle -p(x_{1})u'(x_{1})+a_{1}u(x_{1})=0,\quad {\hbox{and}}\quad p(x_{2})u'(x_{2})+a_{2}u(x_{2})=0.}

These latter conditions are thenatural boundary conditions for this problem, since they are not imposed on trial functions for the minimization, but are instead a consequence of the minimization.

Eigenvalue problems in several dimensions

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Eigenvalue problems in higher dimensions are defined in analogy with the one-dimensional case. For example, given a domainD{\displaystyle D} with boundaryB{\displaystyle B} in three dimensions we may define

Q[φ]=∭Dp(X)∇φ⋅∇φ+q(X)φ2dxdydz+∬Bσ(S)φ2dS,{\displaystyle Q[\varphi ]=\iiint _{D}p(X)\nabla \varphi \cdot \nabla \varphi +q(X)\varphi ^{2}\,dx\,dy\,dz+\iint _{B}\sigma (S)\varphi ^{2}\,dS,}

and

R[φ]=∭Dr(X)φ(X)2dxdydz.{\displaystyle R[\varphi ]=\iiint _{D}r(X)\varphi (X)^{2}\,dx\,dy\,dz.}

Letu{\displaystyle u} be the function that minimizes the quotientQ[φ]/R[φ]{\displaystyle Q[\varphi ]/R[\varphi ]},with no condition prescribed on the boundaryB.{\displaystyle B.} The Euler–Lagrange equation satisfied byu{\displaystyle u} is

−∇⋅(p(X)∇u)+q(x)u−λr(x)u=0,{\displaystyle -\nabla \cdot (p(X)\nabla u)+q(x)u-\lambda r(x)u=0,}

where

λ=Q[u]R[u].{\displaystyle \lambda ={\frac {Q[u]}{R[u]}}.}

The minimizingu{\displaystyle u} must also satisfy the natural boundary condition

p(S)∂u∂n+σ(S)u=0,{\displaystyle p(S){\frac {\partial u}{\partial n}}+\sigma (S)u=0,}

on the boundaryB.{\displaystyle B.} This result depends upon the regularity theory for elliptic partial differential equations; see Jost and Li–Jost (1998) for details. Many extensions, including completeness results, asymptotic properties of the eigenvalues and results concerning the nodes of the eigenfunctions are in Courant and Hilbert (1953).

Applications

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Optics

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Fermat's principle states that light takes a path that (locally) minimizes the optical length between its endpoints. If thex{\displaystyle x}-coordinate is chosen as the parameter along the path, andy=f(x){\displaystyle y=f(x)} along the path, then the optical length is given by

A[f]=∫x0x1n(x,f(x))1+f′(x)2dx,{\displaystyle A[f]=\int _{x_{0}}^{x_{1}}n(x,f(x)){\sqrt {1+f'(x)^{2}}}dx,}

where the refractive indexn(x,y){\displaystyle n(x,y)} depends upon the material.If we tryf(x)=f0(x)+εf1(x){\displaystyle f(x)=f_{0}(x)+\varepsilon f_{1}(x)} then thefirst variation ofA{\displaystyle A} (the derivative ofA{\displaystyle A} with respect toε{\displaystyle \varepsilon }) is

δA[f0,f1]=∫x0x1[n(x,f0)f0′(x)f1′(x)1+f0′(x)2+ny(x,f0)f11+f0′(x)2]dx.{\displaystyle \delta A[f_{0},f_{1}]=\int _{x_{0}}^{x_{1}}\left[{\frac {n(x,f_{0})f_{0}'(x)f_{1}'(x)}{\sqrt {1+f_{0}'(x)^{2}}}}+n_{y}(x,f_{0})f_{1}{\sqrt {1+f_{0}'(x)^{2}}}\right]dx.}

After integration by parts of the first term within brackets, we obtain the Euler–Lagrange equation

−ddx[n(x,f0)f0′1+f0′2]+ny(x,f0)1+f0′(x)2=0.{\displaystyle -{\frac {d}{dx}}\left[{\frac {n(x,f_{0})f_{0}'}{\sqrt {1+f_{0}'^{2}}}}\right]+n_{y}(x,f_{0}){\sqrt {1+f_{0}'(x)^{2}}}=0.}

The light rays may be determined by integrating this equation. This formalism is used in the context ofLagrangian optics andHamiltonian optics.

