Gauge Institute Journal, Volume , No. , H. Vic Dannon
Infinitesimal Variational Calculus H. Vic Dannon [email protected] October, 2011
Abstract We define Variations as Infinitesimal Functions,
and investigate their properties. Variational Calculus is based on Euler’s differential equation, Hence on differentials, and as such it is infinitesimal. But it is derived in an integral context, which imposes unnecessary global restrictions, requires unnecessary arguments, and is ill-equipped to resolve its over hundred-years-old problems. Heuristic conception of Infinitesimals resulted in Ill-defined Variations. Texts of Variational Calculus apply variations inefficiently, or avoids them altogether. We develop Variational Calculus in Infinitesimal context to eliminate unnecessary arguments, to eliminate restrictive global assumptions
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and to give meaning to the poorly understood, and
thus, ill-defined variation δy , and variational derivative
δF . δy We supply a correct proof to the unresolved to date Weierstrass Sufficient condition for a Minimum, and prove its equivalence to Legendre Sufficient Condition for a Minimum. In non-infinitesimal context, the Legendre Sufficient Condition is misunderstood as being insufficient.
Keywords: Infinitesimal, Variation, Infinitesimal Curve
Variation, Infinitesimal Surface Variation, Euler’s Variational Equation, Variational Calculus, Variational Derivative, Functional Derivative, Euler-Lagrange, Legendre Sufficient Condition, Weierstrass Sufficient Condition,
2000 Mathematics Subject Classification; 49-02;26E30;
26E15; 26E20; 26A12; 46S20; 97I40;
Physics and Astronomy Classification Scheme ;45.20.Jj;
4510Db;
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Contents
Introduction 1. Infinitesimal Curve-Variation 2. Euler’s Equation for Fxyy(,, '), and the Variational
Derivative 3. Euler’s Equation and Strong Variations 4. First Integral of Euler’s equation xb= 5. Legendre Sufficient Condition for ∫ Fxyy(,, ') dx to be xa= Minimal xb= 6. Weierstrass Sufficient condition for ∫ Fxyy(,, ') dx to be xa= Minimal 7. Euler’s Equation for Fxyy(,,',") y , and the Variational
Derivative
8. Euler’s Equation forFxy( ,121 , y , y ', y 2 '), and the Variational Derivative 9. Euler’s Equation for f(,txyxy , , , ), and the Variational
Derivative
tt= 1 10. Weierstrass Sufficient Condition for ∫ f (,txyxydt , , , ) to tt= 0 be Minimal
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11. Euler-Lagrange Equations for L(,tqtqtqtqt1212 (), (), (), ()) and the Variational Derivative. 12. Infinitesimal Surface Variation
∂∂yy 13. Euler’s Equation for Fx(,12 x ,, y , ), and the ∂∂xx12 Variational Derivative References
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Introduction
0.1 Global Solutions in the Calculus of Variations
The infinitesimal distance in the plane is
ds=+=+() dx22 () dy 1 (')y 2 dx .
The arc-length of yx() between (,aya()), and (,byb()) is
xb= ∫ 1(')+ ydx2 , xa= and the curve yx() that minimizes the integral has the
shortest arc-length. That curve is the line over the interval [,ab].
The speed of a bead falling a distance x is
2gx , the infinitesimal descent time of a bead along a curve yx() is
ds1(')+ y2 dx dt == v 2gx The descent time of the bead along a curve yx() between
(,aya ()), and (,byb()) is xb= 1(')+ y 2 ∫ dx , xa= 2gx
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and the curve yx() that minimizes the integral has the quickest descent. That curve is an arc of a cycloid over the interval [,ab].
In both problems, finding the curve of shortest path, and finding the curve of quickest descent, the domain of the solution is the interval [,ab].
Thus, a basic problem of the Calculus of Variations is to find a curve
yx()= y0 () x , that will minimize the Functional
xb= Jy()= ∫ Fxyx (,(),'()) y x dx. xa= The solution has to hold globally, over the interval [,ab].
But we are not required to find it by global methods. Since the problem was constructed using infinitesimals, it is amenable to infinitesimal methods. However, for lack of understanding of infinitesimals, variations were poorly understood, and their full power was not used. Instead, the analysis applied to the global problem. We follow with the global derivation that has not been re- examined since Weierstrass time.
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0.2 Global Derivation in the Calculus of Variations
For a curve,
yx()=+ y0 () xεη () x , where ε > 0 is very small, ηη()ab== () 0, the Taylor expansion to second order of
Fxyx(,00 ()++−εη (), xy '() xεη '()) x Fxyxy (,0 (),0 '()) x , about yx0(), is ⎧⎫ ⎪⎪∂∂FF εη⎨⎬⎪⎪++ η'(O ε2). ⎪⎪∂∂yy' ⎩⎭⎪⎪yy==00yy Therefore,
xb= ⎧⎫ ⎪⎪∂∂FF II()εε−= (0) ⎨⎬⎪⎪ η+ η'()dxO +ε2 ∫ ⎪⎪∂∂yy' xa= ⎩⎭⎪⎪yy==00yy Integrating by parts, II()ε −= (0)
xb= ⎪⎪⎧⎫⎛⎞ ⎡ ⎤xb= ⎪⎪∂∂FdF⎜ ⎟ ∂F 2 =−εη∫ ⎨⎬⎜ ⎟ ()xdx ++⎢⎥η () x O (ε ) ⎪⎪∂∂ydxy⎜⎝⎠''⎟ ⎢⎥⎣⎦ ∂y xa= ⎩⎭⎪⎪yy==00yy xa= =0 Hence,
xb= ⎪⎪⎧⎫ II()ε −∂ (0) ⎪⎪ F dF⎛⎞ ∂⎟ =−⎨⎬⎜ ⎟ ηε()xdx+ O (). ε ∫ ⎪⎪∂∂ydxy⎝⎠⎜ ' ⎟ xa= ⎩⎭⎪⎪yy==00yy
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Letting ε ↓ 0 ,
xb= ⎪⎪⎧⎫ ⎪⎪∂∂FdF⎛⎞⎟ Ix'(0) =−⎨⎬⎜ ⎟ η()dx. ∫ ⎪⎪∂∂ydxy⎝⎠⎜ ' ⎟ xa= ⎩⎭⎪⎪yy==00yy
Since IJy(0)= (0 ) is minimal, we have by Fermat Theorem, I '(0)= 0 , and
xb= ⎪⎪⎧⎫ ⎪⎪∂∂FdF⎛⎞⎟ ⎨⎬−=⎜ ⎟ η()xdx 0. ∫ ⎪⎪∂∂ydxy⎝⎠⎜ ' ⎟ xa= ⎩⎭⎪⎪yy==00yy Since η()x is an arbitrary smooth enough function, it seems
plausible that the integrand is identically zero. Namely, ∂∂FdF⎛⎞⎟ −=⎜ ⎟ 0 . ∂∂ydxy⎝⎠⎜ ' ⎟ yy==00yy That is the Euler equation for the extremal curve. To that end, many texts apply the “Fundamental lemma of the Calculus of Variations”. But since the interval [,ab] has no role in the derivation, it is
not necessary to extend it beyond an infinitesimal. An
infinitesimal interval [,xdxxd−+11x] will eliminate the 22 need for the Fundamental lemma.
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Clearly, the interval is irrelevant to the solution of the shortest distance problem: Then, Euler’s Variational Equation is
dy' = 0 , dx 1(')+ y 2
(')y 2 = constant,
yAxB=+, which is a line in the plane.
0.3 Weierstrass Sufficient Condition
In his Lectures on the Calculus of Variations, Weierstrass
xb= observed that a sufficient condition for ∫ Fxyydx(..') to be xa=
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minimal can be given. Weierstrass Lectures were never published, and as the simple reason for the observation was not found, a whole theory was built to substantiate it. That theory is based on the parametric family of extremals that solve the Euler Equation, and constitute a partition of a simply connected domain in the plane.
xb= Hilbert claimed that in that domain, ∫ Fxyy(,, ') dx is path xa= independent, and based on that, a dizzying proof was given for Weierstrass observation.
xb= First, path-independence of ∫ Fxyy(,, ') dx defies the xa= purpose of the Calculus of Variations. An integral that does not depend on the curve is irrelevant in the calculus of Variations. Second, the disjoint extremals do not intersect at any point. For instance, the extremals of the shortest distance problem are parallel lines, each determined by its (,aya()), and
(,byb ()). Therefore, a closed path made of extremals is impossible in the extremals’ field.
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Third, not even a pencil of the disjoint extremals is possible. Pencil pictures that accompany that theory are delusionary. We give here a correct proof, and show that Weierstrass sufficient condition, and Legendre sufficient condition are equivalent.
0.4 Functional Variational Derivatives
There is a false belief that Integrals may have a Variational Derivative, that is called a Functional Derivative. But the first variation of an integral
xb= ⎪⎪⎧⎫ ⎪⎪∂∂FdF⎛⎞⎟ δεJx=−⎨⎬⎜ ⎟ η()dx, ∫ ⎪⎪∂∂ydxy⎝⎠⎜ ' ⎟ xa= ⎩⎭⎪⎪yy==00yy δJ does not allow the definition of a Variational Derivative δy of the Functional J , because it is not clear how δyx() may be pulled from under the integral sign.
δF We can only define the Variational Derivative, of the δy Function F . We proceed with the infinitesimal treatment of variations.
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1.
Infinitesimal Curve-Variation
The differential of the hyper-real function f()x is defined by
df() x= f '() x dx .
It is the product of the infinitesimal dx by the hyper-real function f '(x ), and it vanishes at the zeros of f '(x ).
Graphically, it is the rise of the tangent to f()x at x , over
the infinitesimal run dx . df() x is of infinitesimal size, but since it may vanish, it is not
an infinitesimal. We use the same dx in the definition of all differentials. For instance,
dx()2 = 2 xdx
de()xx= edx We shall call the product of a hyper-real function by an infinitesimal an infinitesimal function.
In Infinitesimal Variational Calculus, we define a variation as an infinitesimal function.
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Denoting the variation of the curve yx() by δyx(),
an infinitesimal by οδ , and a smooth enough hyper-real function by η()x we define 1.1 The Infinitesimal Curve Variation
δοηyx()= δ () x . where,
1.2 We use the same οδ for all curve variations.
the Greek omicron ο , looks like a zero, and was used extensively by Riemann to denote an infinitesimal. In translations, it was sometimes confused with zero, and replaced by it, making Riemann’s considerations evaporate. That is why we subscripted it here with δ . The Greek epsilon have been used extensively by Weierstrass to mean a non-imaginable very small positive quantity, not necessarily an infinitesimal, that may be diminished to a zero limit, while staying positive. In other words, Weierstrass’ epsilon is an infinitesimal on the real line.
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But Infinitesimals exist on the Hyper-real line only. [Dan1]. Thus, there is no such ε , and to prevent confusion, we shall avoid using epsilon here. From the definition 1.1, it is clear that 1.3 The variation of yx(), does not depend on the run dx .
The infinitesimal οδ endows δyx() with infinitesimal size, and οηδ ()x shifts yx() to an infinitesimally close curve: 1.4 The curve yx()+ δ yx () is infinitesimally close to yx().
In particular, 1.5 δyx() does not depend on the size of the x -interval
From the definition 1.1, we have 1.6 δ is a linear operator
Proof:
δο[()yx12+= yx ()]δ [()η 12 x +η ()] x
=+οηδδ12()xx οη ()
=+δδ[yx12 ( )] [ yx ( )].
