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Equivalence groups for second order balance equations
International Journal of Engineering Science 37 (1999) 1901±1925
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Equivalence groups for second order balance equations Erdo gan S. S ß uhubi
*
Department of Engineering Sciences, Faculty of Science and Letters, Istanbul Technical University, Maslak 80626, Istanbul, Turkey Received 28 October 1998; accepted 12 November 1998
Abstract The groups of equivalence transformations for a family of second order balance equations involving arbitrary number of independent and dependent variables are investigated. Equivalence groups are much more general than symmetry groups in the sense that they map equations containing arbitrary functions or parameters onto equations of the same structure but with dierent functions or parameters. Our approach to attack this problem is based on exterior calculus. The analysis is reduced to determine isovector ®elds of an ideal of the exterior algebra over an appropriate dierentiable manifold dictated by the structure of the dierential equations. The isovector ®elds induce point transformations, which are none other than the desired equivalence transformations, via their orbits which leave that particular ideal invariant. The general scheme is applied to a one-dimensional nonlinear wave equation and hyperelasticity. It is shown that symmetry transformations can be deduced directly from equivalence transformations. Ó 1999 Elsevier Science Ltd. All rights reserved.
1. Introduction If a given set of equations contains some arbitrary functions or parameters we have in fact a family of sets of equations of the same structure. Almost all ®eld equations of classical continuum physics possess this property since they describe certain common or similar behaviours of diverse materials. The equivalence groups are de®ned as groups of continuous transformations which leave a given family of equations invariant. In other words, they map an arbitrary member of the family onto another member of the same family and they transform a solution of a set of equations onto a solution of another set in the same family. Although the concept of equivalence transformations is well-known in the theory of dierential equations, both ordinary and partial,
*
Tel.: +0090 212 285 3434; fax: +00 90 212 285 6386. E-mail address: [email protected] (E.S. S ß uhubi)
0020-7225/99/$ ± see front matter Ó 1999 Elsevier Science Ltd. All rights reserved. PII: S 0 0 2 0 - 7 2 2 5 ( 9 9 ) 0 0 0 1 2 - 9
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we owe their ®rst systematic treatment to Ovsiannikov [1] who observed that the usual Lie in®nitesimal invariance approach could be applied to construct equivalence groups as well. However, to obtain all equivalence transformation for a given family of equations is quite a complicated task. Apart from some rather trivial cases we can refer to [2] for special one-dimensional wave equations, to [3] for equations modelling detonation, to [4] simple equations of balance form and to [5] for general one-dimensional nonlinear wave equations of balance form. Some rather special cases can also be found in [6]. The aim of the present work is to study equivalence groups associated with second order system of balance equations given in the following form oRai Ra 0; oxi
i 1; 2; . . . ; n;
a 1; 2; . . . ; N ;
1:1
where Rai and Ra are smooth functions of independent variables xi , dependent variables ua and derivatives ua;i . Throughout the analysis the summation convention on repeated indices is adopted. Subscript comma indicates partial dierentiation. A great number of ®eld equations of continuum physics fall into this category of equations with n 4. Since the solution of Eq. (1.1) is in the form u u x, we have in fact a quasilinear system of second order equations as follows oRai oub;j
ub;ij
oRai b oRai u i Ra 0: oub ;i ox
For such equations symmetry transformation x x x; u;uu x; u imply u oRai x; u; $u oRai x;u;$ a u 0 R x; u; $u 0 ! Ra x;u;$ i i ox ox and they transform a solution u u x of an equation to another solution uu x of the same equation whereas equivalence transformations x x x; u; uu x; u imply u ai oRai x; u; $u oR x;u;$ a u 0 a R x; u; $u 0 ! R x;u;$ i i ox ox and a solution u u x of a given equation is transformed to a solution uu x of another equation of the same family. In order to determine groups of equivalence transformations we employ the geometrical approach which was ®rst formulated in the seminal work of Harrison and Estabrook [7]. This approach is based on Cartan's geometrical formulation of partial dierential equations in terms of exterior dierential forms [8]. Within the framework of exterior calculus, in®nitesimal generators of symmetry groups associated with a given set of partial dierential equations are identi®ed as components of isovector ®elds of an ideal of the exterior algebra over an appropriate dierentiable manifold. Isovector ®elds are certain special vector ®elds in the tangent space of the manifold whose orbits generate transformations which leave the ideal invariant. This scheme has been explored in studying symmetry groups in Refs. [9] and [10] with special emphasis on second
E.S. Sß uhubi / International Journal of Engineering Science 37 (1999) 1901±1925
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order balance equations whereas a general approach for arbitrary system of balance equations of ®nite order was presented in [11]. In order to employ the geometrical approach in studying the equivalence groups of Eq. (1.1), we have ®rst to extend largely the manifold over which symmetry transformations were used to be determined by taking into account Rai , Ra and their ®rst order derivative with respect to their arguments as additional independent variables. Although this extended manifold will usually acquire a huge dimension we see that it would be possible to carry out a large part of intermediary calculations and to obtain quite a reduced number of equations to determine components of isovector ®elds. The appropriate dierential manifold and the fundamental contact and balance forms which generate a closed ideal which is annihilated by solutions of Eq. (1.1) are introduced in Section 2. Section 3 is devoted to the determination of the isovector ®elds of that ideal. The general structure of the isovector components and quite a reduced number of relations to determine them are separately provided for the cases N 6 1 and N 1. As applications of the general scheme a onedimensional nonlinear wave equation and ®eld equations of homogeneous hyperelasticity are considered in Section 4. Section 5 is concerned with deducing determining equations for isovector components corresponding to symmetry transformations from those of equivalence transformations.