Snell's law

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There is a discontinuity of the refractive index when light enters or leaves a lens. Let

n(x,y)={n(−)ifx<0,n(+)ifx>0,{\displaystyle n(x,y)={\begin{cases}n_{(-)}&{\text{if}}\quad x<0,\\n_{(+)}&{\text{if}}\quad x>0,\end{cases}}}

wheren(−){\displaystyle n_{(-)}} andn(+){\displaystyle n_{(+)}} are constants. Then the Euler–Lagrange equation holds as before in the region wherex<0{\displaystyle x<0} orx>0{\displaystyle x>0}, and in fact the path is a straight line there, since the refractive index is constant. At thex=0{\displaystyle x=0},f{\displaystyle f} must be continuous, butf′{\displaystyle f'} may be discontinuous. After integration by parts in the separate regions and using the Euler–Lagrange equations, the first variation takes the form

δA[f0,f1]=f1(0)[n(−)f0′(0−)1+f0′(0−)2−n(+)f0′(0+)1+f0′(0+)2].{\displaystyle \delta A[f_{0},f_{1}]=f_{1}(0)\left[n_{(-)}{\frac {f_{0}'(0^{-})}{\sqrt {1+f_{0}'(0^{-})^{2}}}}-n_{(+)}{\frac {f_{0}'(0^{+})}{\sqrt {1+f_{0}'(0^{+})^{2}}}}\right].}

The factor multiplyingn(−){\displaystyle n_{(-)}} is the sine of angle of the incident ray with thex{\displaystyle x} axis, and the factor multiplyingn(+){\displaystyle n_{(+)}} is the sine of angle of the refracted ray with thex{\displaystyle x} axis.Snell's law for refraction requires that these terms be equal. As this calculation demonstrates, Snell's law is equivalent to vanishing of the first variation of the optical path length.

Fermat's principle in three dimensions

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It is expedient to use vector notation: letX=(x1,x2,x3),{\displaystyle X=(x_{1},x_{2},x_{3}),} lett{\displaystyle t} be a parameter, letX(t){\displaystyle X(t)} be the parametric representation of a curveC,{\displaystyle C,} and letX˙(t){\displaystyle {\dot {X}}(t)} be its tangent vector. The optical length of the curve is given by

A[C]=∫t0t1n(X)X˙⋅X˙dt.{\displaystyle A[C]=\int _{t_{0}}^{t_{1}}n(X){\sqrt {{\dot {X}}\cdot {\dot {X}}}}\,dt.}

Note that this integral is invariant with respect to changes in the parametric representation ofC.{\displaystyle C.} The Euler–Lagrange equations for a minimizing curve have the symmetric form

ddtP=X˙⋅X˙∇n,{\displaystyle {\frac {d}{dt}}P={\sqrt {{\dot {X}}\cdot {\dot {X}}}}\,\nabla n,}

where

P=n(X)X˙X˙⋅X˙.{\displaystyle P={\frac {n(X){\dot {X}}}{\sqrt {{\dot {X}}\cdot {\dot {X}}}}}.}

It follows from the definition thatP{\displaystyle P} satisfies

P⋅P=n(X)2.{\displaystyle P\cdot P=n(X)^{2}.}

Therefore, the integral may also be written as

A[C]=∫t0t1P⋅X˙dt.{\displaystyle A[C]=\int _{t_{0}}^{t_{1}}P\cdot {\dot {X}}\,dt.}

This form suggests that if we can find a functionψ{\displaystyle \psi } whose gradient is given byP,{\displaystyle P,} then the integralA{\displaystyle A} is given by the difference ofψ{\displaystyle \psi } at the endpoints of the interval of integration. Thus the problem of studying the curves that make the integral stationary can be related to the study of the level surfaces ofψ{\displaystyle \psi }. In order to find such a function, we turn to the wave equation, which governs the propagation of light. This formalism is used in the context ofLagrangian optics andHamiltonian optics.

