On phase, action and canonical conservation laws in kinematic-wave theory

Canonical equations of energy and momentum are constructed in the kinematic-wave theory of waves in a continuum. This is done in analogy with what is achieved in nonlinear continuum mechanics. The starting point is a generalized balance of wave action. The standard formulas are recovered when the...

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Дата:	2008
Автор:	Maugin, Gerard A.
Формат:	Стаття
Мова:	English
Опубліковано:	Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України 2008
Назва видання:	Физика низких температур
Теми:	Динамика кристаллической решетки
Онлайн доступ:	http://dspace.nbuv.gov.ua/handle/123456789/117345
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Назва журналу:	Digital Library of Periodicals of National Academy of Sciences of Ukraine
Цитувати:	On phase, action and canonical conservation laws in kinematic-wave theory / Gerard A. Maugin // Физика низких температур. — 2008. — Т. 34, № 7. — С. 721–724. — Бібліогр.: 20 назв. — англ.

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Digital Library of Periodicals of National Academy of Sciences of Ukraine

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spelling	irk-123456789-1173452017-05-23T03:02:48Z On phase, action and canonical conservation laws in kinematic-wave theory Maugin, Gerard A. Динамика кристаллической решетки Canonical equations of energy and momentum are constructed in the kinematic-wave theory of waves in a continuum. This is done in analogy with what is achieved in nonlinear continuum mechanics. The starting point is a generalized balance of wave action. The standard formulas are recovered when the system follows from the averaged-Lagrangian variational formulation of Whitham. 2008 Article On phase, action and canonical conservation laws in kinematic-wave theory / Gerard A. Maugin // Физика низких температур. — 2008. — Т. 34, № 7. — С. 721–724. — Бібліогр.: 20 назв. — англ. 0132-6414 PACS: 75.40.Gb;05.45.–a http://dspace.nbuv.gov.ua/handle/123456789/117345 en Физика низких температур Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection	DSpace DC
language	English
topic	Динамика кристаллической решетки Динамика кристаллической решетки
spellingShingle	Динамика кристаллической решетки Динамика кристаллической решетки Maugin, Gerard A. On phase, action and canonical conservation laws in kinematic-wave theory Физика низких температур
description	Canonical equations of energy and momentum are constructed in the kinematic-wave theory of waves in a continuum. This is done in analogy with what is achieved in nonlinear continuum mechanics. The starting point is a generalized balance of wave action. The standard formulas are recovered when the system follows from the averaged-Lagrangian variational formulation of Whitham.
format	Article
author	Maugin, Gerard A.
author_facet	Maugin, Gerard A.
author_sort	Maugin, Gerard A.
title	On phase, action and canonical conservation laws in kinematic-wave theory
title_short	On phase, action and canonical conservation laws in kinematic-wave theory
title_full	On phase, action and canonical conservation laws in kinematic-wave theory
title_fullStr	On phase, action and canonical conservation laws in kinematic-wave theory
title_full_unstemmed	On phase, action and canonical conservation laws in kinematic-wave theory
title_sort	on phase, action and canonical conservation laws in kinematic-wave theory
publisher	Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України
publishDate	2008
topic_facet	Динамика кристаллической решетки
url	http://dspace.nbuv.gov.ua/handle/123456789/117345
citation_txt	On phase, action and canonical conservation laws in kinematic-wave theory / Gerard A. Maugin // Физика низких температур. — 2008. — Т. 34, № 7. — С. 721–724. — Бібліогр.: 20 назв. — англ.
