PART I PART II VARIATIONAL PRINCIPLES IN CONTINUUM .

-9Equation (8)(7)governs breaking for the dispersive equation ( 1)does for the hyperbolic equation (6).in a similar way the-am:If steepening is to occurpterm must dominatepare therefore needed for this study and, sinceinT]the situation looks hopeful.p(x,theorem [ 7],wheresatisfym (t)s1sl 2and:sx,t)p(x,Bounds ont)is linearUsing the second mean valuecan be written as x.A linear bound forp(x,and the maximum value of the positive slopeIp( x, t) I:Sthis inequality holds for all x.(7)(t).are two numbers which depend on xSzasK( 0)(m ( t)t)andtandin terms ofm (t) is then- m ( t)) ;( 9)The essential difference betweenand (8) is the presence of m (t)in the bound forp .Thefunction m (t) enters in such a way that it makes the right handside of (8) more positive and therefore deters breaking.estimate forAnm ( t) is found by considering its equation,( 1 0)7.See, for example, E. W. Hobson, The Theory of Functions of aReal Variable, Vol. 1, p. 618. New York: Dover {1927).

-10-wherep (t) p(X (t), t) and X (t)satisfies"lxx(X (t), t) 0.From the estimate in (9), the derivatives of m (t) and m (t)can be bounded by dm dt- um2. K( 0) { m - m),( 11)2. -um ( 1 2.) K(O) (m - m ) .ThenIf m m - 2.K(O)/u initially, then m m decreasesfurther and 2.K(O)for all tuUsing this estimate fordm crt- um2. 0.(13)m in (11), we have- 2.K(O)mfrom whichfollows.Let q(t)( 13) is satisfied. - (m (t) ThereforeK O))dq/ dt and note that quq2.and 0 when

-11-lqwhereq q0 0 at tq;::l -: o.la.t qo-Finally, for-uq t0qwe havelro as t - a.qoWe conclude that if the initial wave profile is sufficientlyasymmetrical so that m (O) m (O) will become infinite in a time less than- 2K(O)/a., the slope m (t)ja.m (O) K{O)r1 Thisis a sufficient condition only, not a sharp criterion.For the kernelK {x) a proof that solutions of ( l)gbreak into bores is not yet available.canOne would like to carrythrough similar arguments, but the extension of the proof is difficult;the singularity in K {x) prevents the a priori estimategintegralK (x)gpin terms of slopes.it seems likely thatp-(9)of theHowever, even with the infinity inis dominated by the term2-am in{8) and can not prevent breaking of sufficiently asymmetric waves.A partial indication of this is that if a typical breaking profile issubstituted into the integral it can be shown [ 8]p 8.7 4o{jm j / ) asSee Appendix B.lm l- ro.that

-12-PART IIVARIATIONAL PRINCIPLES IN CONTINUUM MECHANICS4.IntroductionThe variational formulation of classical mechanics providesan elegant setting in which the deep ideas of dynamics, such ascanonical transformations and adiabatic invariants, can be conciselyexpressed.Hopefully, a variational formulation of continuummechanics would be equally as fruitful.But the slow progress 1nthis direction indicates that new problems arise.When Hamilton 1 sprinciple is applied to the particles of a continuum, a variationalformulation does result which correctly gove rns motion of thecontinuum as described by Lagrangian (or particle) coordinates.But in many continuum problems, for example in fluid mechanics,it is preferable to study the equations which correspond to Eulerian(or field) coordinates.To obtain these equations . as they stand asthe Euler equations of some variational principle is generallyimpossible.The reason is that in their original form the continuumequations for field quantities generally do not satisfy a selfadjointnes s condition.This condition is an extension of the usualself-adjointness property to nonlinear operators and wasestablished as a necessary and sufficient condition for Eulerequations by Vainberg [ 9].9.This is why existing variationalM. M. Vainberg, Variational Methods for the StudT of NonlinearOperators.Transl. by A. Fe ins tein,Holden- Day 1964).