Connection with the wave equation
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Thewave equation for an inhomogeneous medium is

utt=c2∇⋅∇u,{\displaystyle u_{tt}=c^{2}\nabla \cdot \nabla u,}

wherec{\displaystyle c} is the velocity, which generally depends uponX{\displaystyle X}. Wave fronts for light are characteristic surfaces for this partial differential equation: they satisfy

φt2=c(X)2∇φ⋅∇φ.{\displaystyle \varphi _{t}^{2}=c(X)^{2}\,\nabla \varphi \cdot \nabla \varphi .}

We may look for solutions in the form

φ(t,X)=t−ψ(X).{\displaystyle \varphi (t,X)=t-\psi (X).}

In that case,ψ{\displaystyle \psi } satisfies

∇ψ⋅∇ψ=n2,{\displaystyle \nabla \psi \cdot \nabla \psi =n^{2},}

wheren=1/c{\displaystyle n=1/c}. According to the theory offirst-order partial differential equations, ifP=∇ψ,{\displaystyle P=\nabla \psi ,} thenP{\displaystyle P} satisfies

dPds=n∇n,{\displaystyle {\frac {dP}{ds}}=n\,\nabla n,}

along a system of curves (the light rays) that are given by

dXds=P.{\displaystyle {\frac {dX}{ds}}=P.}

These equations for solution of a first-order partial differential equation are identical to the Euler–Lagrange equations if we make the identification

dsdt=X˙⋅X˙n.{\displaystyle {\frac {ds}{dt}}={\frac {\sqrt {{\dot {X}}\cdot {\dot {X}}}}{n}}.}

We conclude that the functionψ{\displaystyle \psi } is the value of the minimizing integralA{\displaystyle A} as a function of the upper end point. That is, when a family of minimizing curves is constructed, the values of the optical length satisfy the characteristic equation corresponding the wave equation. Hence, solving the associated partial differential equation of first order is equivalent to finding families of solutions of the variational problem. This is the essential content of theHamilton–Jacobi theory, which applies to more general variational problems.

Mechanics

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Main article:Action (physics)

In classical mechanics, the action,S,{\displaystyle S,} is defined as the time integral of the Lagrangian,L{\displaystyle L}. The Lagrangian is the difference of energies,

L=T−U,{\displaystyle L=T-U,}

whereT{\displaystyle T} is thekinetic energy of a mechanical system andU{\displaystyle U} itspotential energy.Hamilton's principle (or the action principle) states that the motion of a conservative holonomic (integrable constraints) mechanical system is such that the action integral

S=∫t0t1L(x,x˙,t)dt{\displaystyle S=\int _{t_{0}}^{t_{1}}L(x,{\dot {x}},t)\,dt}

is stationary with respect to variations in the pathx(t){\displaystyle x(t)}.The Euler–Lagrange equations for this system are known as Lagrange's equations:

ddt∂L∂x˙=∂L∂x,{\displaystyle {\frac {d}{dt}}{\frac {\partial L}{\partial {\dot {x}}}}={\frac {\partial L}{\partial x}},}

and they are equivalent to Newton's equations of motion (for such systems).

The conjugate momentaP{\displaystyle P} are defined by

p=∂L∂x˙.{\displaystyle p={\frac {\partial L}{\partial {\dot {x}}}}.}

For example, if

T=12mx˙2,{\displaystyle T={\frac {1}{2}}m{\dot {x}}^{2},}

thenp=mx˙.{\displaystyle p=m{\dot {x}}.}

Hamiltonian mechanics results if the conjugate momenta are introduced in place ofx˙{\displaystyle {\dot {x}}} by a Legendre transformation of the LagrangianL{\displaystyle L} into the HamiltonianH{\displaystyle H} defined by

H(x,p,t)=px˙−L(x,x˙,t).{\displaystyle H(x,p,t)=p\,{\dot {x}}-L(x,{\dot {x}},t).}

The Hamiltonian is the total energy of the system:H=T+U{\displaystyle H=T+U}.Analogy with Fermat's principle suggests that solutions of Lagrange's equations (the particle trajectories) may be described in terms of level surfaces of some function ofX{\displaystyle X}. This function is a solution of theHamilton–Jacobi equation:

∂ψ∂t+H(x,∂ψ∂x,t)=0.{\displaystyle {\frac {\partial \psi }{\partial t}}+H\left(x,{\frac {\partial \psi }{\partial x}},t\right)=0.}