δα[yx ()]=== οαηδδ () x αοη () x αδ [()] yx .,
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Therefore, the operators d , and δ commute. Namely, 1.7 dyx(δδ ( ))= ( dyx ( ))
Proof: dyx[()]δδ=+− yx ( dx ) δ yx ()
=+−δ[(yx dx ) yx ()]
= δ[()dy x ].,
d Therefore, the operators , and δ commute. Namely, dx dd 1.8 (δδyx ( ))= ( yx ( )) dx dx
1.9 IfFx(,(),'())yx y x is a hyper-real function on an x -interval
Then, its’ Taylor series to third order about yx0() is
Fxy(,00++=δδ yy , ' y ') Fxy (,00 , y ')+ ⎧⎫ ⎪⎪∂∂FF ++⎨⎬⎪⎪δδyx() y'() x ⎪⎪∂∂yy' ⎩⎭⎪⎪yy==00yy ⎧⎫ ⎪⎪∂∂22FF ∂ 2 F ++⎪⎪[δδyx ()]22 2yx ()δ y '() x + (δy '()) x ⎨⎬22 ⎪⎪∂∂yy∂∂yy' (') ⎩⎭⎪⎪yy00 y 0
3 +O()οy .
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Define 1.10 the First Variation of Fxyy(,, ')
∂∂FF δδFxyy(,, ')=+ yδ y' ∂∂yy'
⎛⎞∂∂⎟ =+⎜δδyy' ⎟F ⎝⎠⎜ ∂∂yy' ⎟
1.11 δx = 0
Proof: Fxyy(,, ')=⇒ xδ F =0.,
Define 1.12 the Second Variation of Fxyy(,, ')
2 ⎛⎞∂∂ δδδ2Fxyy(,, ')=+⎜ y y' ⎟ F ⎝⎠⎜ ∂∂yy' ⎟
∂∂22FF ∂ 2 F =+()δδyy22 2 ()(')δy + (δy ') ∂∂yy22∂∂yy' (')
Just as dx2 = 0, we have
1.13 δ2yx()= 0
Proof: Fxyy(,, ')=⇒ yδ2 F =0.,
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Then, 1.8 becomes 1.14 IfFx(,(),'())yx y x is hyper-real function on an x -interval
Then, its’ Taylor series to third order about yx0() is
23 Fxy(,0++=δδ yy ,' 0 y ')(,,') Fxy00 y ++δ Fxy (,,')00 y + δ Fxy (,,')(00 y + Oοy )
Therefore, 1.15 IfFx(,(),'())yx y x is hyper-real function on an x -interval,
xb= and Jy()= ∫ Fxyx (,(),'()) y x dx xa=
Then, Taylor series to third order of Jy(0 + δ y) about
yx0() is
xb=
Jy()00+=δ y∫ Fxy (,(),'()) x y0 x dx+ xa=
xb=
+ ∫ δFxy(,00 , y ') dx xa=
xb= 2 + ∫ δ Fxy(,00 , y ') dx xa=
3 +O()οy .
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Define xb= 1.16 the First Variation of ∫ Fxyx(,(),'()) y x dx xa= xb==xb δδ∫∫Fxyy(,, ') dx= Fxyy (,, ') dx xa==xa
Define xb= 1.17 the Second Variation of ∫ Fxyx(,(),'()) y x dx xa=
xb==xb δδ22∫∫Fxyy(,, ') dx= Fxyy(,, ') dx. xa==xa
Then, 1.15 becomes 1.18 IfFx(,(),'())yx y x is hyper-real function on an x -interval,
xb= and Jy()= ∫ Fxyx (,(),'()) y x dx. xa=
Then, Taylor series to third order of Jy() about yx0() is
23 Jy(0000+=δδδ y ) Jy () + Jy () + Jy () + O ()οy .
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2.
Euler’s Equation for Fxyy(,, '), and the Variational Derivative
We apply variations, over infinitesimal interval, to derive Euler’s Variational Equation in Infinitesimal Variational Calculus.
2.1 Euler’s Variational Equation
xb=
If yx0()minimizes Jy()= ∫ Fxyx (,(),'()) y x dx xa=
∂∂FdF⎛⎞⎟ Then −=⎜ ⎟ 0 . ∂∂ydxy⎝⎠⎜ ' ⎟ yy==00yy Proof: Since by 1.5, δyx() does not depend on the size of the x -
interval, we will use an infinitesimal interval
[,xdxxd−+11x], ax< Then, yx0() minimizes 19 Gauge Institute Journal, Volume , No. , H. Vic Dannon ξ=+xdx1 2 dI()οδδ =+= dJ ( y00 y )∫ F (,() ξξδξ y + y , y0 '() +δξ y ') d , ξ=−xdx1 2 over all the curves y0 + δy that cross y0()ξ at the endpoints of [,xdxxd−+11x]. That is, 22 δδyx()(−=+11 dx yx dx). 22 The Taylor expansion to second order of Fy(,ξξδξξδξ00 ()++− y (), y '() y '()) Fxyy (,0 (), ξ0 '()) ξ, about y0()ξ , is ∂∂FF δδyy++'(Oο2). ∂∂yy' δ yy==00yy Therefore, ξ=+xdx1 2 ⎧⎫ ⎪⎪∂∂FF dJ()() y+−δδ y dJ y = ⎨⎬⎪⎪y+δ y'( dξ + O ο2 ) 00∫ ⎪⎪∂∂yy' δ ξ=−xdx1 ⎩⎭⎪⎪yy00 2 d By 1.8, δy' = δy, and integrating by parts, dx dJ()() y00+−δ y dJ y = ξ=+xdx1 ξ=+xdx1 2 ⎪⎪⎧⎫⎡⎤⎛⎞ ⎛⎞ 2 ⎪⎪∂∂FdF ⎢⎥ ∂F 2 =−⎨⎬⎜⎜⎟⎟δξξyd() + δξy()+ O (ο ) ∫ ⎪⎪⎜⎜⎟⎟⎢⎥δ 1 ∂∂ydyξ ⎝⎠''⎢⎥ ⎝⎠ ∂y ξ=−xdx⎩⎭⎣⎦⎪⎪yy00 y0ξ=−xdx1 2 2 =0 The integration sum over the infinitesimal interval yields 20 Gauge Institute Journal, Volume , No. , H. Vic Dannon ⎪⎪⎧⎫ ⎪∂∂FdF⎛⎞⎟ ⎪ only the term, ⎨− ⎜ ⎟ ⎬δyxdx() , and we obtain ⎪∂∂ydxy⎝⎠⎜ ' ⎟ ⎪ ⎩⎭⎪⎪yy00 ⎪⎪⎧⎫⎛⎞ ⎪⎪∂∂FdF⎟ 2 dJ()() y+−δδ y dJ y =⎨⎬ −⎜ ⎟ y()( x dx+ O ο) 00⎪⎪∂∂ydxy⎝⎠⎜ ' ⎟ δ ⎩⎭⎪⎪yy00 Substituting by 1.1, δοηyx()= δ () x , ⎪⎪⎧⎫⎛⎞ ⎪⎪∂∂FdF⎟ 2 dI()οο−= dI (0) ⎨⎬ −⎜ ⎟ η()x dx+ O ()ο, δδ⎪⎪∂∂ydxy⎝⎠⎜ ' ⎟ δ ⎩⎭⎪⎪yy00 ⎪⎪⎧⎫⎛⎞ dI()οδ − dI (0) ⎪⎪∂∂FdF⎟ =−⎨⎬⎜ ⎟ ηο()xdx + O ( ), ο ⎪⎪∂∂ydxy⎝⎠⎜ ' ⎟ δ δ ⎩⎭⎪⎪yy00 Since this hold for any infinitesimal οδ , dI()ε has a derivative at ε = 0 , which is, ⎪⎪⎧⎫ ⎪⎪∂∂FdF⎛⎞⎟ dI '(0) =−⎨⎬⎜ ⎟ η()x dx . ⎪⎪∂∂ydxy⎝⎠⎜ ' ⎟ ⎩⎭⎪⎪yy00 Since dI(0)= dJ ( y0 ) is minimal, we have by Fermat Theorem, dI '(0)= 0 , and since dx > 0, ⎪⎪⎧⎫ ⎪⎪∂∂FdF⎛⎞⎟ ⎨⎬−=⎜ ⎟ η()x 0. ⎪⎪∂∂ydxy⎝⎠⎜ ' ⎟ ⎩⎭⎪⎪yy00 Since η()x is an arbitrary smooth enough function, ∂∂FdF⎛⎞⎟ −=⎜ ⎟ 0 ., ∂∂ydxy⎝⎠⎜ ' ⎟ yy==00yy 21 Gauge Institute Journal, Volume , No. , H. Vic Dannon From the derivation of Euler’s equation, 2.2 The First variation of Fxyy(,, ') is ∂∂FF δδFxyy(,, ')=+ yδ y' ∂∂yy' ⎪⎪⎧⎫∂∂FdF =−⎨⎬⎪⎪δy . ⎩⎭⎪⎪∂∂ydxy' =∂−∂()(δyDyxy' )F Therefore, we define 2.3 The First Variational Derivative of Fxyy(,, ') δFFdF∂∂⎛⎞⎟ =−⎜ ⎟ δyydxy∂∂⎝⎠⎜ ' ⎟ =∂−()yxyDF ∂' ξξ==xx 2.4 δξξξξδξξξ∫∫Fy(,(),'()) y d= Fy (,(),'()) y dx. ξξ==aa xb= 2.5 If yx0()minimizes Jy()= ∫ Fxyx (,(),'()) y x dx xa= δF Then = 0, δy y0 22 Gauge Institute Journal, Volume , No. , H. Vic Dannon δF = 0 , y0 δJ = 0 . y0 2.6 The Second Variation of Fxyy(,, ') ⎛⎞2 2 ∂∂⎟ δδδFxyy(,, ')=+⎜ y y' ⎟ F ⎝⎠⎜ ∂∂yy' ⎟ ⎛⎞222 ⎜ 22∂∂∂⎟ =+⎜()δδδδyyyy 2()(') + ( ') ⎟F ⎝⎠⎜ ()∂∂yy22∂∂yy' ( ')⎟ 2.7 Second Variational Derivative of Fxyy(,, ') By 2.3, δ =∂ −D ∂ , δy yxy' Applying this operator consecutively 2 δ 2 =∂−()yxyD ∂' ()δy 2 and we may define the second Variational Derivative of Fxyy(,, ') by 2 δ F 222 =∂−(2yxyyxyDD ∂∂+'' ∂ )F ()δy 2 But we know of no use to this proposition. 