-13principles usually appear in term s of auxiliary quantities (e. g ., thevelocity potential and stream function in fluids or the vector andscaler potentials in electromagne tism) whose equations are selfadjoint .Equations which are not self-adjoint can only come indirectlyfrom variational principles .formulations are possible.Several different approaches to indirectA formulation in which the Lagrangianfor any equation of the formMu( x ) is immediately known , 1s to vary u( 14)0in the functional(15)Sinc eJ( u]takes on its minimum value zero for solutions of ( 14),its Euler equation must be satisfied by solutions of ( 14).th e actual Euler equation forequation ( 14).J( u]However,1s more complicated thanFor a linear operatorLthe functionalJ( u]b e comes the inner product (Lu, Lu) whose Euler equation isL .' Lu 0,whereL* istheoperatoradjointtoof the heat equation,the functionalJ( u)isff uxx - ut) 2 dxdtand the corresponding Euler equation isL.Inthecase

-14--c: :,)(a: :, ) 0 .uThus, in this approach the functional to be varied is easilyformulated and the difficulty is that information about the originalequation is hidden in a more complicated Euler equation.Another indirect variational technique is to vary only partof the Lagrangian.To illustrate this approach we consider theequation"V (A.( T) "V T) (16)0which arises in problems of steady-state heat conduction in asolid.In such problemsdistributionT (x)o-(16) is to be solved for the temperaturein a body.The kind of variational principlethat we have in mind now is to find a functionalJ' 0pending on the two functionsandT,ci [ T , T], deofor which the condition( 17)leads to equation (16).varied in ci ( T , T)0whileThe meaning of (17)T0Tisis treated as a known function.Then, in the resulting Euler equation,functional which leads tois thatTis set equal toT One0( 16) in this way is( 18)

-15-Awhere0 A.(T }.0which becomes(16)also leads to (16}VaryingwithTTin (18} T.0givesAnother functional whichby this method is given by the integralfA0\7 T0· T dx-.( 19)The above example is taken from a paper by Glansdorff andPrigogine[ 10]the functionalwhere a theory is presented which arrives at r (T , T)0given in (18).Indeed some basis forchoosing the most advantageous functional for a given equation wouldbe desirable.In this kind of variational technique the equations ob-tained are precisely the ones of interest but the functionals useddependon the solutions of the problems.The approach to be considered here for finding indirectvariational principles is to introduce new dependent variables bymeans of "potential" representations which result in self-adjointequations.Variational principles in terms of the potentials arethen found which have Euler equations equivalent to the originalnon-self-adjoint set.The ideas of adiabatic invariants andcanonical transformations do occur in continuum problems forwhich potential type variational formulations are used.10.In hisP. G1ansdorff and I. Prigogine, Physica, Vol. 30 ( 1964),pp. 351-374.

-16averaging method for nonlinear dispersive equations, Whitham [ ll]finds that a prohibitive amount of algebra is avoided if the wholetheory is obtained from a variational formalism.The variationalprinciples he uses are of this potential repr esentation type and theideas of his theory then follow the lines of adiabatic invariants inclassical La grangian - Hamiltonian mechanics .Clebsch [ 12] givesthe first variational formulation of rotational fluid flow byintroducing a velocity representation of the formUnder this transformation the new equations of motion can be putin a canonical form which resembles the classical Hamiltonequations [ 13].The difficult steps in the potential representationmethod are to find a suitable representation and to find aLagrangian which leads to the appropriate equations for thepotentials .required.A very general (one to one) representation is not alwaysFor special problems such as irrotational flow , thelimited velocity potential representation u \Tpis adequate andthe work in fi nding t he variational formulation is reduced.There11.G. B . Whitham, Proc. Roy. Soc. A, Vol. 283 ( 1965), pp . 238261. J. Fluid Mech ., Vol. 22 (1965), pp. 273 - 283.12.A. Clebsch, J. reine u. angew. math ., Vol. 56 (1859), pp. 1-10.13.See , for example , H . Lamb, Hydrodynamics , 6 ed ., p. 249,New York: Dover (19 32 ).

-17are, however, interesting examples for which more generalpotential representations are required.This is true for Rossbywave s and internal wave s which are examples of nonlineardispersive waves that can only exist in rotational fluid flows.Themodel equations used to study these waves are approximate formsof general equations for an inviscid fluid.Variational formulationsfor these examples are found (Section 4, 6) by first studying thevariational formulation for a general inviscid fluid (Section 3) andthen suitab ly modifying the potential repre s entation andLagran g ian .The disc us sion begins {Section 2) with a brief account ofVainber g 1 s necessary and sufficient self-adjointnes s condition forEuler equations and its application to differential equa t ions.