Further applications

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Further applications of the calculus of variations include the following:

Variations and sufficient condition for a minimum

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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. Thefirst variation[l] is defined as the linear part of the change in the functional, and thesecond variation[m] is defined as the quadratic part.[26]

For example, ifJ[y]{\displaystyle J[y]} is a functional with the functiony=y(x){\displaystyle y=y(x)} as its argument, and there is a small change in its argument fromy{\displaystyle y} toy+h,{\displaystyle y+h,} whereh=h(x){\displaystyle h=h(x)} is a function in the same function space asy{\displaystyle y}, then the corresponding change in the functional is[n]

ΔJ[h]=J[y+h]−J[y].{\displaystyle \Delta J[h]=J[y+h]-J[y].}

The functionalJ[y]{\displaystyle J[y]} is said to bedifferentiable if

ΔJ[h]=φ[h]+ε‖h‖,{\displaystyle \Delta J[h]=\varphi [h]+\varepsilon \|h\|,}

whereφ[h]{\displaystyle \varphi [h]} is a linear functional,[o]‖h‖{\displaystyle \|h\|} is the norm ofh,{\displaystyle h,}[p] andε→0{\displaystyle \varepsilon \to 0} as‖h‖→0.{\displaystyle \|h\|\to 0.} The linear functionalφ[h]{\displaystyle \varphi [h]} is the first variation ofJ[y]{\displaystyle J[y]} and is denoted by,[30]

δJ[h]=φ[h].{\displaystyle \delta J[h]=\varphi [h].}

The functionalJ[y]{\displaystyle J[y]} is said to betwice differentiable if

ΔJ[h]=φ1[h]+φ2[h]+ε‖h‖2,{\displaystyle \Delta J[h]=\varphi _{1}[h]+\varphi _{2}[h]+\varepsilon \|h\|^{2},}

whereφ1[h]{\displaystyle \varphi _{1}[h]} is a linear functional (the first variation),φ2[h]{\displaystyle \varphi _{2}[h]} is a quadratic functional,[q] andε→0{\displaystyle \varepsilon \to 0} as‖h‖→0.{\displaystyle \|h\|\to 0.} The quadratic functionalφ2[h]{\displaystyle \varphi _{2}[h]} is the second variation ofJ[y]{\displaystyle J[y]} and is denoted by,[32]

δ2J[h]=φ2[h].{\displaystyle \delta ^{2}J[h]=\varphi _{2}[h].}

The second variationδ2J[h]{\displaystyle \delta ^{2}J[h]} is said to bestrongly positive if

δ2J[h]≥k‖h‖2,{\displaystyle \delta ^{2}J[h]\geq k\|h\|^{2},}

for allh{\displaystyle h} and for some constantk>0{\displaystyle k>0}.[33]

Using the above definitions, especially the definitions of first variation, second variation, and strongly positive, the following sufficient condition for a minimum of a functional can be stated.

Sufficient condition for a minimum:

The functionalJ[y]{\displaystyle J[y]} has a minimum aty=y^{\displaystyle y={\hat {y}}} if its first variationδJ[h]=0{\displaystyle \delta J[h]=0} aty=y^{\displaystyle y={\hat {y}}} and its second variationδ2J[h]{\displaystyle \delta ^{2}J[h]} is strongly positive aty=y^.{\displaystyle y={\hat {y}}.}[34][r][s]