23 Gauge Institute Journal, Volume , No. , H. Vic Dannon 3. Euler’s Equation, and Strong Variations In the introduction, we presented the textbooks’ derivation of Euler’s equation, with variations defined as εη()x , where ε > 0 is very small, and is allowed to decrease to zero. Such an epsilon is, in fact, an infinitesimal on the real line, and no such infinitesimal exists. Calculus of Variations Texts call such variations “Weak Variations”. We defined these variations correctly, as hyper-real infinitesimal functions, [Dan2], and applied them to derive Euler’s equation in the Infinitesimal Calculus of Variations. For instance, if Fxyy(,, ')=+ ( y ')23 ( y '), 01≤≤x , Euler’s Equation is d ()2'yy+= 3(')2 0, dx 24 Gauge Institute Journal, Volume , No. , H. Vic Dannon yx0()= 0, 01≤≤x is a solution, hence, an extremal, and x =1 23 IJyyy()οδδ == ()∫ {} ( δδ ')( + ')dx x =0 xx==11 =+οη2233(')dx οη (')dx δδ∫∫ xx==00 Therefore, xx==11 23 Id''(οηοηδδ )=+ 2∫∫ ( ')x 6 ( ') dx. xx==00 Hence, x =1 Id''(0)=> 2∫ (η ')2 x 0 , x =0 x =1 23 and yx0()= 0, minimizes Jy()=+∫ {} ( y ') ( y ') dx. x =0 Calculus of variations Texts consider “Strong Variations”, that have derivatives that are infinite hyper-reals [Dan2]. Such strong variation is the saw-tooth y()θ [Pars, p.46] 25 Gauge Institute Journal, Volume , No. , H. Vic Dannon It is bounded by sinθθ cos , that is infinitesimal for θ infinitesimal. But for θ infinitesimal, y ' close to x = 1, has an infinite hyper-real jump, and the integral is an infinite hyper-real. Indeed, Jy( )=+ (tan23223θθθθθ tan )cos +− (cot cot )sin2θ =−12cot2θ is a negative infinite hyper-real for θ infinitesimal. The purpose of using such y()θ is unclear, because we define variations in a form that will be useful to us. In particular, Euler’s Equation is derived only with variations that are infinitesimal functions. Strong variations are inapplicable to the derivation of Euler’s Equation, and Hence irrelevant to the calculus of variations. To date, no other derivation of the Euler Equation is known except the one that uses infinitesimal variations. 3.1 The derivation of Euler’s Variational Equation requires infinitesimal variations, and the Equation holds only if derived with infinitesimal variations. 26 Gauge Institute Journal, Volume , No. , H. Vic Dannon Consequently, 3.2 Euler’s Variational Equation prohibits arbitrary, non- infinitesimal variations. But Euler’s Variational Equation is the main result of the Calculus of Variations, and without Euler’s equation, there is no Calculus of Variations. The mention of “strong variations” is pointless because with such variations, Euler’s Equation disappears, and the Calculus of variations has no results. 3.3 Euler’s Equation, and hence, the Calculus of Variations, are incompatible with strong variations Thus, when we talk about variations we mean only the infinitesimal variations. The irrelevant Strong Variations, defy the fundamental role of Euler’s Variational Equation in the foundations of Dynamics, and Theoretical Physics. 27 Gauge Institute Journal, Volume , No. , H. Vic Dannon 4. First Integral of Euler’s Equation Forsyth presents a derivation of the first Integral of the Euler’s equation for Fxyy(,, '). To be well defined, the extremals that solve the Euler equation for Fxyy(,, '), need to be disjoint curves, infinitesimally close to each other, but non-intersecting. We parametrize the family of extremals by α , and denote them by yxα(). At each plane point (,xy), the slope of an extremal yα is a function of (,xy). That is dy α = pxy(, ) dx Thus, Fxyy(,, ')=Φ (,, xyp ) and the Euler Equation transforms into a first order equation for pxy(, ) 28 Gauge Institute Journal, Volume , No. , H. Vic Dannon 5.1 First Integral of Euler’s equation ypxy'(,= ) transforms Euler’s equation for y , into a first order equation for p ⎛⎞22 ∂pp ∂1 ⎜ ∂Φ ∂Φ ∂Φ ⎟ +=pp⎜ − − ⎟ ∂∂xy2 ⎜ ∂∂∂∂∂ yxpyp⎟ ∂Φp ⎝⎠ which characteristic equations are dy dp dx == . p 1 ⎛⎞∂Φ ∂Φ22 ∂Φ ⎟ ⎜ −−p ⎟ 2 ⎜ ∂∂∂∂∂yxpyp⎟ ∂Φp ⎝⎠ Proof: Euler’s Equation for y is ∂Φdxy ∂Φ(,,p ) = ∂∂ydxp ∂Φ222dx ∂Φ dy ∂Φ dp =++ ∂∂x p dx ∂∂ y p dx()∂p 2 dx Substituting dy = p , dx dp(, x y ) ∂∂ p dx p dy =+, dx∂∂ x dx y dx ∂∂pp =+p , ∂∂xy 29 Gauge Institute Journal, Volume , No. , H. Vic Dannon we obtain Euler’s equation for p , 22 2⎛⎞ ∂Φ ∂Φ ∂Φ ∂Φ⎜ ∂pp ∂ ⎟ =+pp +⎜ +⎟. ∂∂∂∂∂yxpyp()∂p 2 ⎝⎠⎜ ∂∂ xy⎟ Reorganizing it as a first order equation for pxy(, ), we obtain ⎛⎞22 ∂pp ∂1 ⎜ ∂Φ ∂Φ ∂Φ ⎟ +=pp⎜ − − ⎟. ∂∂xy2 ⎜ ∂∂∂∂∂ yxpyp⎟ ∂Φp ⎝⎠ The characteristic equations are dx dy dp == ., 1 p 1 ⎛⎞∂Φ ∂Φ22 ∂Φ ⎟ ⎜ −−p ⎟ 2 ⎜ ∂∂∂∂∂yxpyp⎟ ∂Φp ⎝⎠ 5.2 Surface of Revolution The area of the surface generated by the revolution of yyx= () about the x axis between x = 0 , and x = 1, is x =1 21(')π ∫ yy+ 2dx, x =0 Taking Fxyy(,, ')=+ y 1 ( y ')2 , we have Φ=+(,,xyp ) y 1 p2 , ∂Φ 1 =+(1p2 )2 , ∂y 30 Gauge Institute Journal, Volume , No. , H. Vic Dannon ∂Φ p = y 1 , ∂p (1+ p2 )2 ∂Φ2 = 0, ∂∂xp ∂Φ2 p = 1 , ∂∂yp (1+ p2 )2 22⎛⎞⎟ ∂Φ ⎜ 1 py⎟ =−=y ⎜ 13⎟ 3, 2 ⎜ 22⎟ 2 ()∂p ⎝+⎜(1pp )22 (1 + ) ⎠⎟ (1 +p )2 3 ⎛⎞22 22 ⎛⎞ 1(∂Φ ∂Φ ∂Φ⎟ 1 +pp)⎜ 2 1 ⎟ ⎜ −−pp⎟ =⎜(1 +− )2 p⎟ ⎜⎟⎟ ⎜ 1 2 ⎜ ∂∂∂∂∂yxpyp⎟ y⎜ 2 ⎟ ∂Φp ⎝⎠⎜⎝+(1p )2 ⎠⎟ 1 + p2 = . y Thus, the characteristic equations are dy dp dx== y p 1 + p2 Hence, 1 p dy= dp , y 1 + p2 logyp=++1 log(12 ) log α , 2 1 yp=+α(12 )2 , dy 1 y =+α(1 ( )2 )2 , dx 31 Gauge Institute Journal, Volume , No. , H. Vic Dannon dy 1 = dx , y22− α α 1 log(yy+−=+22αβ) x, α 1 x yy+−=22α Ceα . Otherwise, solving Euler’s equation directly, ∂F 1 =+[1 (y ')2 ]2 , ∂y ∂Fy' = y 1 , ∂y ' [1+ (y ')2 ]2 ⎛⎞⎟ 22 dy⎜ '⎟ ( yyyyy ')+ " ( ') ⎜y 11⎟ =−3, ⎜ 22⎟ 2 dx ⎝+⎜ [1 (yy ')]22 ⎠⎟ [1 + ( ')] [1 + (y ')]2 Euler’s equation is 22 1 (')yyyyy+ " (') 2 2 [1+= (y ') ] 13 − , [1++ (yy ')22 ]22 [1 ( ') ] yy(')2 1(')(')+=+−yyyy22 " , 1(')+ y 2 yy(')2 1"=−yy , 1(')+ y 2 1+= (yyyyyy ')22 "[1 +− ( ') ] ( ')2, 32 Gauge Institute Journal, Volume , No. , H. Vic Dannon which seems to be more difficult to solve. 