Conservation laws a simple application to the telegraph equation

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1 J Comput Electron : DOI /s Conservation laws a simple application to the telegraph equation Uwe Norbrock Reinhol Kienzler Publishe online: 1 May 2008 Springer Scienceusiness Meia LLC 2008 Abstract Conservation laws are a recognize tool in physical engineering sciences. The classical proceure to construct conservation laws makes use of Noether s Theorem. It requires the existence of a Lagrange-function for the system uner consieration. Two unknown sets of functions have to be etermine. A broaer class of such laws is obtaine, if Noether s Theorem is applie together the essel-hagen extension, raising the number of sets of unknown functions to three. The same conservation laws can be obtaine by using the Neutral-Action metho the avantage that only one set of unknown functions is require. Moreover, the Neutral-Action metho is also applicable in the absence of a Lagrangian, since for this proceure only the governing ifferential equations are neee. y this, the Neutral-Action metho appears to be the most useful tool in constructing conservation laws for systems issipation. The intention of this paper is to give a reference for further research on this topic rather than showing some etaile calculation on a special subject. Thus, the Neutral-Action metho is introuce in etail is applie to a simple example the telegraph equation to show the usefulness of this proceure for further applications in electronics. Keywors Conservation laws Telegraph equation Neutral-Action metho U. Norbrock R. Kienzler Department of Prouction Engineering, University of remen, IW3-F4, PO ox , remen, Germany uwenor@uni-bremen.e R. Kienzler rkienzler@uni-bremen.e 1 Introuction When investigating mathematical problems, the use of conservation laws is quite common in physical engineering sciences. In electroynamics the Maxwell-equations in integral form can be interprete as conservation laws. Kirchhoff s noal loop laws in electronics are conservation laws for currents voltages, respectively. Recently, a broaer class of these conservation laws, so-calle material conservation laws, were establishe [1 3]. In the application to the fiel of piezoelectricity has been iscusse. Especially, the literature on these conservation laws in electronics is sparse. One reason might be that it is not always possible to fin a physical interpretation of the conservation laws establishe. ut even though, conservation laws form a reliable tool for testing new calculation schemes numerical programming. They are employe in the iscussion of global existence theorems, stability of solutions others [4]. The classical metho to construct conservation laws is base on Noether s Theorem. However, it can only be applie in certain restricte cases. In her classical approach, Noether [5] assume that a Lagrangian function is available for the system of interest. This assumption exclues the application of these methos to systems for which a Lagrangian function oes not exist, ue to, e.g., issipation. y using the Neutral-Action metho [6], this requirement can be isregare, since a given set of governing partial ifferential equations is sufficient to construct conservation laws. It, therefore, follows that a systematic treatment of issipative systems in terms of conservation laws becomes possible. ut even if a Lagrangian function is available, it can be shown that the Neutral-Action metho elivers consierably less effort the same results as Noether s metho in combination essel-hagen s extension [3, 7, 8].