See also

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Notes

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  1. ^Whereaselementary calculus is aboutinfinitesimally small changes in the values of functions without changes in the function itself, calculus of variations is about infinitesimally small changes in the function itself, which are called variations.[1]
  2. ^"Euler waited until Lagrange had published on the subject in 1762 ... before he committed his lecture ... to print, so as not to rob Lagrange of his glory. Indeed, it was only Lagrange's method that Euler called Calculus of Variations."[7]
  3. ^SeeHarold J. Kushner (2004): regarding Dynamic Programming, "The calculus of variations had related ideas (e.g., the work of Caratheodory, the Hamilton-Jacobi equation). This led to conflicts with the calculus of variations community."
  4. ^The neighborhood off{\displaystyle f} is the part of the given function space where|y−f|<h{\displaystyle |y-f|<h} over the whole domain of the functions, withh{\displaystyle h} a positive number that specifies the size of the neighborhood.[14]
  5. ^ Note the difference between the terms extremal and extremum. An extremal is a function that makes a functional an extremum.
  6. ^ For a sufficient condition, see sectionVariations and sufficient condition for a minimum.
  7. ^The following derivation of the Euler–Lagrange equation corresponds to the derivation on pp. 184–185 of Courant & Hilbert (1953).[18]
  8. ^Note thatη(x){\displaystyle \eta (x)} andf(x){\displaystyle f(x)} are evaluated at thesame values ofx,{\displaystyle x,} which is not valid more generally in variational calculus with non-holonomic constraints.
  9. ^The productεΦ′(0){\displaystyle \varepsilon \Phi '(0)} is called the first variation of the functionalJ{\displaystyle J} and is denoted byδJ.{\displaystyle \delta J.} Some references define thefirst variation differently by leaving out theε{\displaystyle \varepsilon } factor.
  10. ^ As a historical note, this is an axiom ofArchimedes. See e.g. Kelland (1843).[19]
  11. ^The resulting controversy over the validity of Dirichlet's principle is explained by Turnbull.[25]
  12. ^ The first variation is also called the variation, differential, or first differential.
  13. ^ The second variation is also called the second differential.
  14. ^Note thatΔJ[h]{\displaystyle \Delta J[h]} and the variations below, depend on bothy{\displaystyle y} andh{\displaystyle h}. The argumenty{\displaystyle y} has been left out to simplify the notation. For example,ΔJ[h]{\displaystyle \Delta J[h]} could have been writtenΔJ[y;h].{\displaystyle \Delta J[y;h].}[27]
  15. ^A functionalφ[h]{\displaystyle \varphi [h]} is said to belinear ifφ[αh]=αφ[h]{\displaystyle \varphi [\alpha h]=\alpha \varphi [h]}   and  φ[h+h2]=φ[h]+φ[h2],{\displaystyle \varphi \left[h+h_{2}\right]=\varphi [h]+\varphi \left[h_{2}\right],} whereh,h2{\displaystyle h,h_{2}} are functions andα{\displaystyle \alpha } is a real number.[28]
  16. ^ For a functionh=h(x){\displaystyle h=h(x)} that is defined fora≤x≤b,{\displaystyle a\leq x\leq b,} wherea{\displaystyle a} andb{\displaystyle b} are real numbers, the norm ofh{\displaystyle h} is its maximum absolute value, i.e.‖h‖=maxa≤x≤b|h(x)|.{\displaystyle \|h\|=\displaystyle \max _{a\leq x\leq b}|h(x)|.}[29]
  17. ^ A functional is said to bequadratic if it is a bilinear functional with two argument functions that are equal. Abilinear functional is a functional that depends on two argument functions and is linear when each argument function in turn is fixed while the other argument function is variable.[31]
  18. ^ For other sufficient conditions, see inGelfand & Fomin 2000,
    • Chapter 5: "The Second Variation. Sufficient Conditions for a Weak Extremum" – Sufficient conditions for a weak minimum are given by the theorem on p. 116.
    • Chapter 6: "Fields. Sufficient Conditions for a Strong Extremum" – Sufficient conditions for a strong minimum are given by the theorem on p. 148.
  19. ^ One may note the similarity to the sufficient condition for a minimum of a function, where the first derivative is zero and the second derivative is positive.

