ZAMM 67, T 222 -T 223 (197'7) X. W. BAUER Ein verallgemeinertes S t o k e s -B el t r a m i -System qpw, = w:, A?fN = -w, aL GemaW der Transformation von TAEQENDRE: werden dann generalisierte Impulse pi E F.-eingefuhrt, welche imRahmen der HAmLToNschen Formulierung mit dcr Funktion I1 IE -L(q, p ; t ) + (a@, p ; t ) . p ) auf folgendes k-nonischc System von 2n Differcntialgleichungen fuhren : C P i (i = 1, 2, ... , n) .T 224 Angewandte Analysis und mathematische Physik Dieses System (2) ermoglicht unter Heranziehung weiterer theoretischer Hilfsmittel (HAWLToN-JacoBI-Gleichung, Aussagen iiber LAGRANGE-Klammern, kanonische Transformationen u.s.w.) und Kenntnisse iiber ein H,-System die Anwendung der kanonischen Storungstheorie als optimale Vereinfachung der Methode der Variation der Konstanten. Obgleich nun mit diesem Verfahren in der Himmelsmechanik bedeutende Erfolge erreicht wurden, findet in anderen Disziplinen nur vereinzelt eine Anwendung statt. I n diesem Bericht wird mit ,,TAuLoa-Anfangswerten" als speziellen kanonisch konjugierten Integrationskonstanten ein Zugang zur kanonischen Storungstheorie angegeben, welcher auf beliebige Systeme von expliziten Differentialgleichungen 1. Ordnung anwendbar ist. Ein ausfiihrlicherer Bericht insbesondere betreffs Anwendungen im Maximumprinzip [ 31 ist vorgesehen. 2. Kanonisehe Storungstheorie im Maximumprinzip Bei Optimierungsaufgaben in der Variationsrechnung [3] treten Probleme in der Form J f o ( x , u ; t ) dt --+ Extr. T 226 Angewandte Analysis und mathematische Physik 4. Picard -Iteration und genHherte Liisung Im Rahmen einer PICARD-Iteration zum Zusatzsystem (17 ; G Go, F , + Fx,o) fuhrt der generell vorgeschriebene Anfangswert G,,(O) = E als idealer Iterationsansatz zum Matrizanten von -FX,o als Losung (vergl. [5], 8. 443-4):G",+I = -Fx,oG:t t 71 t 71 1 s Go(x,u;t) = E -J S x , O d z~+ J I p , , O J F x , o d t z d t i -SFx,oS Fx,oSk',,odgdtzdti +...-fi:(--Fx,~), 0 0 0 0 0 0 worin Q:( -Fx, o) nur bei einem in x nichtlinearen Ho-System von x abhangig ist. Die Integraldarstellung zurn a-System (25) nimmt damit die zweckmiiflige Form t t a@) = x(0) + J Of dz + J
In 1895 Macdonald [lo] gave a representation of the first and second Green's functions G, and G, of the potential equation for a wedge of an arbitrary angle a. As is well known, the first Green's function G, is equivalent with the electrostatic potential in a point P induced by an electric point charge located a t Q (in the three dimensional case) or a uniformly charged (infinitely long) line source parallel to the edge of the wedge passing through Q (in the two dimensional case), provided the walls of the wedge are of infinite conductivity (see Fig. 1).Similarly, the second Green's function G, represents the velocity potential of the motion of liquid in a point P due to a point or a line source located a t Q for rigid walls of the wedge. It will be remembered that the success of the familiar process of taking images, often used in electrostatics, depends on the fact that the angle CY of the wedge must be of the form ?rim where m is a positive integer. Macdonald's solution for any angle of the wedge is given in the form of a definite integral and the reduction of this integral to known forms is effected in certain cases. Later Macdonald [12] mas able to obtain expressions for the two Green's functions G, and G, of the wave equation Au + k'u, = 0 for a wedge of any angle a. Here a solution in series is first deduced from the corresponding solutions in series for the potential equation and then transformed into an integral representation. This is equivalent with the solution of the following diffraction problems. (The special case of the half-plane, i.e. the case of a wedge with a = 2n, was previously given by Carslaw 121.)(1) Sound waves from a point source or from a line source with axis parallel to the rigid walls of the wedge (Green's function G, for the three or two dimensional case respectively).(2) Electromagnetic waves from either a Hertz oscillator or line current with axis parallel to the edge of the wedge for infinite conductivity of the walls (Green's function G, for the three or two dimensional case).The case of an inci'dent plane wave is obtained when the point Q (source) tends to infinity. The analysis of Macdonald's method is rather involved and uses the conventional methods of nineteenth century mathematical physics exclusively.In 1896 Sommerfeld [16,17] introduced his theory of the multivalued sohtions of the wave equation and showed how by this method the two Green's functions of the wave equation can be obtained for a wedge of an angle of the
No abstract
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.
hi@scite.ai
334 Leonard St
Brooklyn, NY 11211
Copyright © 2024 scite LLC. All rights reserved.
Made with 💙 for researchers
Part of the Research Solutions Family.