series	Физика низких температур
work_keys_str_mv	AT maugingerarda onphaseactionandcanonicalconservationlawsinkinematicwavetheory
first_indexed	2025-07-08T12:04:11Z
last_indexed	2025-07-08T12:04:11Z
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fulltext	Fizika Nizkikh Temperatur, 2008, v. 34, No. 7, p. 721–724 On phase, action and canonical conservation laws in kinematic-wave theory Gerard A. Maugin Université Pierre et Marie Curie (Paris-6), Institut Jean Le Rond d’Alembert UMR 7190 CNRS, Case 162, 4 place Jussieu, 75252 Paris Cedex 05, France E-mail : gam@ccr.jussieu.fr Received November 26, 2007 Canonical equations of energy and momentum are constructed in the kinematic-wave theory of waves in a continuum. This is done in analogy with what is achieved in nonlinear continuum mechanics. The starting point is a generalized balance of wave action. The standard formulas are recovered when the system follows from the averaged-Lagrangian variational formulation of Whitham. PACS: 75.40.Gb Dynamic properties (dynamic susceptibility, spin waves, spin diffusion, dynamic scaling, etc.); 05.45.–a Nonlinear dynamics and chaos. Keywords: waves, phase, action, momentum, conservation laws. 1. Introduction During the last two decades many works following a line opened by J.D. Eshelby have set forth the essential role played by conservation laws issued from the applica- tion of Noether’s theorem in modern continuum mechan- ics (see, e.g., Maugin [1,2]), especially in the study of the evolution of singularity sets in crystal elasticity (disloca- tions, cracks, phase-transition fronts) where components of the divergence of the so-called energy-momentum ten- sor intervening in the conservation (or non strict conser- vation) of canonical momentum provide the driving force on these «objects». Until recently, it was thought that this could be formulated only for conservative systems deriv- able from a Lagrangian (hence the application of Noether’s theorem). But in recent works the author has shown how the canonical equations of momentum and en- ergy could be constructed systematically even in a ge- neral dissipative framework exhibiting both intrinsic dis- sipation and heat conduction (e.g., Maugin [3,4]). Remember that these two equations are related to the sys- tem’s invariance (or lack of invariance) with respect to space-time translations, space being parametrized by the material coordinates X of finite-strain continuum me- chanics and time t is Newton’s time (to remain in a non relativistic framework). Of course one is tempted to examine what happens with the wavelike quantities usually associated in lin- ear-wave mechanics, i.e., material wave vector K and frequency � since these combine with X and t in a duality exhibited by the usual phase function � �� K X t. The use of generally defined wave number and frequency by Lighthill [5], Whitham [6] and others led to the concept of kinematic-wave theory which in turn has provided astute ways to deal with certain problems of nonlinear wave propagation (in particular, for weakly nonlinear and dis- persive systems yielding nonlinear Schr�dinger equa- tions and the notions of bright and dark solitons) in the expert hands of authors such as Benney [7] and Newell [8]. We have produced a rare but aesthetically pleasing il- lustration in the case of surface solitons [9]. Kinema- tic-wave theory is discussed and illustrated in books (Whitham [10], Maugin [11], Ostrovsky and Potapov [12]). In recent works, we naturally pondered the transcrip- tion of Eshelby’s ideas in this general nonlinear-wave framework. At first we again applied this only to the con- servative case, along with a due application of Noether’s theorem (Maugin [13,14]). Going one step further in the present paper, we give the expression of canonical equa- tions for wave-kinematics for general systems since waves also propagate in inhomogeneous dissipative con- tinua including even source terms. Accordingly, a rarely considered notion in continuum mechanics, that of ac- tion, is introduced and plays a fundamental role. © Gerard A. Maugin, 2008 2. Generalized phase function A general smooth motion of a continuum in Euclid- ean-Newtonian space-time is represented by the smooth vector-valued function x x X� ( , )t , where x stands for the actual position and X represents the set of material coor- dinates on the material manifold [1]. The phase of a plane linear wave in a continuum is defined in this material de- scription by � � � �( , ) ~( , )X K K Xt t� � � � , (1) where K is the material wave vector and � is the associ- ated circular frequency. But in the kinematic-wave theory a general phase function � �� ( , )X t (2) is introduced from which the material wave vector K and the frequency � are defined by K X � � � � � � �R , � � � � � �t . (3) Whence there follows at once the two equations (curl-free nature of K, and conservation of wave vector) � � �R K 0, (4) � � � � K t R� 0 . (5) In particular, (3) are trivially satisfied for plane wave solutions for which the last of (1) holds true. For an inhomogeneous rheonomic linear behavior with disper- sion we have the dispersion relation � � ( ; , )K X t . (6) Accordingly, the conservation of wave vector (5) becomes � � �� K V K Xt g R expl , V K g � � � , (7) and thus the Hamiltonian system [15] D Dt X K � � � , D Dt K X � � � � expl , (8) where we have set D Dt