References

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  1. ^abCourant & Hilbert 1953, p. 184
  2. ^abGoldstine, Herman H. (1980).A History of the Calculus of Variations from the 17th Through the 19th Century. Springer New York. pp. 7–21.ISBN 978-1-4613-8106-8.
  3. ^abFerguson, James (2004),A Brief Survey of the History of the Calculus of Variations and its Applications,arXiv:math/0402357,Bibcode:2004math......2357F
  4. ^abRowlands, Peter (2017).Newton and the Great World System.World Scientific Publishing. pp. 36–39.doi:10.1142/q0108.ISBN 978-1-78634-372-7.
  5. ^Torres, Delfim F. M. (2021-07-29)."On a Non-Newtonian Calculus of Variations".Axioms.10 (3): 171.arXiv:2107.14152.doi:10.3390/axioms10030171.ISSN 2075-1680.
  6. ^Gelfand, I. M.;Fomin, S. V. (2000). Silverman, Richard A. (ed.).Calculus of variations (Unabridged repr. ed.). Mineola, New York: Dover Publications. p. 3.ISBN 978-0486414485.
  7. ^abThiele, Rüdiger (2007)."Euler and the Calculus of Variations". In Bradley, Robert E.; Sandifer, C. Edward (eds.).Leonhard Euler: Life, Work and Legacy. Elsevier. p. 249.ISBN 9780080471297.
  8. ^Goldstine, Herman H. (2012).A History of the Calculus of Variations from the 17th through the 19th Century. Springer Science & Business Media. p. 110.ISBN 9781461381068.
  9. ^abcvan Brunt, Bruce (2004).The Calculus of Variations. Springer.ISBN 978-0-387-40247-5.
  10. ^abFerguson, James (2004). "Brief Survey of the History of the Calculus of Variations and its Applications".arXiv:math/0402357.
  11. ^Dimitri Bertsekas. Dynamic programming and optimal control. Athena Scientific, 2005.
  12. ^Bellman, Richard E. (1954)."Dynamic Programming and a new formalism in the calculus of variations".Proc. Natl. Acad. Sci.40 (4):231–235.Bibcode:1954PNAS...40..231B.doi:10.1073/pnas.40.4.231.PMC 527981.PMID 16589462.
  13. ^"Richard E. Bellman Control Heritage Award".American Automatic Control Council. 2004. Archived fromthe original on 2018-10-01. Retrieved2013-07-28.
  14. ^Courant, R;Hilbert, D (1953).Methods of Mathematical Physics. Vol. I (First English ed.). New York: Interscience Publishers, Inc. p. 169.ISBN 978-0471504474.{{cite book}}:ISBN / Date incompatibility (help)
  15. ^Gelfand & Fomin 2000, pp. 12–13
  16. ^Gelfand & Fomin 2000, p. 13
  17. ^Gelfand & Fomin 2000, pp. 14–15
  18. ^Courant, R.;Hilbert, D. (1953).Methods of Mathematical Physics. Vol. I (First English ed.). New York: Interscience Publishers, Inc.ISBN 978-0471504474.{{cite book}}:ISBN / Date incompatibility (help)
  19. ^Kelland, Philip (1843).Lectures on the principles of demonstrative mathematics. p. 58 – via Google Books.
  20. ^Weisstein, Eric W."Euler–Lagrange Differential Equation".mathworld.wolfram.com. Wolfram. Eq. (5).
  21. ^Kot, Mark (2014). "Chapter 4: Basic Generalizations".A First Course in the Calculus of Variations. American Mathematical Society.ISBN 978-1-4704-1495-5.
  22. ^Manià, Bernard (1934). "Sopra un esempio di Lavrentieff".Bollenttino dell'Unione Matematica Italiana.13:147–153.
  23. ^Ball & Mizel (1985). "One-dimensional Variational problems whose Minimizers do not satisfy the Euler-Lagrange equation".Archive for Rational Mechanics and Analysis.90 (4):325–388.Bibcode:1985ArRMA..90..325B.doi:10.1007/BF00276295.S2CID 55005550.
  24. ^Ferriero, Alessandro (2007). "The Weak Repulsion property".Journal de Mathématiques Pures et Appliquées.88 (4):378–388.doi:10.1016/j.matpur.2007.06.002.
  25. ^Turnbull."Riemann biography". UK: U. St. Andrew.[permanent dead link]
  26. ^Gelfand & Fomin 2000, pp. 11–12, 99
  27. ^Gelfand & Fomin 2000, p. 12, footnote 6
  28. ^Gelfand & Fomin 2000, p. 8
  29. ^Gelfand & Fomin 2000, p. 6
  30. ^Gelfand & Fomin 2000, pp. 11–12
  31. ^Gelfand & Fomin 2000, pp. 97–98
  32. ^Gelfand & Fomin 2000, p. 99
  33. ^Gelfand & Fomin 2000, p. 100
  34. ^Gelfand & Fomin 2000, p. 100, Theorem 2

Further reading

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