-185.The Self-Adjointnes s Condition of VainbergNecessary and sufficient conditions for a class of nonlinearequations to be the Euler equations of a variational principle havebeen given by Vainberg [ 14].Although Vainberg's work deals mostlywith nonlinear integral operators, his results are easily applied tothe partial differential equations of physics with which we areconcerned.This theory will now be discussed in terms of a singleordinary differential equation of the formF(u , u , uXwhereFXX, x) ( 20)0 ,1s a polynomial function of the dependent variablefirst and second derivatives, and the independent variablex.u, itsSucha limited model example can be used because extension of the ideasto systems of nonlinear partial differential equations is straightforward.Vainberg found that the conditions for(20)to result fromthe variation of some functional are analogous to the conditions fora vector function to be the gradient of some scalar function.It iswell known that in three dimensions a vector can be expressed asthe gradient of a scalar if its curl is zero or, by Stoke's theorem,if the line in te gral of the vector around any closed curve is zero.To generalize the ''curl'' condition the notion of the derivative of anoperator is needed.Here it is useful to definelim J--014 .Loc. cit.N (u Jh)(}- N(u)DN(u;h)

-19as the derivative of the operatorFor operators such asjust the operatorFFNat the point u , in the direction h.in ( 20 }, onelinearized about urecognizes thatoperating on h .analogy to the "curl 11 condition is that for an operatorfrom varying some functional,FFDF(u;h} 1sTheto comemust satisfy the symmetrycondition(21)at each point uand for all hare usual integrals overx.1and h .2The inner products in (21}Condition ( 21)means that when Fislinearized about any point the resulting operator is self-adjoint.linear operatorsF Lu,DF(u;h}usua l definition of self-adjointness. Lh, and (21}Forreduces to theVainberg also defines theconcept of a curvelinear integral in function space for an operatorand shows that the independence of this integral on path determineswhen the operator is an Euler equation.The definitions of thesefunction space concepts will not be needed here.motivation that if the operatorFWe just note forhas the path independence property,its integral along any curve in function space, between the points0and u, say, must equal the straight line integral in function spacebe tween0and u.But this straight line integral can be writtenexplicitly as'I0l(F(A.u), u} dA. J [ u] ,( 2 2)

-20where again the inner product is an integral overfunctionalJ[ uJ1n (22)Lagrangian forx.Thus theshould be an excellent candidate for aF.(21)It will now be shown thatsufficient condition for(20}is indeed a necessary andto come from a variational principle,and that the Lagrangian is given in (22}.the self-adjointnes s condition ( 21 }.Suppose thatComputing6J[ uJFsatisfiesfrom( 22}gives1 x2 OJ[ u]If0 xk( -u, kux' kuxx' x)Ou1By the sel -adjointness of Fit follows thatx2x2f kuDF(ku;Ou) ( dxdkku DF(ku; Ou)dx JMu DF( -u, u)dxxlxlF(Au, AU , AuXXX, x} 6u dx;henceiJ. l x20 J[ u] 0 x1lF(ku, kux' kuxx' x). -.& F (ku, kux' kuxx' x)i Oudxd -

-21-Assuming that the variationsou vanish at xand x , interchange21the order of integration and integrate the second term by parts withrespect to .to obtainx2 oJ( u]J.F ( u, u X , u XX , x) ou dx .xlThe variational principleits Euler equation. 0 therefore leads to (20) as6J[ u) Conversely, suppose that F0 - is the Eulerequation from the variational principlex2ofo((u,ux,x)dx 0xlThen Fmust have the. formF( u, u X , u XX , x) ;(.U -The deriva tive of FDF(u;h) ;;(.U)X (23)X1s now given byX h XUUUUXhX-( ;;:u X uhXIt is a straightforward calculation to show thath1DF(u;h) h DF(u;h ) ,1from which the self-adjointness of Ffollows.hx)XXX

-22When Fis the expression in ( 23)J[ u]becom esThe second term can be integrated by parts to giveJ( u] uX;tUXSince the second and third integrals do not contribute to thevariations,J[ u J is equivalent to the original Lagrangian.To extend the above theory to systems of nonlinear partialdifferential equations it is convenient to introduce vector notation.Denote a system of partial differential equations by( 24)where

-23-TheF.are again polynomial functions of the dependent vari a bles1u , u , . , un,12 IXn their derivatives and the independent variablesThe derivative of the vector operatordefined byDF.(u:h) 1--lim T-0F.(u1- Th)-F .(u)1-0"The self-adjointness condition forFwhere the inner product is defined byis thenF1s now

-24Using this definition of the inner product, the generalization ofthe functionalJ [ u]in (22)is1J[ u] J(F(X. ). )dX.0With these definitions the results obtained fortrue for( 24).( 20)are also