33 Gauge Institute Journal, Volume , No. , H. Vic Dannon 5. Legendre Sufficient Condition for xb= ∫ Fxyy(,, ') dx to be Minimal xa= 5.1 δ2F > 0 is sufficient for a minimum y0 Proof: Since by 1.18 23 Jy(0000+=δδδ y ) Jy () + Jy () + Jy () + O ()οy , and since at the minimum δJy()0 = 0, a sufficient condition for Jy()(00+>δ y Jy), is 2 δ Jy()0 > 0. This condition can be guaranteed by δ2F > 0. y0 5.2 Legendre Sufficient Condition for a Minimum 34 Gauge Institute Journal, Volume , No. , H. Vic Dannon ∂2F > 0 . (')∂y 2 y0 Proof: ξ=+xdx1 2 δξ22Fyydx(, , ') = ∫ δξ Fyyd(, , ')ξ ξ=−xdx1 2 xdx+1 2 ⎪⎪⎧⎫∂∂22FF∂2F =++⎨⎬⎪⎪()δδδδyyyy22 2()(') ( ') dξ, ∫ ⎪⎪()∂∂yy22∂∂yy' ( ') xdx−1 ⎩⎭⎪⎪ 2 where δy vanishes at the integration end points. d Since 2(δδyy )( ')== 2( δδ yy )( ) ' ( δ y )2 , we have dξ xdx++11xdx xdx + 1 22∂∂22FdyF()δ 2 2∂2F =+()δξyd22dξδ +( y ') dξ ∫∫()∂∂yy22dyyξ ∂∂ ' ∫( ') xdx−−11xdx xdx − 1 22 2 Integration by parts gives xdx++1 xdx+1 xdx1 2 dy()δ 22∂∂ F ⎡⎤2F 2 2 d∂2 F dyξδ=−⎢⎥()22()δy dξ ∫∫dyyξξ∂∂ ''⎢⎥∂∂ yy dyy∂∂ ' xdx−−1 ⎣⎦xdx−1 xdx1 2 2 2 =0 Hence, xdx++11xdx 22⎡⎤∂∂22FdF ∂2F δξ22Fyydx(, , ')=−+ (δ y )⎢⎥dξ( δ y ')2 dξ ∫∫⎢⎥()∂∂yy22dyyξ ∂∂ ' ( ') xdx−−11⎣⎦xdx 22 35 Gauge Institute Journal, Volume , No. , H. Vic Dannon xdx++11xdx 22∂∂⎡⎤FdF ∂ ∂2F =−+()δξydy22⎢⎥(δ ') dξ ∫∫∂∂yydy⎢⎥ξ ∂' (')∂y 2 xdx−−11⎣⎦xdx 22 ∂∂⎡⎤FdF ∂ ∂2F =−+()δδyd22⎢⎥x(y ') dx ∂∂yydxy⎢⎥⎣⎦ ∂' (')∂y 2 ∂∂FdF At yy= , −=0, and we have, 0 ∂∂ydxy' ⎡ 2 ⎤ 22⎢⎥∂ F δξFyydxy(,00 , ')= ( δ ') dx. ⎢⎥(')∂y 2 ⎣ ⎦y0 Since dx > 0, it cancels on both sides, and ⎡ 2 ⎤ 22⎢ ∂ F ⎥ δξFyy(,00 , ')= ( δ y ') . ⎢(')∂y 2 ⎥ ⎣ ⎦y0 Since (')0δy 2 > in the infinitesimal integration interval, ⎡ ∂2F ⎤ ⎢⎥> 0, ⎢⎥(')∂y 2 ⎣ ⎦y0 2 guarantees δξFyy(,00 , ')> 0., 5.3 Shortest Distance For Fxyy(,, ')=+ 1 ( y ')2 , the Euler equation solution is a line with slope y' = m, and 36 Gauge Institute Journal, Volume , No. , H. Vic Dannon applying Legendre Sufficient Condition ⎡ ⎤ ⎡⎤∂2 11 ⎢⎥1(')+=y 2 ⎢⎥ = >0. ⎢⎥2 ⎢⎥33 (')∂y ⎢⎥2222 ⎣⎦y0 ⎣[1++ (ym ') ]⎦ym'= [1 ] That is, the line is the shortest distance. 37 Gauge Institute Journal, Volume , No. , H. Vic Dannon 6. Weierstrass Sufficient Condition, xb= for ∫ Fxyy(,, ') dx to be Minimal xa= In his Lectures on the Calculus of Variations, Weierstrass xb= observed that a sufficient condition for ∫ Fxyy(,, ') dx to be xa= minimal can be given. A bizarre theory to substantiate Weierstrass observation, did not produce a correct proof, and to date Weierstrass result remains unproved. We give here a one line proof. 6.1 Weierstrass Sufficient Condition for a Minimum ∂F Fxyx(,(),'()) y x−−− Fxy (, (),'()) x y x (' y y ')> 0. 00∂y ' y0 Proof: 38 Gauge Institute Journal, Volume , No. , H. Vic Dannon Let y0()ξ , the extremal that solves the Euler equation, intersect the curve y()ξ at x . Since yx()= y0 () x , the Taylor expansion of Fxyy(,, ') to second order about (,xy00 (), x y '()) x is ∂F Fxyx(,(),'()) y x−− Fxy (, (),'()) x y x (' y− y ')= 00∂y ' y0 ∂2F =−+('yy ')23 O (ο ). (')∂y 2 0 δ y0 ∂2F Therefore, Legendre’s sufficient condition > 0 , is (')∂y 2 y0 equivalent to Weierstrass sufficient Condition ∂F Fxyx(,(),'()) y x−−− Fxy (, (),'()) x y x (' y y ')> 0., 00∂y ' y0 From the proof of 6.1, we have 39 Gauge Institute Journal, Volume , No. , H. Vic Dannon 6.2 Weierstrass’ and Legendre’s conditions are equivalent 6.3 The Shortest Distance For Fxyy(,, ')=+ 1 ( y ')2 , the Euler equation solution is a line with slope y' = m, and applying Weierstrass Sufficient Condition ∂F Fxyx(,(),'()) y x−− Fxy (, (),'()) x y x (' y− y ')= 00∂y ' y0 ⎡ 2 ⎤ 22⎢⎥∂+1(')y =+1(')1(')yy −+00 −⎢⎥(' y−y ') ⎢⎥∂y ' ⎣ ⎦y0 22y0 ' =+1(')1(')yy −+00 −1 (' y −y ') 2 2 [1+ (y0 ') ] 2 1''+ yy0 =+1(')y − 1 , 2 2 [1+ (y0 ') ] 2 which is > 0, for yy≠ 0 , because ('yy−>0 ') 0., That is, the line is the shortest distance. 40 Gauge Institute Journal, Volume , No. , H. Vic Dannon 7. Euler’s Equation for Fxyyy(,, ' ") and the Variational Derivative We apply variations, over infinitesimal interval, to derive Euler’s Variational Equation for Fxyyy(,, ' ")in Infinitesimal Variational Calculus. 7.1 Euler’s Variational Equation for Fxyyy(,, ' ") xb= If yx0()minimizes Jy()= ∫ Fxyyy (,, ' ") dx xa= ⎛⎞2 ⎛ ⎞ ∂∂FdFdF⎜⎜⎟⎟ ∂ Then −+⎜⎜⎟⎟=0 . ∂∂ydxy⎝⎠⎜⎜'"⎟⎟()dx 2 ⎝ ∂ y ⎠ yy00 y0 Proof: Since by 1.5, δyx() does not depend on the size of the x - interval, we will use an infinitesimal interval [,xdxxd−+11x], ax< Then, yx0() minimizes 41 Gauge Institute Journal, Volume , No. , H. Vic Dannon ξ=+xdx1 2 dI()οδδ =+= dJ ( y00 y )∫ F (, ξδδδ y ++ y , y0 ' y '," y0 + y ") dξ, ξ=−xdx1 2 over all the curves y0 + δy with δδyx()(−=+11 dx yx dx), 22 δδyx'(−=11 dx ) yx '( + dx ) 22 The Taylor expansion to second order of Fy(ξξδξξδξ ,00 ( )++ y ( ), y '( ) y '( ), y 0 "( ξδξ ) +− y "( )) Fxyy ( ,00 ( ξ ), '( ξ ), y0 "( ξ )), about y0()ξ , is ∂∂FF ∂ F δδyy++'" δ yO +()ο2 . ∂∂yy'" ∂ y δ yy00 y 0 Therefore, dJ()() y00+−δ y dJ y = ξ=+xdx1 2 ⎧⎫ ⎪⎪∂∂FF ∂ F =++⎨⎬⎪⎪δδyy'" δξ ydO+()ο2 ∫ ⎪⎪∂∂yy'" ∂ y δ ξ=−xdx1 ⎩⎭⎪⎪yy00 y 0 2 d Since δyy' = δ, and δδyx()(−=+11 dx yx dx), dx 22 integrating by parts, ξξ=+xdx1 xdx+ 1 =+xdx1 2 ∂∂FF⎡⎤⎛⎞ 2 2 d⎛⎞∂F δyd'( ξ=−⎢⎥⎜⎜⎟⎟ δξy ) δξξy() d ∫∫⎢⎥⎜⎜⎟⎟ 1 ∂∂yy''⎢⎥⎝⎠ 1 dξ ⎝⎠∂y' ξξ=−xdx yy00⎣⎦xdx−1 =−xdx y0 2 2 2 =0 42 Gauge Institute Journal, Volume , No. , H. Vic Dannon d 2 Since δδyy" = , and δδyx'(−=11 dx ) yx '( + dx ), two ()dx 2 22 integrations by parts give, ξξ=+xdx1 xdx+ 1 =+xdx1 2 ∂∂FF⎡⎤⎛⎞ 2 2 d⎛⎞∂F δyd"' ξ=−⎢⎥⎜⎜⎟⎟ δξy() δξξy'() d ∫∫⎢⎥⎜⎜⎟⎟ 1 ∂∂yy""⎢⎥⎝⎠ 1 dξ ⎝⎠∂y" ξξ=−xdx yy00⎣⎦xdx−1 =−xdx y0 2 2 2 =0 xdx+ 1 ξ=+xdx1 ⎡⎤⎛⎞∂∂Fd2 2 2 ⎛⎞F =−⎢⎥⎜⎜⎟⎟δξyy() + δξξ()d ⎢⎥⎜⎜⎟⎟∫ 2 ⎢⎥⎝⎠∂∂yy""1 ()dξ ⎝⎠ ⎣⎦yy00xdx−1 ξ=−xdx 2 2 =0 Hence, dJ()() y00+−δ y dJ y = ξ=+xdx1 2 ⎪⎪⎧⎫⎛⎞2 ⎛ ⎞ ⎪⎪∂∂FdFdF ∂ 2 =−+⎨⎬⎜⎜⎟⎟δξξyd()+ O ( ο ). ∫ ⎪⎪∂∂ydyξ ⎝⎠⎜⎜'"⎟⎟()dξ 2 ⎝ ∂ y ⎠ δ ξ=−xdx1 ⎪⎪yy00 y 0 2 ⎩⎭ The integration sum over the infinitesimal interval yields ⎪⎪⎧⎫⎛⎞2 ⎛ ⎞ ⎪⎪∂∂FdFdF⎜⎜⎟⎟ ∂ ⎨⎬−+⎜⎜⎟⎟δyxdx() , and we obtain ⎪⎪∂∂ydxy⎝⎠⎜⎜'"⎟⎟()dξ 2 ⎝ ∂ y ⎠ ⎩⎭⎪⎪yy00 y 0 ⎪⎪⎧⎫⎛⎞2 ⎛ ⎞ ⎪⎪∂∂FdFdF⎜⎜⎟⎟ ∂ 2 dJ()() y00+−δδ y dJ y =⎨⎬ −⎜⎜⎟⎟ + y()( x dx+ O οδ ) ⎪⎪∂∂ydxy⎝⎠⎜⎜'"⎟⎟()dξ 2 ⎝ ∂ y ⎠ ⎩⎭⎪⎪yy00 y 0 Substituting by 1.1, δοηyx()= δ () x , ⎪⎪⎧⎫⎛⎞2 ⎛ ⎞ ⎪⎪∂∂FdFdF⎜⎜⎟⎟ ∂ 2 dI()οοδ −= dI (0) ⎨⎬ −⎜⎜⎟⎟ + δδη()x dx+ O ()ο, ⎪⎪∂∂ydxy⎝⎠⎜⎜'"⎟⎟()dξ 2 ⎝ ∂ y ⎠ ⎩⎭⎪⎪yy00 y 0 43 Gauge Institute Journal, Volume , No. , H. Vic Dannon ⎪⎪⎧⎫⎛⎞2 ⎛ ⎞ dI()οδ − dI (0) ⎪⎪∂∂FdFdF ∂ =−⎨⎬⎜⎜⎟⎟ + ηο()xdx+ O ( ). ⎪⎪⎜⎜⎟⎟2 δ οδ ∂∂ydxy⎝⎠'"()dξ ⎝ ∂ y ⎠ ⎩⎭⎪⎪yy00 y 0 Since this hold for any infinitesimal οδ , dI()ε