t g R� � � ��V . (9) Simultaneously, we have the Hamilton-Jacobi equation (compare eq. (3)) � � � � � � � � � � � � � t t X K X , ; 0 . (10) If we now consider a wave in an inhomogeneous rheo- nomic dispersive nonlinear material, the frequency will also depend on the amplitude. Let a the n-vector of R n R n that characterizes this small slowly varying amplitude of a complex system (in general with several degrees of freedom). Thus, now, � � ( ; , , )K X at . (11) Accordingly, the second of Hamilton’s equations (7) will now read [15] D Dt TK X A a� � � � � � expl R( ) , A a : � � � � . (12) In the studies of Newell [8] and Maugin and Hadouaj [9], one is even led to considering a nonlinear «disper- sive» dispersion relation in which the assumed slowly varying quantities such as space and time derivative of the amplitude are involved in the function , which rela- tion becomes a true «wave equation» itself for the ampli- tude. 3. Generalized action function In full similarity with (1), the scalar quantity called the action density per unit reference volume is classically defined by S t S H Ht( , ) ~ ( , )X P P X� � � � , (13) where P is a material momentum and H is an energy (Hamiltonian). It is on the basis of this and the invariance of S that L. de Broglie deduced his celebrated relation P K� � , if Planck’s relation H � � � applies, so that S � �� in this essentially linear theory (here � is Planck’s reduced unit of action). Forgetting about the expression given in (13), consider a general smooth function S S t� ( , )X and define general material momentum and energy by (com- pare to (3)) P � � R S , H S t X � � � � , (14) where � R is the material gradient and d dt t X / : /� � � is the material time derivative. Obviously then, � � �R P 0 , d dt HR P � �� . (15) From the second of these it follows that if the first is valid initially, it remains valid in time. The case (13) satisfies (14) trivially. In standard analytical mechanics, the first of (14) is none other than the Jacobi equation of motion; it would be de Broglie’s «guidance» equation in the causal interpretation of quantum mechanics; see post scriptum below. In nonlinear (of course conservative) dynamic inho- mogeneous (but scleronomic = no explicit time depend- ence) elasticity we know the expression of H, for instance as a sufficiently regular function 722 Fizika Nizkikh Temperatur, 2008, v. 34, No. 7 Gerard A. Maugin H H N� �( , , , )P X X , (16) where � � � X F 1 is the spatial gradient of the «inverse motion» [1,2], and N is the entropy density. As a matter of fact, we more precisely have H E N� � � � �1 2 0 1 1 � P C P X F( , , ) , (17) where �0 is the reference (possibly X dependent) matter density, E is the internal energy per reference volume, P C V� ��0 , C F F� �T , and V F v� � ��1 , with v x� � �/ t, F x� � R . This is not the standard formulation of elastic- ity, but the one using the so-called «inverse motion». We let the reader do the tricky exercise that the second of (15) then yields the canonical equations of energy and momen- tum in the form (cf. [1]): d N d t � 0 , d dt R P b f� div inh , (18) wherein b 1 T F� � �( )L , L H K E� � � � �P V , T F� � �E / , f X inh expl � � � L , (19) where the explicit gradient in the last quantity extracts the X explicit dependence of the «Lagrangian»L in the case of material inhomogeneity via the density and the internal energy. Tensor b is called the material Eshelby stress ten- sor. Had we started from a Lagrangian variational princi- ple, equations (18) would have followed in this very form after application of Noether’s theorem under time and (material) space translations. Works [3,4] have shown how to establish the generalizations of (18) for a general continuum in the presence of dissipation and all types of additional effects. Remark. The action is seldom considered as a basic quantity in continuum mechanics. However, a conserva- tion-like equation is obtained for this action — as defined in (13) — for a group of simultaneous space and time transformations (expansions or scaling) as shown by the author [1] (Chapter 4) and also Lazar [16]. 4. Canonical energy equation for wave mechanics Imagine that we have to start with a local scalar bal- ance law of the form d S dt ER d�� W . (20) Later on S will be identified with an action, and W as an action flux, while Ed is an external energy input. For the time being we formally multiply both sides of (20) by � , a circular frequency, and subtract from both sides of the re- sulting equation the quantity dL dt ~ / , where the scalar quantity ~ L will be specified later. After some manipula- tion we obtain thus the formal equation d dt S L h E S t d L dt R d R( ~ ) ( ) : ~ .� � � � �� W W (21) Because of the meaning granted to Ed this is an equation of energy balance. In particular, if we were working in the kinematic-wave theory of Lighthill and Whitham, ~ L would be the so-called averaged (over the phase) Lag- rangian such that [10,13,14] S L � � � ~ � , W K � � � ~ L , ~ ~ ( , , , )L L t� K X� , (22) for an inhomogeneous rheonomic system. Then (21) yields the energy balance as d H dt h h hR t ~ ~ ~ :�� Q ext , (23) wherein energy density, effective heat flux and internal and external energy sources are given by ~ ~ H S L� �� , ~ Q W� � , h L t t � � � � ~ expl , h Ed ext � �. (24) In the absence of external heat source, (23) follows from the variational formulation of Whitham of the averaged Lagrangian after application of Noether’s theorem for time translations. 