2 48 J Comput Electron : Fig. 1 Definition of conservation laws 2 Definition of conservation laws A mechanical system is consiere that is governe by a system of q ifferential equations β x i,ν, ν 0, β 1, 2,...,q 1 x k x i i, k 1, 2,...,m inepenent variables, ν 1, 2,...,μ epenent variables, in which β enotes an operator acting on the inepenent epenent variables, as well as on their erivatives, representing some ifferential equation. If any set of m associate functions P i, i 1, 2,...,m 2 satisfies P i 0 3 along solutions of 1, then 3 is enote a conservation law P i being the conserve current. Equation 3 can be interprete as a local formulation of a conservation law, because the ivergence of the conserve current occurs. A conservation law may also be written in integral form. Let be a boy infinitesimal volume element V, which is enclose by a surface S area element A unit outwar normal vector n i Fig. 1. y using the ivergence theorem we can write P i V P i n i A 0 4 S leaing to a conservation law in global form. 3 The Neutral-Action metho As mentione above the methoology to establish conservation laws is ifferent epening upon whether the system consiere is Lagrangian or not. The classical way of constructing conservation laws for Lagrangian systems has been establishe by the mathematician Emmy Noether [5] was extene by essel-hagen [7] in It starts from the action integral efine as the integral of the Lagrangian over an arbitrary omain in the space of inepenent variables. Noether s theorem guarantees the existence of a conservation law, if transformations of the epenent inepenent variables exist leaving the action integral invariant. Such transformations are calle variational symmetries. A etaile elaboration on this topic can be foun in [4] [3]. The extension of essel-hagen is base on the fact that the Lagrangian is not unique. It is rather possible to a a term which satisfies the Euler-Lagrange equations ientically, thus leaving the equations of motion unchange. Such an extension is also calle gauge function or Null-Lagrangian. y aing this Null-Lagrangian to the original Lagrange-function, further conservation laws can be erive. For systems out a Lagrangian, no proceure existe for a systematic construction of conservation laws, until the Neutral-Action metho was avance [6]. Most recently, this metho has been applie to the subject of material or configurational mechanics [3] aswellas for ynamics [9] ispersive wave motion [10]. All what is require is the set of ifferential equations governing the system β x i,ν, ν 0. 5 x k First, the concept of the alreay mentione Null Lagrangian will be introuce. In the following, let E enote the Euler-operator cf., e.g., [4] [ E ν 2. 6 x j j1 2 ν x j If a Lagrange function can be written as L F i 7 F i F i x k,ν, i, j, k 1,...,m 8 it follows that L F i E L 0, 9 i.e., this special Lagrange function satisfies the Euler- Lagrange equation ientically. On the other h, if the Euler-Lagrange equation is satisfie ientically, then it follows that the associate Lagrange function is a ivergence.

3 J Comput Electron : The proof is quiet simple. For convenience, we only prove this relationship for the first orer Euler-Lagrange equation. It shoul be mentione that F i may also epen on the erivatives of the epenent variables ν, 2 ν x j,... In this case the Euler-Lagrange equation has to be expe [4]. Consier the Lagrangian L L x i,ν, ν F i x i,ν F i x i 1 The first orer Euler-Lagrange equation is L L ν For the first term in 11 we fin L F i 2 F i x i F i ν β1 while for the secon term it follows L ν [ j1 β1 2 F i ν β ν β 12 ν ν F j F i 2 F i x i F j x j j1 Fi ν β ν β x j x i β1 2 F i ν β ν β. 13 ecause of permutability of secon orer erivatives, equations yiel the same result. Thus the Euler- Lagrange equation is satisfie ientically. L is calle a Null Lagrangian. Setting the variation of the action integral A LV 14 of such a Null Lagrangian to zero, one obtains δa 0 E L 0, 15 where δa enotes the variation of the epenent variables. This means that the action integral A oes not epen on the explicit functional form gx insie the omain of integration but only on the values at the bounary of. So the iea is to seek after characteristic functions f β such that q P i f β β. 16 β1 β1 From 9 15, it follows q m E f β β P i E A LV E L 0 δa 0 17 q f β β V. 18 β1 The characteristics f β have to be etermine from 17. The action integral behaves neutrally uner its variation, so the formalism is calle Neutral-Action metho. 4 Application to the telegraph-equation The metho is illustrate a rather classical example of electronics. The wave equation a amping term is a reasonable mathematical moel for a variety of evolution processes in many areas of physics. One special application in electromagnetism is the so-calle telegraph equation, in which positive amping occurs, which, in turn, correspons to issipation. Telegraph wires can be moele as an electrical circuit, which consists of a resistor of resistance R a coil of inuctance L. Furthermore, it is suppose that current is getting lost from the wire to the groun, either through a resistor of conuctance G or through a capacitor of capacitance C Fig. 2. The electric voltage ux, t at position x time t can be etermine by solving the partial ifferential equation [11] 2 u RGu RC LGu x2 t LC 2 u t 2 19 which is referre to as the telegraph-equation. For the electric current, an analogous equation can be erive. If, for convenience, we set x,