has a derivative at ε = 0 , which is, ⎪⎪⎧⎫⎛⎞2 ⎛ ⎞ ⎪⎪∂∂FdFdF⎜⎜⎟⎟ ∂ dI '(0) =−⎨⎬⎜⎜⎟⎟ + η()x dx . ⎪⎪∂∂ydxy⎝⎠⎜⎜'"⎟⎟()dξ 2 ⎝ ∂ y ⎠ ⎩⎭⎪⎪yy00 y 0 Since dI(0)= dJ ( y0 ) is minimal over the infinitesimal interval, we have by Fermat Theorem, dI '(0)= 0 , and since dx > 0, ⎪⎪⎧⎫⎛⎞2 ⎛ ⎞ ⎪⎪∂∂FdFdF⎜⎜⎟⎟ ∂ ⎨⎬−+⎜⎜⎟⎟η()x = 0. ⎪⎪∂∂ydxy⎜⎜⎝⎠'"⎟⎟()dξ 2 ⎝ ∂ y ⎠ ⎩⎭⎪⎪yy00 y 0 Since η()x is an arbitrary smooth enough function, ⎛⎞2 ⎛ ⎞ ∂∂FdFdF⎜⎜⎟⎟ ∂ −+⎜⎜⎟⎟ =0., ∂∂ydxy⎝⎠⎜⎜'"⎟⎟()dξ 2 ⎝ ∂ y ⎠ yy==00yy y0 From the derivation of the Euler’s equation, 7.2 The First variation of Fxyy(,,',") y is ∂∂FF ∂ F δδδFxyy(,,',") y=++ y y'"δ y ∂∂yy'" ∂ y 44 Gauge Institute Journal, Volume , No. , H. Vic Dannon ⎪⎪⎧⎫∂∂FdFd2 ⎛⎞ ∂ F =−+δy ⎨⎬⎪⎪⎜ ⎟ 2 ⎜ ⎟ ⎪⎪⎩⎭∂∂ydxy'"()dx ⎝⎠ ∂ y 2 =∂−∂+∂()(δyDDyxyxy'")F. Therefore, we define 7.3 The First Variational Derivative of Fxyy(,,',") y 2 ⎛⎞ δFFdFdF∂∂⎜ ∂⎟ =− + ⎜ ⎟ δyydxy∂∂'"()dx 2 ⎝⎠⎜ ∂ y⎟ 2 =∂−()yxyxyDD ∂'" + ∂ F ξξ==xx 7.4 δξ∫∫Fyyyd(,,',") ξ= δξ Fyyydx (,,',") . ξξ==aa xb= Fxyx(,(),'(),"()) y x y x dx 7.5 If yx0()minimizes ∫ xa= δF Then = 0, δy y0 δF = 0 , y0 δJ = 0 . y0 45 Gauge Institute Journal, Volume , No. , H. Vic Dannon 7.6 The Second Variation of Fxyy(,,',") y ⎛⎞2 2 ∂∂ ∂⎟ δδδδFxyy(,, ',") y=++⎜ ( y ) ( y ') ( y ") ⎟ F ⎝⎠⎜ ∂∂yy'" ∂ y⎟ ∂∂22FF ∂ 2 F =+()δδyy22 ( ') + ( δ y ") 2 + ()∂∂yy22 ( ') ( ∂ y ") 2 ∂∂22FF∂2F ++2(δδyy )( ') 2(δδyy )( ") +2(δy ')( δ y ") ()(')∂∂yy ()(")∂∂yy (')("∂y ∂ y) 46 Gauge Institute Journal, Volume , No. , H. Vic Dannon 8. Euler’s Equation for Fxy( ,121 , y , y ', y 2 ') and the Variational Derivative The infinitesimal distance in 3 dimensional space is ds=++=++() dx222 () dy () dz 1 (')(')y 22 z dx . The arc-length of the curves yx(), and zx(), between (,aya (),()) za , and (,byb(),()) zb is xb= ∫ 1(')(')++yz22dx, xa= and the curves yx(), and zx() that minimize the integral have the shortest arc-length. These curves are the line between (,aya(),()) za , and (,byb (),()) zb , over the interval [,ab]. Then, the problem of the Calculus of Variations is to find curves yx()= y () x, 110 yx()= y () x 220 that will minimize the Functional 47 Gauge Institute Journal, Volume , No. , H. Vic Dannon xb= Jy(,12 y )= ∫ Fxy (,,121 y , y ',') y 2 dx. xa= We apply variations, over infinitesimal interval, to derive Euler’s Variational Equation in Infinitesimal Variational Calculus. 8.1 Euler’s Equation for Fxy( ,121 , y , y ', y 2 ') xb= If yx(), and yx()minimize Jy(,12 y )= Fxy (,,121 y , y ',') y 2 dx 10 20 ∫ xa= ∂∂FdF⎛⎞ Then −=⎜ ⎟ 0, ⎜ ⎟ ∂∂ydxy11yy⎝⎠' 1100 ∂∂FdF⎛⎞ −=⎜ ⎟ 0 ⎜ ⎟ ∂∂ydxy22yy⎝⎠' 2200 Proof: Since by 1.5, δyx1(), and δyx2() do not depend on the size of the x -interval, we will use an infinitesimal interval [,xdxxd−+11x], ax< dI()οδ=+ dJ ( y y , y +δ y )= δ 112200 48 Gauge Institute Journal, Volume , No. , H. Vic Dannon ξ=+xdx1 2 =++++Fy(,ξδ yy , δ yy , ' δ yy ', ' δ yd ')ξ, ∫ 11221122000 0 ξ=−xdx1 2 over all the curves y+ δy, and y+ δy, that cross y ()ξ 10 1 20 2 10 and y ()ξ at the endpoints of [,xdxxd−+11x]. That is, 20 22 δδyx()(−=11 dx yx + dx), 1122 δδyx()(−=11 dx yx + dx). 2222 The Taylor expansion to second order of Fy(ξξδ , ( )++ yy , ( ξδ ) yy , '( ξδ ) + yy ', '( ξδ ) + y '), 1122112000 02 about the curves y ()ξ , and y ()ξ is 10 20 ∂∂∂FFF ∂ F 2 δδδδyyyy1122++++''O(οδ ). ∂∂yy11yyyy'' ∂ y 2 ∂ y 2 11220000 Therefore, dJ(, y++−δδ y y y )(, dJ y y )= 112200 1200 ξ=+xdx1 2 ⎪⎪⎧⎫ ⎪⎪∂∂∂FFF ∂ F =+++⎨⎬⎪⎪δδδδyyyyd''x+O(ο2) ∫ ⎪⎪1122δ 1 ∂∂yy11'' ∂ y 2 ∂ y 2 ξ=−xdx⎪⎪yyyy1122 2 ⎩⎭⎪⎪0000 d By 1.8, δy' = δy, and integrating by parts, dx dJ(, y++−δδ y y y )(, dJ y y )= 112200 1200 49 Gauge Institute Journal, Volume , No. , H. Vic Dannon 1 1 xdx+ ξ=+xdx 2 2 ⎪⎪⎧⎫⎡⎤ ⎪⎪∂∂FdF⎛⎞⎢⎥ ⎛⎞ ∂F =−⎨⎬⎜⎜⎟⎟δξξyd() +⎢⎥ δξy() + ∫ ⎪⎪⎜⎜⎟⎟11 1 ∂∂ydy11ξ ⎝⎠''⎢⎥ ⎝⎠ ∂y1 ξ=−xdx⎪⎪yy11 y11 2 ⎩⎭⎣⎪⎪00⎢⎥0⎦xdx− 2 =0 1 1 xdx+ ξ=+xdx⎧⎫⎡⎤2 2 ⎪⎪⎛⎞ ⎛⎞ ⎪⎪∂∂FdF ⎢⎥ ∂F 2 +−⎨⎬⎜⎜⎟⎟δξξyd() +⎢⎥ δξy()+ O ( ο ) ∫ ⎪⎪⎜⎜⎟⎟21δ 1 ∂∂ydy22ξ ⎝⎠''⎢⎥ ⎝⎠ ∂y2 ξ=−xdx⎪⎪yy22 y21 2 ⎩⎭⎣⎦⎪⎪00⎢⎥0xdx− 2 =0 The integration over the infinitesimal interval yields only the terms, ⎪⎪⎪⎧⎫⎧ ⎪⎫ ⎪⎪⎪∂∂FdF⎛⎞ ∂∂FdF⎛⎞⎪ ⎨⎬⎨−+⎜⎜⎟⎟δδy() x dx −⎬y() x dx . ⎪⎪⎪⎜⎜⎟⎟12⎪ ⎪⎪⎪∂∂ydxy11yy⎝⎠'' ∂∂ydxy22 yy⎝⎠⎪ ⎩⎭⎩⎪⎪⎪1100 2 0 2 0⎭⎪ Substituting by 1.1, δοηyx11()= δ () x, δοηyx22()= δ () x, dI()οδ −= dI (0) ⎛⎧ ⎫ ⎧ ⎫ ⎞ ⎜⎪⎪⎛⎞ ⎪⎛⎞ ⎪⎟ ⎪⎪∂∂FdF⎟⎟⎪ ∂∂ FdF ⎪⎟ 2 =−⎜⎨⎬⎪⎪⎜⎜⎟⎟ηη()xx +−⎨⎪ ⎬⎪()⎟οdx+O (ο ) ⎜⎪⎪⎜⎜⎟⎟12⎪ ⎪⎟ δδ ⎜⎪⎪∂∂y11yy dxy⎝⎠''⎪ ∂∂y 2 yy dxy⎝⎠ 2 ⎪⎟ ⎝⎠⎩⎭⎪⎪1100⎩⎪ 2 0 2 0 ⎭⎪ dI()ο − dI (0) δ = οδ ⎛⎧⎪⎪⎫ ⎧⎪⎫ ⎪ ⎞ ⎜⎪⎪∂∂FdF⎛⎞⎟⎟⎪ ∂∂ FdF⎛⎞ ⎪⎟ =−⎜⎨⎬⎪⎪⎜⎜⎟⎟ηη()xx +−⎨⎪ ⎬⎪()⎟dx+O (ο ) ⎜⎪⎪⎜⎜⎟⎟12⎪ ⎪⎟ δ ⎜⎪⎪∂∂ydxy11yy⎝⎠''⎪ ∂∂ ydxy 2 y⎝⎠ 2 y ⎪⎟ ⎝⎠⎩⎭⎪⎪1100⎩⎪ 2 0 2 0 ⎭⎪ 50 Gauge Institute Journal, Volume , No. , H. Vic Dannon Since this hold for any infinitesimal οδ , dI()ξ has a derivative at ξ = 0, which is, ⎛⎧⎪⎪⎫ ⎧⎪ ⎫⎪ ⎞ ⎜⎪⎪∂∂FdF⎛⎞⎟⎟⎪ ∂∂ FdF⎛⎞⎪⎟ dI '(0)=−⎜⎪⎪⎨⎬⎜⎜⎟⎟ηη()x +−⎪⎨ ⎬⎪()x⎟ dx . ⎜⎪⎪⎜⎜⎟⎟12⎪ ⎪⎟ ⎜⎪⎪∂∂y11yy dxy⎝⎠''⎪ ∂∂y 2 y dxy⎝⎠ 2 y⎪⎟ ⎝⎠⎩⎭⎪⎪1100⎩⎪ 2 0 2 0⎭⎪ Since dI(0)= dJ ( y , y ) is minimal, we have by Fermat 1200 Theorem, dI '(0)= 0 , and since dx > 0, ⎪⎪⎧⎫⎪⎧ ⎪⎫ ⎪⎪∂∂FdF⎛⎞⎟⎟⎪ ∂∂ FdF⎛⎞ ⎪ ⎨⎬⎪⎪−+−⎜⎜⎟⎟ηη()xx⎨⎪ ⎬⎪()= 0. ⎪⎪⎜⎜⎟⎟12⎪ ⎪ ⎪⎪∂∂ydxy11yy⎝⎠''⎪ ∂∂ ydxy 22 y⎝⎠ y ⎪ ⎩⎭⎪⎪1100⎩⎪ 2 0 2 0 ⎭⎪ Since η1()x and η2()x arbitrary smooth enough functions, independent of each other, ∂∂FdF⎛⎞ −=⎜ ⎟ 0, ⎜ ⎟ ∂∂ydxy11yy⎝⎠' 1100 ∂∂FdF⎛⎞ −=⎜ ⎟ 0 ., ⎜ ⎟ ∂∂ydxy22yy⎝⎠' 2200 For the shortest distance in 3 dimensional space, the equations are y ' = C1 , 1(')(')++yz22 z ' = C2 . 