5. Canonical momentum equations for wave mechanics We proceed just like in the previous paragraph but multiply both sides of the a priori set equation (20) by a material (co-)vector which is none other than a wave vec- tor K and add to both sides of the resulting vectorial equa- tion the material co-vector � RL ~ . After some manipula- tions we arrive at the following co-vectorial balance equation d dt E S LR d R R R ~ ~ : ~P b f K W K� � � � � � �� div � , (25) where we have set ~ P K� S , ~ ( ~ )b 1 W K� � � �L . (26) If S is the action, then the first of the last two equations defines the material wave momentum and is a continuum generalization of de Broglie’s relation. The mixed mate- rial tensor defined in (26) we called the material wave Eshelby tensor. If we are within the framework of the Lighthill-Whitham theory of the averaged Lagrangian ~ L , then equations (22) apply and computing the material gra- dient of ~ L and substituting in the right-hand side of (25) On phase, action and canonical conservation laws in kinematic-wave theory Fizika Nizkikh Temperatur, 2008, v. 34, No. 7 723 we arrive at the balance of material wave momentum in the form d dt R ~ ~P b f f� � div inh ext , (27) with «material forces» of inhomogeneity and external ori- gin given by f X inh expl � � � ~ L , f K ext � Ed . (28) In the absence of external heat source, (27) follows from the variational formulation of Whitham of the averaged Lagrangian after application of Noether’s theorem for material space translations. 6. Amplitude dependence If we are in the strict framework of the averaged Lagrangian variational method, even supposing from the beginning that the averaged Lagrangian depends explic- itly on the amplitude a of the studied wave process (as supposed in the generalized dispersion relation (11)), it is shown that one of Euler-Lagrange in fact results in the equation of amplitude independence 0 � � � ~ /L a . (29) unless, of course, there exists an external source f a bal- ancing this. In this case equations (23) and (27) will con- tain additional terms in their right hand side, given by h a a f a� � � , f f a a aext. ( )� � � R T . (30) 7. Conclusion Of course there are essential differences between the elasticity case recalled at the end of Section 3 and the wave case of Section 4. In the elasticity case, the depar- ture point in the absence of variational formulation is the local balance of physical linear momentum, i.e., the basic balance law of continuum mechanics written in the actual configuration and involving the Cauchy stress or the first Piola-Kirchhoff stress T, along with a statement of the first law of thermodynamics [4]; and all agree on these. Furthermore, the nonlinear case and the case involving characteristic internal length scales (hence a weak nonlocality), such as in so-called gradient elasticity, are automatically included [1,16] and prove extremely useful for studying directly nonlinear waves in crystals [11,17]. In the kinematic-wave theory briefly discussed in this pa- per, equation (21) is hardly conceived as a basic equation, although this is not forbidden but a little farfetched and we do not clearly see what nonlinearity and nonlocality mean. For sure, however, the momentum equation (25) should yield the time evolution of Brenig’s wave momen- tum [18] since, for instance, ~ P Ka� � 2(up to a factor of modulus one) for a one-dimensional harmonic wave-like motion (see equation (6.3) in [14]) P.S. Some of the above- made considerations may be close to those found in the causal re-interpretation of quantum mechanics; see Jammer [19] Sections 2.5 and 2.6; Holland [20] Chapter 2. 1. G.A. Maugin, Material Inhomogeneities in Elasticity, Chap- man and Hall, London (1993). 2. G.A. Maugin, Trans. ASME Appl. Mech. Rev. 48, 213 (1995). 3. G.A. Maugin, Mech. Res. Commun. 33, 705 (2006). 4. G.A. Maugin, Arch. Appl. Mech. 75, 723 (2006). 5. J.M. Lighthill, J. Inst. Maths. Applics. 1, 269 (1965). 6. G.B. Whitham, J. Fluid Mech. 22, 273 (1965). 7. D.J. Benney and A.C. Newell, J. Math. and Phys. 46, 133 (1967). 8. A.C. Newell, Solitons in Mathematics and Physics, S.I.A.M., Philadelphia (1985). 9. G.A. Maugin and H. Hadouaj, Phys. Rev. B44, 1266 (1992). 10. G.B. Whitham, Linear and Nonlinear Waves, Interscien- ce-John Wiley, New York (1974). 11. G.A. Maugin, Nonlinear Waves in Elastic Crystals, Oxford University Press, Oxford (1999). 12. L.A. Ostrovsky and A.I. Potapov, Modulated Waves, Johns Hopkins University Press, Baltimore (1999). 13. G.A. Maugin, Wave Motion 44, 472 (2007). 14. G.A. Maugin, J. Phys. Conf. Series 62, 72 (2007). 15. G.A. Maugin, Proc. Est. Acad. Sci. Phys. Math. 52/1, 5 (2003). 16. M. Lazar and C. Anastassiadis, J. Elasticity 88, 5 (2007). 17. A.M. Kosevich, The Crystal Lattice : Phonons, Solitons, Dislocations, Berlin-Wiley-VCH (1999). 18. W. Brenig, Zeit. Phys. 143, 168 (1955). 19. M. Jammer, The Philosophy of Quantum Mechanics, J. Wiley-Science, New York (1974). 20. P.R. Holland, The Quantum Theory of Motion, Cambridge University Press, Cambridge, UK (1993). 724 Fizika Nizkikh Temperatur, 2008, v. 34, No. 7 Gerard A. Maugin

On phase, action and canonical conservation laws in kinematic-wave theory

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