4 50 J Comput Electron : A simple calculation involving equations results in Ef f k 1 f k 2 f k 3 f For any f which satisfies 23, the associate conserve currents can be calculate using 16 q 1: Fig. 2 Schematic of a telegraph wire t use the abbreviations k 1 LC, k 2 RC LG, k 3 RG, we can rewrite 19as u k 1 ü k 2 u k 3 u In the following, we use this representation of the telegraph equation to iscuss the construction of conservation laws in electronics. 5 Calculation of conservation laws Since issipation occurs in the telegraph equation, the classical methos in constructing conservation laws are not applicable. As mentione above, the Neutral-Action metho appears to be the most useful tool in constructing conservation laws for these kin of systems. The conition for the existence of a conservation law for the telegraph equation 20 requires the fulfillment of the equation [ Ef u x 2 t 2 ü u 2 t u x 2 u ] f In 21, the mixe secon erivative of u is roppe, since 20 oes not epen on u. Now, we have to specify the epenence of the characteristic f whichwetaketobe f ft,x. 22 f fu k 1 ü k 2 u k 3 u P t t The conserve currents are P x x. 24 P t k 2 fu k 1 fu f u 25 P x f u fu 26 which, in fact, fulfill P t P x 0. In particular, one such f can be foun by the ansatz ft,x gxht 27 leaing to g K 2 g 0 28 ḧ k 2 k 1 ḣ k 3 K 2 k 1 h 0 29 the solutions for K 2 > 0 gx A cos Kx sin Kx 30 ht Ce λ 1t De λ 2t 31 λ 1,2 k 2 ± k2 2 4K2 k 3 k 1. 2k 1 Here A,, C, D only three beeing linearly inepenent K are arbitrary constants. The corresponing currents P t P x are P t [ k 2 u k 1 u Ce λ1t De λ 2t k 1 Cλ1 e λ1t Dλ 2 e λ 2t u ] [Acos Kx sin Kx] 32 P x [K A sin Kx cos Kxu A cos Kx sin Kxu ][ Ce λ 1t De λ 2t ] 33

5 J Comput Electron : in which u is the solution of 20. Unfortunately an interpretation of this conservation law in electronical terms has not been foun so far. 6 Conclusions The classical proceure of constructing conservation laws is via Noether s theorem. It requires the existence of a Lagrangian for the system uner consieration. Furthermore, this metho ems the knowlege of infinitesimal transformations, which have to be calculate in a separate step. Further conservation laws can be obtaine by using essel- Hagen s extension, since the equations of motion are left unchange when a so calle gauge function is ae to the Lagrangian. This gauge functions have to be etermine aitionally. The same conservation laws as above can be obtaine by using the Neutral-Action metho, where only one set of unknown functions f β have to be calculate. Moreover the Neutral-Action metho can also be applie in the absence of a Lagrangian, since only the governing ifferential equations are require for this proceure. Thus it is possible to calculate conservation laws even for issipative systems. A conservation law for such a issipative system, the telegraph equation, has been erive the Neutral-Action metho. As regars the value un usefulness of conservation balance laws in a general way, reference may be mae to an evaluation of such laws by Olver [4]. It may suffice to mention here the applicability of conservation balance laws in numerics. eing incorporate into various algorithms, the accuracy of the numerical results can be valiate by checking whether or not the conservation laws are satisfie ientically. If the equations are not satisfie, so-calle spurious material noal forces occur in finite-element calculations, which can be use to improve the finite-element mesh by shifting the noes in such a way as to eliminate the spurious forces [12, 13]. It seems that a systematic treatment even more conservation laws can be obtaine for this problem. Stuies along this line are in progress will be ealt in a forth-coming paper. References 1. Maugin, G.A.: Material Inhomogeneities in Elasticity. Chapman & Hall, Lonon Gurtin, M.E.: Configurational Forces as asic Concepts of Continuum Physics. Springer, New York Kienzler, R., Herrmann, G.: Mechanics in Material Space. Springer, erlin Olver, P.J.: Applications of Lie Groups to Differential Equations. Springer, New York Noether, E.: Nachr. Ges. Wiss. Göttingen, Math.-Phys. Kl Honein, T., Chien, N., Herrmann, G.: Phys. Lett. A 155, essel-hagen, E.: Math. Ann Chien, N., Honein, T., Herrmann, G.: Int. J. Solis Struct Norbrock, U., Kienzler, R.: J. Soun Vib. 259, Herrmann, G., Kienzler, R.: Wave Motion Hinsch, H.: Elektronik. Springer, erlin raun, M.: Proc. Est. Aca. Sci. Phys. Math Müller, R., Maugin, G.A.: Comput. Mech