1(')(')++yz22 51 Gauge Institute Journal, Volume , No. , H. Vic Dannon Hence, z ' = C , y ' dz= Cdy , which yields a line. From the derivation of Euler’s equation, 8.2 The First Variation of Fxy( ,121 , y , y ', y 2 ') is ∂∂∂∂FFFF δδFy=+112 δδ y'' ++ yδy2 ∂∂yy11'' ∂∂ yy 2 2 ⎪⎪⎪⎧⎫⎧∂∂FdF ∂∂ FdF⎪⎫ =−⎨⎬⎨⎪⎪⎪δδyy +− ⎬⎪. ⎪⎪⎪12⎪ ⎩⎭⎩⎪⎪⎪∂∂ydxy11'' ∂∂ ydxy 22⎭⎪ =∂−∂+∂−∂δδyD()( yD)F. {}1'2yxy11 yxy 22' Therefore, we define 8.3 First Variational Derivatives of Fxy( ,121 , y , y ', y 2 ') δFFdF∂∂⎛⎞ =−⎜ ⎟, ⎜ ⎟ δyydxy11∂∂⎝⎠ 1' δFFdF∂∂⎛⎞ =−⎜ ⎟. ⎜ ⎟ δyydxy22∂∂⎝⎠ 2' 52 Gauge Institute Journal, Volume , No. , H. Vic Dannon ξξ==xx 8.4 δξ∫∫Fyyyydx( ,121 , , ', 2 ') = δξ Fyyyyd( ,121 , , ', 2 ') ξ. ξξ==aa 8.5 If yx(), and yx() minimize 10 20 xb= Jy(,12 y )= ∫ Fxy (,,121 y , y ',') y 2 dx xa= δF δF Then = 0 , = 0 δy1 yy, δy2 yy, 1200 1200 δF = 0 , yy, 1200 δJ = 0 . yy, 1200 8.6 The Second Variation of Fxy( ,121 , y , y ', y 2 ') 22 2 ⎡⎤⎡ ⎤ δδFy=∂+∂+∂+∂()11'22'yy ( δ y ') () δ y yy ( δ y ') F {}⎣⎦⎢⎥11⎣⎢ 2 2⎦⎥ 22 22 =∂+∂∂+∂{}()δδδδyyyy111'1 2()(')yy ( ') F yy1111 ' 22 22 +∂+∂∂+∂{}()δδδδyyyy222'2 2()(')yy ( ') F yy2222 ' 53 Gauge Institute Journal, Volume , No. , H. Vic Dannon 9. Euler’s Equation for f (,txyxy , , , ) and the Variational Derivative The infinitesimal distance in the plane is ds=+=+() dx22 () dy ()dx 2 ()dy 2dt . dt dt The arc-length of the curve with coordinates xt(), and yt(), between ((xt00 ),( yt )), and ((xt11 ),( yt )) is tt= 1 ()dx 22+ ()dy dt , ∫ dt dt tt= 0 and the curve with xt() , and yt() that minimize the integral has the shortest arc-length, between ((xt00 ),( yt )), and ((xt11 ),( yt )), over the interval [,]tt01. The Parametric problem is to find the curve with xt()= x0 () t , yt()= y0 () t tt= 1 that will minimize the Functional Kxy(,)= ∫ ftxyxydt (,,,,) . tt= 0 54 Gauge Institute Journal, Volume , No. , H. Vic Dannon By 8.1, 9.1 Euler’s Equation for f (,txyxy , , , ) tt= 1 If xt0(), and yt0() minimize Kxy(,)= ∫ ftxyxydt (,,,,) tt= 0 ∂∂f df⎛⎞⎟ Then −=⎜ ⎟ 0, ∂∂xdtx⎝⎠⎜ ⎟ xy00,,xy00 ∂∂f df⎛⎞⎟ −=⎜ ⎟ 0 ∂∂ydty⎝⎠⎜ ⎟ xy00,,xy00 For the shortest distance in the plane, the equations are x = C1 , xy22+ y = C1 . xy22+ Hence, y = C , x dy= Cdx , which yields a line. From the derivation of Euler’s equation, 8.2 The First Variation of f (,txyxy , , , ) is 55 Gauge Institute Journal, Volume , No. , H. Vic Dannon ∂∂∂∂f fff δδδδf =+++xxy δy ∂∂∂∂xxyy ⎪⎪⎪⎧⎫⎧∂∂f df ∂∂ f df⎪⎫ =−⎨⎬⎨⎪⎪⎪δδxy +− ⎬⎪. ⎩⎭⎩⎪⎪⎪∂∂x dtx ∂∂ y dty ⎭⎪ ={}δδxD()( ∂−∂+xtx yD ∂−∂ yty )f. Therefore, we define 9.3 First Variational Derivatives of f (,txyxy , , , ) δf ∂∂fdf =− , δxxdt∂∂x δf ∂∂fdf =− . δyydt∂∂y ττ==tt 9.4 δττττττδτττττ∫∫fxyxyd(,(),(),(),()= fxyxyd (,(),(),(),())τ. ττ==tt00 tt= 1 9.5 If xt0(), yt0() minimize Kxy(,)= ∫ ftxyxydt (,,,,) tt= 0 δf δf Then = 0 , = 0 δx δy xy00, xy00, δf = 0, xy00, 56 Gauge Institute Journal, Volume , No. , H. Vic Dannon δK = 0 . xy00, 9.6 The Second Variation of f (,txyxy , , , ) 2 2 δδδ2f =∂+∂+∂+∂⎡⎤()xx () ⎡⎤ () δδ yy () f {}⎣⎦xx ⎣⎦ yy 22 22 =∂+∂∂+∂{}()δδδδxxxxxx 2()()x ()xf 22 22 +∂+∂∂+∂{}()δδδδyyyyyy 2()()y ()yf 57 Gauge Institute Journal, Volume , No. , H. Vic Dannon 10. Weierstrass Sufficient Condition, tt= 1 for ∫ f (,txyxydt , , , ) to be Minimal tt= 0 Weierstrass sufficient condition for a parametric curve, can be derived by translation of the condition for the curve yx(). But it follows immediately from the Euler Identity for f (,txyxy , , , ). For t = λτ , f (xt ( ), yt ( ),λλ xt ( ), yt ( )) dt= λττττ f ( x ( ), y ( ), x ( ), y ( )) dτ Differentiating both sides with respect to λ , ∂∂fxtfyt(λλ ( )) ∂∂( ( )) dt +=dt f((),(),(),()) xττττ y x y dτ ∂∂∂∂(λλxt ( )) ( λλyt ( )) ∂∂fxtfyt(λλ ( )) ∂∂( ( )) λτdd+=λτf((),(),(),())x τy τx τy τd τ ∂∂(())λλxt ∂∂(()) λλyt 11∂∂ffxt() yt() λλ∂∂xt() yt() and we obtain 10.1 Euler’s Identity 58 Gauge Institute Journal, Volume , No. , H. Vic Dannon ∂∂f f t =⇒λτ xt()+= yt () fx ((ττττ ),( y ),( x ),( y )) ∂∂xt() yt () 10.2 Weierstrass Sufficient Condition for Minimum ∂∂f f fx(,,,) y xy−− x y >0, 00 ∂∂xy xyxy0000,,, xyxy0000,,, or, ⎛⎞⎛⎞ ⎜⎜∂∂ff⎟⎟ ∂∂ ff xy ⎜⎜−+−⎟⎟ >0. ⎜⎜∂∂xx⎟⎟ ∂∂ yy ⎝⎠xyxy00,,, xyxy0000 ,,, ⎝xyxy00,,, xyxy0000 ,,, ⎠ Proof: Let x0()τ ,y0()τ , the parametric extremal that solves the Euler equation, intersect the parametric curve x()τ ,y()τ at τ , so that xx()τ= 0 ()τ, and yy()ττ= 0 (). Using the Euler Identity in 10.1, either inequality ensures f (xyxy00 (),(),(),())ττττ−> fxyxy (0000 (),(),(),()) ττττ 0., 10.3 The shortest distance f (,,xyxy , )=+ x22 y Weierstrass sufficient condition 59 Gauge Institute Journal, Volume , No. , H. Vic Dannon xy xyx22+−00 − y >0 , 22 22 xy00++ xy 00 holds if 2222 2 ()()(xyxy++>+00 xxyy 0 0) which is guaranteed to hold since 2 ()xy00−> xy 0., 60 Gauge Institute Journal, Volume , No. , H. Vic Dannon 11. Euler-Lagrange Equations for L(,tqtqtqtqt1212 (), (), (), ()), and the Variational Derivative At time t , a particle with mass m , and speed (,xyz,), has kinetic energy 1 mx(22++ y z2), potential energy Uxyz(,,), 2 and Lagrangian L =++−1 mx()(222 y z Uxyz,,). 2 Its path of least action is the curve ((),(),())xtytzt000 that tt= 1 minimizes the Action Integral, ∫ Ldt . tt= 0 The Euler equation of 7.1 with respect to x is ∂∂LLd = , ∂∂xdtx ∂Ud −=()mx , ∂xdt dp F = x , x dt 61 Gauge Institute Journal, Volume , No. , H. Vic Dannon dp dp Similarly, F = y , and F = z . y dt z dt Therefore, the path of least action satisfies G G dp F = dt which is Newton’s Force Law. For two masses, m with kinetic energy 1 mx()222++ y z , 1 2 11 1 1 and m with kinetic energy 1 mx(22++ y z2), in a potential 2 2 22 2 2 Ux(,,,111222 y z x , y , z ), the Lagrangian is L =+++++−11mx()()(,,,,222 y z mx 222 y z Uxyzxyz,). 221111 2222 111222 The possible paths may depend on say, two generalized coordinates qq12, , such as distances, and angles, so that xxqq111= (,2 ); xxqq221= (,2 ) yyqq111= (,2 ); yyqq221= (,2 ) zzqq111= (,2 ); zzqq221= (,2 ). Then, dx∂∂ x dq x dq dx∂∂ x dq x dq 1111=+2; 2212=+2 dt∂∂ q12 dt q dt dt∂∂ q12 dt q dt dy∂∂ y dq y dq dy∂∂ y dq y dq 1111=+2; 2212=+2 dt∂∂ q12 dt q dt dt∂∂ q12 dt q dt 62 Gauge Institute Journal, Volume , No. , H. Vic Dannon dz∂∂ z dq z dq dz∂∂ z dq z dq 1111=+2; 2212=+2 dt∂∂ q12 dt q dt dt∂∂ q12 dt q dt and the Lagrangian is ⎡⎧⎪⎪⎛⎞⎛⎞⎛⎞222⎫ ⎧⎪ ⎛⎞⎛⎞⎛⎞ 222⎪⎫⎤ ⎢⎥⎪⎪⎜⎜⎜∂∂∂xyz111⎟⎟⎟⎪ ⎜⎜⎜ ∂∂∂ xyz 222 ⎟⎟⎟⎪2 L =+++++⎢⎥mm12⎨⎬⎜⎜⎜⎟⎟⎟⎨ ⎜⎜⎜ ⎟⎟⎟⎬q1 ⎢⎥⎪⎪⎜⎜⎜⎝⎠⎝⎠⎝⎠∂∂∂qqq⎟⎟⎟⎪ ⎝⎠⎝⎠⎝⎠⎜⎜⎜ ∂∂∂ qqq ⎟⎟⎟⎪ ⎣⎩⎪⎪111⎭ ⎩⎪ 111⎪⎭⎦ ⎡⎧⎪⎪⎛⎞⎛⎞⎛⎞222⎫ ⎧⎪ ⎛⎞⎛⎞⎛⎞ 222⎪⎫⎤ ⎢⎥⎪⎪⎜⎜⎜∂∂∂xyz111⎟⎟⎟⎪ ⎜⎜⎜ ∂∂∂ xyz 222 ⎟⎟⎟⎪2 ++++++⎢⎥mm12⎨⎬⎜⎜⎜⎟⎟⎟⎨ ⎜⎜⎜ ⎟⎟⎟⎬q2 ⎢⎥⎪⎪⎜⎜⎜⎝⎠⎝⎠⎝⎠∂∂∂qqq⎟⎟⎟⎪ ⎝⎠⎝⎠⎝⎠⎜⎜⎜ ∂∂∂ qqq ⎟⎟⎟⎪ ⎣⎩⎪⎪222⎭ ⎩⎪ 222⎪⎭⎦ −Uq(,12 q ). Thus, the Lagrangian depends on qqqq1212,,,. That is, LL= (,tqtqtqtqt1212 (), (), (), ()). The Euler-Lagrange problem is to find the curve qtqt(), (), 1200 tt= 1 that minimizes the Action ∫ Ldt , over the interval [,]tt01. tt= 0 By 9.1, we have, 11.1 Euler-Lagrange Equation tt= 1 If qtqt(), () minimizes L(,tqtqtqtqtdt (), (), (), ()) 1200 ∫ 1212 tt= 0 ∂∂LLd ⎛⎞⎟ Then −=⎜ ⎟ 0 , ⎜ ⎟ ∂∂qdtq11qq,,⎝⎠qq 1200 1200 63 Gauge Institute Journal, Volume , No. , H. Vic Dannon ∂∂LLd ⎛⎞⎟ −=⎜ ⎟ 0 ⎜ ⎟ ∂∂qdtq22qq,,⎝⎠qq 1200 1200 By 9.2, 11.2 The First Variation of L(,tq1212 , q , q , q ) ∂∂∂∂LLLL δδδδL =+++qqq112δq2 ∂∂∂∂qqqq112 2 ⎪⎪⎪⎧⎫⎧∂∂LLdd ∂∂ LL⎪⎫ =−+−δδqq⎨⎬⎨⎪⎪⎪⎬⎪. 