-256.A Variational Formulation of Inviscid Fluid MechanicsVariational principles for non-dissipative fluid flows de-scribed in terms of Lagrangian (or particle) coordinates are knownto be the generalization of the classical Hamilton's principle fora system of particles.The appropriate Lagrangian density hasthe form of kinetic energy -internal energy for a fluid particleand is a function of the position and velocity of the particle.Inthese variational principles the particle paths themselves arevaried to give the necessarily self-adjoint Lagrangian equationsfor the fluid.Variational principles for fluid flows described inthe more usual Eulerian (field) coordinates are more difficult.The fluid equations in Eulerian coordinates are not self-adjoint asthey stand and therefore cannot be the Euler equations of avariational principle in which the field quantities are varied .Forthis reason the most common variational principles in fluidmechanics appear for special flows in which new variables, suchas the velocity potential or stream function , can be introduced. Theadvantage is that when the Eulerian equations are written in termsof these auxiliary variables, they often become self-adjoint .Although one usually looks at specialized flows, knowing a changein variables which leads to variational principles for the generalfluid equations is useful when treating new problems.Clebsch [ 15] introduced the representation15.Loc. cit.

-26-(25)and was successful in finding a variational formulation forincompressible rotational flow.the potentialsp,a., Pfaff's theorem [ 16]He was motivated to introducein the form of ( 25) from the results ofwhich states that the differential formud.x vdy (26)wdz1s reducible to a form{27)Equating the expressions mrepresentation ( 25 ).(26) and {27) leads to Clebsch'sClebsch finds that a variational formulationof the equationsDuDt - \7 p ( 28)\7 u 0 whereuDDt16. {u, v, w)a at u· \7'See, for example, A. R. Forsyth, Theory of DifferentialEquations, Vol. l, New York: Dover {1959).

-27-is obtained by introducing the representation(29)( 3 0)for the dependent variables.From ( 28)equations to be satisfied by . a,DaDt 0.!2Q. Dt0'V · u whereu13it follows that theare0stands for the expression in (29).follow from varying . a,13( 31)Equations(31)in the variational principleHere, as in all of the subsequent variational principles considered,the Lagrangian is an integral over a fixed region of( ,t)spac eand the variations are assumed to vanish on the boundary of thisregion.This is the first instance in which the pressure appearsas the Lagrangian density.As will be seen in the later examples,

-28the pressure, expressed in terms of the potentials used, is the desiredLagrangian density for problems in fluid mechanics described byEulerian coordinates, much the way the kinetic energy -potentialenergy, expressedin terms of coordinates and velocities, is theLagrangian for problems in classical mechanics.The potentials a., f3account for circulation in the flow.Expressions for the vorticity components in terms of a.s w'l us vw The intersection of surfacesrepresents a vortex line .that a.and f3yZX- vz- w- uy( ,1), s)a. X 8(a.,f3)8(y,z) 8(a.,f3)8( z, x) 8(a.,f3)a(x, y) Y'a.and f3X'Vf3 .constant,f3 constantThe first two equations in (31)are constant following particles.indicateTherefore vortexlines move with the fluid and always carry the same particles.:role of the potentialsLamb [ i 7].17.Loc. cit.a., f3areTheas ca,nonical variables is discussed byThis propel'ty is alsod:l.scussed by

-29Clebsch [ 18)potentials.with his ideas on canonical transformations of theseHere, it will only be remarked that if an arbitraryfunction H(a.,13,t) 1s added to the pressure in (30)the equationsto be satisfied by the new potentials take on the canonical form Dt -H (a., . t) ,a.and follow from varying the integral of the new expression for thepressure.An extension of the above variational formulation to acompressible fluid is given by Bateman [ 19].His results are fora so-called baratropic fluid in which the pressure is a function ofthe density .The equations for compressible flow areDuPnt 'V p ,p 'V· u ( 3 2)0,(33)18.Loc. cit.19.H. Bateman, Partial Differential Equations, p . 164, Cambridge:Cambridge University Press ( 1964).

-30-and for conveniencep(p) 1s taken to be of the formpI p f ( p) - f( p)To obtain a variational formulation for( 34)(32)and (33), Batemanuses the representationu 'Vp a'\713,( 3 5)(36)Now the potentialsp,a, 13 must satisfy the equationsDaDt 0 ' Dt 0 'l2 . Dtwhereofp,p '\7 u(37) u · stands for the expression in ( 35 ).a1n (36)andl3Again, the variationin the integral of the expression for the pressureleads to the set (37).leads to0 'The variation ofpin this integral( 34).The first

II. VARIATIONAL PRINCIPLES IN CONTINUUM MECHANICS 4. Introduction 12 5. The Self-Adjointness Condition of Vainberg 18 6. A Variational Formulation of In viscid Fluid Mechanics . . 25 7. Variational Principles for Ross by Waves in a Shallow Basin and in the "13-P.lane" Model . 37 8. The Variational Formulation of a Plasma . 9.