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Infinitesimal Variational Calculus H 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 1 Gauge Institute Journal, Volume , No. , H. Vic Dannon 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; 2 Gauge Institute Journal, Volume , No. , H. Vic Dannon 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 3 Gauge Institute Journal, Volume , No. , H. Vic Dannon 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 4 Gauge Institute Journal, Volume , No. , H. Vic Dannon 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 5 Gauge Institute Journal, Volume , No. , H. Vic Dannon 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. 6 Gauge Institute Journal, Volume , No. , H. Vic Dannon 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 7 Gauge Institute Journal, Volume , No. , H. Vic Dannon 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. 8 Gauge Institute Journal, Volume , No. , H. Vic Dannon 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= 9 Gauge Institute Journal, Volume , No. , H. Vic Dannon 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. 10 Gauge Institute Journal, Volume , No. , H. Vic Dannon 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. 11 Gauge Institute Journal, Volume , No. , H. Vic Dannon 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. 12 Gauge Institute Journal, Volume , No. , H. Vic Dannon 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. 13 Gauge Institute Journal, Volume , No. , H. Vic Dannon 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 .
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