12⎪⎪⎪⎪ ⎩⎭⎩⎪⎪⎪∂∂qdtq11 ∂∂ qdtq 22⎭⎪ =δδqD()( ∂−∂+ qD ∂−∂)L . {}12qtq11 qtq 22 Therefore, we define 11.3 First Variational Derivatives of L(,tq1212 , q , q , q ) δLL∂∂d L =− , δqqdt11∂∂q1 δLL∂∂d L =− . δqqdtq22∂∂12 ττ==tt 11.4 δ∫∫LL( ττττττ ,qqqqd1212 ( ), ( ), ( ), ( )) = δτττττ( , qqqqd1212 ( ), ( ), ( ), ( )) τ. ττ==tt0 0 64 Gauge Institute Journal, Volume , No. , H. Vic Dannon tt= 1 11.5 If qtqt(), ()minimizes Kqq(12 , )= L ( tqtqtqtqtdt ,1212 ( ), ( ), ( ), ( )) 1200 ∫ tt= 0 δL δL Then = 0 , = 0 δq1 qq, δq2 qq, 1200 1200 δL = 0 , qq, 1200 tt= 1 δ ∫ Ldt = 0 . tt= 0 qq, 1200 11.6 The Second Variation of L(,tq1212 , q , q , q ) 22 2 ⎡⎤⎡ ⎤ δδLL=∂+∂+∂+∂()qq11qq () δ () δ qq 22 qq () δ {}⎣⎦⎢⎥11⎣⎢ 2 2⎦⎥ =∂+∂∂+∂()δδδδqqqq22 2()() ()22 L {}1111qq qq1111 +∂+∂∂+∂()δδδδqqqq22 2()() ()22 L {}2222qq qq2222 65 Gauge Institute Journal, Volume , No. , H. Vic Dannon 12. Infinitesimal Surface Variation Denoting the variation of the surface yx(,12 x ) by δyx(,12 x ), an infinitesimal by οδ , and a smooth enough hyper-real function by η(,xx12 ) we define 12.1 The Infinitesimal Surface variation δοηyx(,12 x )= δ (, x 12 x ). where, 12.2 We use the same οδ for all surface variations. From the definition 8.1, it is clear that 12.3 Surface variation does not depend on dx1 , or dx2 12.4 The surface yx(,12 x )+ δ yx (, 12 x ) is infinitesimally-close to the surface yx(,12 x ). 66 Gauge Institute Journal, Volume , No. , H. Vic Dannon In particular, 12.5 δyx(,12 x ) does not depend on dx1 or dx2 Similarly to 1.6, 12.6 δ is a linear operator Similarly to 1.7, 12.7 d , and δ commute Therefore, 12.8 ∂ , and δ commute. x1 ∂ , and δ commute. x2 ∂∂yy 12.9 If Fx(,12 x ,, y , ) is a hyper-real function on a ∂∂xx12 surface yx(,12 x ) Then, its’ Taylor series to third order about the surface yxx012(, ) is ∂∂yy00∂∂yy Fx(,120 x , y++δδ y , , + δ )= ∂∂∂∂xxxx1122 ∂∂yy00 =+Fx(,120 x , y , , ) ∂∂xx12 67 Gauge Institute Journal, Volume , No. , H. Vic Dannon ⎪⎪⎧⎫ ⎪⎪∂∂FF ∂ F ++⎪⎪δδy ∂∂yy + δ ⎨⎬∂∂yy∂∂xx ⎪⎪∂y ∂∂12 ⎪⎪y0 ⎪⎪∂∂xx12 ⎩⎭⎪⎪yy00 ⎪⎧ ⎪ 22 2 ⎪ ∂∂FF∂∂yy ∂ F ++⎨⎪ ()δδyy22 2 δ +()δ ⎪ 2 ∂∂yy∂∂xx112 ⎪()∂y y ∂∂y ()∂ ⎪ 0 ∂∂xx11 ⎩⎪ yy00 ⎪⎫ ∂∂∂222FFF⎪ ++2()2δδy ∂∂yyδ δ∂y +(δ∂y)2 ⎪ ∂∂yy∂∂xx∂y∂x ∂ y∂x⎬ ∂∂y 21∂ ∂ 2()∂ 2 2⎪ ∂∂xx21∂x2 ∂ x 2⎪ yy00y0⎭⎪ 3 +O()οy . Define ∂∂yy 12.10 the First Variation of Fx(,12 x ,, y , ) ∂∂xx12 ∂∂FF ∂ F δδδFy=+∂∂yy + δ ∂∂yy∂∂xx ∂y ∂∂12 ∂∂xx12 ⎪⎪⎧⎫ ⎪⎪∂∂ ∂ =+⎪⎪δδyF∂∂yy + δ ⎨⎬∂∂xx∂∂yy ⎪⎪∂y 12∂∂ ⎩⎭⎪⎪∂∂xx12 12.11 δδxx12==0 ∂∂yy Proof: Fx(,12 x ,, y , )=⇒ x1 δ F =0., ∂∂xx12 68 Gauge Institute Journal, Volume , No. , H. Vic Dannon Define ∂∂yy 12.12 The Second Variation of Fx(,12 x ,, y , ) ∂∂xx12 ∂∂22FF ∂ 2 F δδ22Fy=+() 2 δ yδ∂∂yy +() δ2+ 2 ∂∂yy∂∂xx112 ()∂y y ∂∂y ()∂ 0 ∂∂xx11 yy00 ∂∂∂222FFF ++2()2δδy ∂∂yyδ δ∂y +(δ∂y)2 ∂∂yy∂∂xx∂y∂x ∂ y∂x ∂∂y 21∂ ∂ 2()∂ 2 2 ∂∂xx21∂x2 ∂ x 2 yy00y0 Just as dx2 = 0, we have 2 12.13 δ yx(,12 x )= 0 ∂∂yy 2 Proof: Fx(,12 x ,, y , )=⇒ yδ F =0., ∂∂xx12 Then, 12.9 becomes ∂∂yy 12.14 If Fx(,12 x ,, y , ) is hyper-real function on a ∂∂xx12 surface yx(,12 x ) Then, its’ Taylor series to third order about the surface yxx012(, ) is 69 Gauge Institute Journal, Volume , No. , H. Vic Dannon ∂∂yy00∂∂yy Fx(,120 x , y++δδ y , , + δ )= ∂∂∂∂xxxx1122 ∂∂yy00 23 =+Fx(,120 x , y , , ) δδFFO++() οy ∂∂xx12 Therefore, ∂∂yy 12.15 IfFx(,12x ,, y , ) is hyper-real function on the surface ∂∂xx12 xbxb2211== ∂∂yy yx(,12 x ) and Jy()= Fx(,,,,)12 x y dxdx12 ∫∫ ∂∂xx12 xaxa2211== Then, Taylor series to third order of Jy(0 + δ y) about the surface yxx012(, ) is xbxb2211== ∂∂yy00 Jy()01+=δ y Fx (,,,,) x2 y0 dxdx12+ ∫∫ ∂∂xx12 xaxa2211== xbxb2211== ∂∂yy00 + δF(, x120 x , y , , ) dx12 dx ∫∫ ∂∂xx12 xaxa2211== xbxb2211== 2 ∂∂yy00 + δ Fx(,120 x , y , , ) dxdx12 ∫∫ ∂∂xx12 xaxa2211== 3 +O()οy . Define 70 Gauge Institute Journal, Volume , No. , H. Vic Dannon xbxb2211== ∂∂yy 12.16 The First Variation of Fx(,12 x ,, y , ) dxdx12 ∫∫ ∂∂xx12 xaxa2211== xbxb2211== xbxb2211== ∂∂yy ∂∂yy δδFx(,12 x ,, y , ) dxdx12= Fx(,12 x ,, y , ) dxdx12 ∫∫ ∂∂xx12 ∫∫ ∂∂xx12 xaxa2211== xaxa2211== Define xbxb2211== ∂∂yy 12.17 The Second Variation of Fx(,12 x ,, y , ) dxdx12 ∫∫ ∂∂xx12 xaxa2211== xbxb2211== xbxb2211== 22∂∂yy ∂∂yy δδF( x12 , x ,, y , ) dx12 dx = F( x12 , x ,, y , ) dx12 dx . ∫∫ ∂∂xx12 ∫∫ ∂∂xx12 xaxa2211== xaxa2211== Then, 12.15 becomes ∂∂yy 12.18 IfFx(,12x ,, y , ) is hyper-real function on the surface ∂∂xx12 xbxb2211== ∂∂yy yx(,12 x ) and Jy()= Fx(,,,,)12 x y dxdx12 ∫∫ ∂∂xx12 xaxa2211== Then, Taylor series to third order of Jy() about yx0() is 23 Jy(0000+=δδδ y ) Jy () + Jy () + Jy () + O ()οy . 71 Gauge Institute Journal, Volume , No. , H. Vic Dannon 13. ∂∂yy Euler’s Equation for Fx(,12 x ,, y , ) ∂∂xx12 and the Variational Derivative At distance x , and at time t , let uxt(,) be the amplitude of a vibrating string, over the x -interval [0,]l , and the time interval [0,T ]. The density of the string at x is ρ()x . At time t , the material point at x has mass ρ()xdx, speed 2 ∂u , and kinetic energy 1 ρ()xdx ∂u . ∂t 2 (∂t ) 2 The potential energy density at x is 1 kx()∂u dx. 2 ()∂x Thus, the Lagrangian density is 22 L(,,xtu ,∂∂uu , )=−11ρ ()xdx∂u kx() ∂u dx. ∂∂xt 22()∂t ()∂x Then, the surface uxt(,)that minimizes the Action integral tTxl== L(,,xtu ,∂∂uu , ) dxdt ∫∫ ∂∂xt tx==00 satisfies Euler’s equation of 9.1,which is the wave equation 72 Gauge Institute Journal, Volume , No. , H. Vic Dannon ∂−∂=()k ∂∂uu ()ρ 0. xt∂∂xt ∂∂yy 13.1 Euler’s Variational Equation for Fx(,12 x ,, y , ) ∂∂xx12 xbxb2211== ∂∂yy If yx0()minimizes Jy()= Fx(,,,,)12 x y dxdx12 ∫∫ ∂∂xx12 xaxa2211== ⎛⎞ ⎛⎞ ⎜⎜⎟⎟ ∂∂∂∂∂FF⎜⎜⎟⎟ F Then −−⎜⎜⎟⎟=0 . ∂∂yx⎜⎜∂∂∂∂yy⎟⎟ ∂ x y0 12⎜⎜⎟⎟ ⎝⎠∂∂xx12 ⎝⎠ yy00 Proof: Since by 12.5, δyx(,12 x ) does not depend on the x1 , or the x2 intervals, we will use infinitesimal intervals [,xdxxdx−+11], ax< [,xdxxdx−+11], ax< Then, yx0() minimizes dI()οδδ =+ dJ ( y0 y )= ξξ=+xdxxdx11 =+ 2222 211 1 ∂∂∂∂yyyy =+Fyy(,ξξ120 , δ ,+ δ ,+ δ )dd ξξ12, ∫∫ ∂∂∂∂ξξξξ1122yy ξξ=−xdxxdx11 =− 00 2222 211 1 over all the surfaces y0 + δy that cross y012(,ξξ ) at the curves through the endpoints of the infinitesimal intervals. That is, δξδyx(,)(−=+11 dx yx dx,)ξ, xd−<<+11xξ xdx 112222 222222222 73 Gauge Institute Journal, Volume , No. , H. Vic Dannon δξy(, x−=11 dx )δξ y (, x + dx ), xdxxd−<<+11ξ x 1222 2 22 2 1111221 The Taylor expansion to second order of ∂∂∂∂yyyy ∂∂ yy Fyy(,ξξ120 ,++ δ , δ , +− δ )Fy (, ξξ120 , , , ), ∂∂∂∂ξξξξ1122 ∂∂ ξξ12 yy00 y0y0 about y012(,ξξ ), is ∂∂FF ∂ F δδyO+++∂∂yy δ()ο2 . ∂∂yy∂∂ξξδ ∂y ∂∂12 y0 ∂∂ξξ12 yy00 Therefore, dJ()() y00+−δ y dJ y = xdxxdx++11⎪⎪⎧⎫ 221122⎪⎪∂∂FF ∂ F =++⎪⎪δδyd∂∂yy δξξd+O()ο2 ∫∫⎨⎬∂∂yy∂∂ξξ12 δ ⎪⎪∂y ∂∂12 xdxxdx−−11⎪⎪y0 2211⎪⎪∂∂ξξ12 22⎩⎭⎪⎪yy00 By 12.8, δ , and ∂ commute. Thus, ξ1 -integrating by parts, ξ1 xdx+ 1 ⎪⎪⎧⎫ 112 ⎪⎪∂∂FF ∂ F ⎪⎪δδyd++∂∂yy δξ= ∫ ⎨⎬∂∂yy∂∂ξξ1 ⎪⎪∂y ∂∂12 xdx−1 ⎪⎪y0 11⎪⎪∂∂ξξ12 2 ⎩⎭⎪⎪yy00 xdx+ 1 1 ⎧⎫⎡⎤112 xdx11+ ⎪⎪⎛⎞ 2 ⎪⎪∂∂∂∂FFF⎢⎥⎜ ∂ F⎟ =−++⎪⎪δδδξδyy∂y d⎢⎥⎜ ⎟ y(,ξξ ) ∫ ⎨⎬∂∂yy∂ξ 11⎢⎥⎜ ∂y⎟ 2 ⎪⎪∂∂y ξ ∂∂2 ⎜ ∂⎟ xdx−1 ⎪⎪y0 1 ⎢⎥⎜ ⎟ 11⎪⎪∂∂ξξ12 ⎝⎠∂ξ1 2 ⎩⎭⎪⎪yy00 ⎣⎢⎥y0⎦xdx−1 112 =0 Interchanging the order of integration, dJ()() y00+−δ y dJ y = 74 Gauge Institute Journal, Volume , No. , H. Vic Dannon xdxxdx++11⎪⎪⎧⎫ 112222⎪⎪∂∂∂∂FFF =−+⎪⎪δδδξyy∂y ddξ+O()ο2 ∫∫⎨⎬∂∂yy∂ξ 21 δ ⎪⎪∂∂y ξ ∂∂2 xdxxdx−−11⎪⎪y0 1 1122⎪⎪∂∂ξξ12 22⎩⎭⎪⎪yy00 By 12.8, δ , and ∂ commute. Thus, ξ2 -integrating by parts, ξ2 xdx+ 1 ⎪⎪⎧⎫ 222 ⎪⎪∂∂∂∂FFF ⎪⎪δδδyy−+∂y dξ= ∫ ⎨⎬∂∂yy∂ξ 2 ⎪⎪∂∂y ξ ∂∂2 xdx−1 ⎪⎪y0 1 22⎪⎪∂∂ξξ12 2 ⎩⎭⎪⎪yy00 xdx+ 1 1 ⎧⎫⎡⎤222 xdx22+ ⎪⎪⎛⎞ 2 ⎪⎪∂∂∂∂∂FF F⎢⎥⎜ ∂ F⎟ =−−⎪⎪δξyd +⎢⎥⎜ ⎟ δξξy(, ) ∫ ⎨⎬∂∂yy21⎢⎥⎜ ∂ y⎟ 2 ⎪⎪∂∂y ξξ∂∂ ∂ ⎜ ∂⎟ xdx−1 ⎪⎪y0 12 ⎢⎥⎜ ⎟ 22⎪⎪∂∂ξξ12⎝⎠ ∂ ξ 2 2 ⎩⎭⎪⎪yy00⎣⎢⎥ y0⎦xdx−1 222 =0 Thus, dJ()() y00+−δ y dJ y = xdxxdx++11⎪⎪⎧⎫ 112222⎪⎪ ⎪⎪∂∂∂∂∂FF F 2 =−−∫∫⎨⎬δξξyd12 d+ O() οδ ⎪⎪∂∂y ξξ∂∂∂∂yy ∂ xdxxdx−−11⎪⎪y0 12 112222⎪⎪∂∂ξξ12 ⎩⎭⎪⎪yy00 The integration sum over the infinitesimal interval yields ⎧⎫ ⎪⎪ ⎪⎪∂∂∂∂∂FF F only the term, ⎨⎬−− δydx12 dx , ⎪⎪∂∂yx∂∂∂∂yy ∂ x ⎪⎪y0 12 ⎪⎪∂∂xx12 ⎩⎭⎪⎪yy00 and we obtain dJ()() y00+−δ y dJ y = 75 Gauge Institute Journal, Volume , No. , H. Vic Dannon ⎧⎫ ⎪⎪ ⎪⎪∂∂∂∂∂FF F 2 =−⎨⎬ − δοydx12 dx+ O()δ ⎪⎪∂∂yx∂∂∂∂yy ∂ x ⎪⎪y0 12 ⎪⎪∂∂xx12 ⎩⎭⎪⎪yy00 Substituting by 12.1, δοηyx(,12 x )= δ (, x 12 x ), dI()οδ −= dI (0) ⎧⎫ ⎪⎪ ⎪⎪∂∂∂∂∂FF F 2 =−⎨⎬ − οηδδ(,x12 x ) dx 12 dx+ O ()ο , ⎪⎪∂∂yx∂∂∂∂yy ∂ x ⎪⎪y0 12 ⎪⎪∂∂xx12 ⎩⎭⎪⎪yy00 ⎧⎫ ⎪⎪ dI()οδ − dI (0) ⎪⎪∂∂∂∂∂FF F =−⎨⎬ − ηο(,x12 x ) dx 12 dx+ O ()δ , ο ⎪⎪∂∂yx∂∂∂∂yy ∂ x δ ⎪⎪y0 12 ⎪⎪∂∂xx12 ⎩⎭⎪⎪yy00 Since this hold for any infinitesimal οδ , dI()ε has a derivative at ε = 0 , which is, ⎧⎫ ⎪⎪ ⎪⎪∂∂∂∂∂FF F dI '(0) =−⎨⎬ − η(,)x12 x dx 12 dx . ⎪⎪∂∂yx∂∂∂∂yy ∂ x ⎪⎪y0 12 ⎪⎪∂∂xx12 ⎩⎭⎪⎪yy00 Since dI(0)= dJ ( y0 ) is minimal, we have by Fermat Theorem, dI '(0)= 0 , and since dx > 0, ⎛⎞⎟ ⎜ ⎟ ⎜ ∂∂∂∂∂FF F⎟ ⎜ −−⎟η(,xx12 )= 0. ⎜ ∂∂yx∂∂∂∂yy ∂ x ⎟ y0 12⎟ ⎜ ∂∂xx12⎟ ⎝⎠yy00 Since η(,xx12 ) is an arbitrary smooth enough function, 76 Gauge Institute Journal, Volume , No. , H. Vic Dannon ∂∂∂∂∂FF F −− =0 ., ∂∂yx∂∂∂∂yy ∂ x y0 12 ∂∂xx12 yy00 From the derivation of Euler’s equation, ∂∂yy 13.2 The First variation of Fx(,12 x ,, y , ) is ∂∂xx12 ∂∂FF ∂ F δδFx(, x ,, y ∂∂yy , )=+y δ∂y +δ∂y 12 ∂∂xx ∂∂yy∂x ∂x 12 ∂y ∂∂1 2 ∂∂xx12 ⎛⎞ ⎜ ∂∂∂∂∂FF F⎟ =−⎜ − ⎟δy ⎜ ∂∂yy⎟ ⎜ ∂∂yx12∂∂ ∂ x ⎟ ⎝⎠∂∂xx12 ⎛⎞ ⎜ ⎟ =∂−∂∂−∂∂δyF⎜ ∂∂yy⎟ ⎜ yx12 x ⎟ ⎝⎠∂∂xx12 Thus, we define ∂∂yy 13.3 The First Variational Derivative of Fx(,12 x ,, y , ) ∂∂xx12 δFF∂∂∂∂∂ F F =− − ∂∂yy δyyx∂∂12∂∂ ∂ x ∂∂xx12 ⎛⎞ ⎜ ⎟ =∂−∂∂⎜ ∂∂yy −∂∂ ⎟F ⎜ yx12 x ⎟ ⎝⎠∂∂xx12 77 Gauge Institute Journal, Volume , No. , H. Vic Dannon ξξ12==xx 2 1 1 ξξ12==xx 2 1 1 ∂∂yy ∂∂yy 13.4 δξξξξδξξF(,12 ,, y , ) dd12= F(,12 ,, y , ) ddξξ12. ∫∫ ∂∂ξξ12 ∫∫ ∂∂ξξ12 ξξ2211==aa ξξ2211==aa 13.5 If yxx012(, )minimizes ξξ12==xx 2 1 1 ∂∂yy Jx(,12 x ,) y = F(,,,ξξ12 y , ) d ξξ12 d ∫∫ ∂∂ξξ12 ξξ2211==aa δF Then = 0, δy y0 δF = 0 , y0 δJ = 0 . y0 ∂∂yy 13.6 The Second Variation of Fx(,12 x ,, y , ) is ∂∂xx12 ⎛⎞2 2 ⎜ ∂∂yy⎟ δδδFy=∂+∂+∂⎜()y ( )∂∂yy δ⎟ F ⎜ ∂∂xx12⎟ ⎝⎠∂∂xx12 78 Gauge Institute Journal, Volume , No. , H. Vic Dannon References [Arthurs] Arthurs, A. M., “Calculus Of Variations”, Routledge, 1975. [Byerly1] Byerly, William, Elwood, “Introduction to the Calculus of Variations”, Harvard, 1933 [Byerly2] Byerly, William, Elwood, “an introduction to the use of Generalized Coordinates in Mechanics and Physics”, Dover, (1916 reprint) [Dan1] Dannon, H. Vic, “Infinitesimals” in Gauge Institute Journal Vol.6 No 4, November 2010; [Dan2] Dannon, H. Vic, “Infinitesimal Calculus” in Gauge Institute Journal Vol.7 No 4, November 2011 [Elsgolts] Elsgolts, L. “Differential Equations and the Calculus of Variations”, MIR, 1970. [Forsyth] Forsyth, A. R., “Calculus of Variations”, Cambridge, 1927. [Fox] Fox, Charles, “An Introduction to the calculus of Variations”, Oxford, 1950. [Hancock] Hancock, Harris, “Lectures on the Calculus of Variations (The Weierstrassian Theory)” University of Cincinnati, 1904. [Pars] Pars, L. A., “An Introduction to the Calculus of Variations”, Heinemann, 1962 [Sagan] Sagan, Hans, “Introduction to the Calculus of Variations”, McGraw- Hill, 1969. [Todhunter] Todhunter, I. “A History of the Calculus of Variations during the nineteenth century” Chelsea, (1861 reprint) [Weinstock] Weinstock, Robert, “Calculus of Variations with Applications to Physics and Engineering” McGraw-Hill, 1952. 79