2. Fundamental equations We ®rst reduce Eq. (1.1) to a ®rst order system of dierential equations by introducing the variables vai ua;i :
2:1
To construct equivalence groups we have to consider Rai and Ra as independent variables. To recognise their functional dependence we further de®ne the following variables sai j
oRai ; oxj
rai b
oRai ; oub
saij b
oRai ovbj
;
tia
oRa ; oxi
sab
oRa ; oub
tbai
oRa ovbi
:
2:2
Let M Rn denote the space of independent variables with the cartesian coordinate cover fxi g and G Rn RN the graph space with a coordinate cover fxi ; ua g. We call K the extended manifold which has the coordinate cover n o aij ai a a ai xi ; ua ; vai ; Rai ; Ra ; sai 2:3 ; r ; s ; t ; s ; t b j b i b b : One can easily observe that the dimension of K is n n 1 n 2N n 12 N 2 which is quite a large number even for rather simple systems. Let K K denote the exterior algebra over the manifold K. We next de®ne contact 1-forms
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ra dua ÿ vai dxi 2 K1 K; aij b 1 j ai b Xai dRai ÿ sai j dx ÿ rb du ÿ sb dvj 2 K K;
2:4
Xa dRa ÿ tia dxi ÿ sab dub ÿ tbai dvbi 2 K1 K and balance n-forms xa dRai ^ li Ra l Kn K;
2:5
where the symbol ^ expresses the exterior product and l; li and lji ; which will be utilised later, are bases of form spaces Kn M; Knÿ1 M and Knÿ2 M, respectively, and they are given by Edelen [10] l dx1 ^ dx2 ^ ^ dxn ; li
o yl; oxi
lji
o o o yl y yl ÿlij ; oxj i oxj oxi
2:6
lijk
o yl ; oxi jk
where y represents the interior product of a vector with an exterior form. They satisfy the following identities dxi ^ lj dij l; dxk ^ lji dkj li ÿ dki lj ;
2:7a dxm ^ lijk dmi ljk dmj lki dmk lij :
2:7b
Let us now consider the ideal I of K K generated by the forms xa ; ra ; Xai ; Xa ; dra ; dXai and dXa where the operator d represents the exterior dierentiation. We have dra ÿdvai ^ dxi 2 K2 K; aij b 2 j ai b dXai ÿdsai j ^ dx ÿ drb ^ du ÿ dsb ^ dvj 2 K K;
2:8
dXa ÿdtia ^ dxi ÿ dsab ^ dub ÿ dtbai ^ dvbi 2 K2 K: The ideal I is closed. In fact, it is a simple exercise to show that dxa dRa ^ l Xa ^ l sab rb ^ l ÿ tbai drb ^ li 2 I: Next we consider a map / : M ! K which induces a map / : K K ! K M annihilating the generators of the ideal I, i.e.,
E.S. Sß uhubi / International Journal of Engineering Science 37 (1999) 1901±1925
/ ra 0;
/ Xai 0;
/ Xa 0;
/ xa 0:
1905
2:9
In fact, the solution map / : M ! K is a composite map given by / /2 /1 where /1 : M ! H , /2 : H ! K such that the induced maps transforms the following exterior algebras: / K K ! K M;
/ /1 /2 ;
/2 K K ! K H ; /1 K H ! K M; where the space H is de®ned by the coordinate cover xi ; ua ; vai . We choose /2 in such a way that /2 ra ra ; /2 Xai
/2 Xa
/2 xa
ai oRai oR ai j ai ÿ sj dx ÿ rb dub oxj oub
a oRa oR a i a ÿ ti dx ÿ sb dub oxi oub
oRai ovbj oRa ovbi
! ÿ saij dvbj 0; b
2:10a
! ÿ tbai dvbi 0;
oRai oRai b oRai b a R l du ^ l dvj ^ li : i oxi oub ovbj
2:10b
2:10c
By applying then /1 we require: / ra /1 /2 ra ua;i ÿ vai dxi 0;
2:11a
/ Xai /1 /2 Xai 0;
2:11b
/ Xa /1 /2 Xa 0; / wa /1 /2 wa
oRai oub;j
!
ub;ij
oRai b oRai u i Ra l 0: oub ;i ox
2:11c 2:11d
Due to the well-known commutation rule / d d/ ; the exterior derivatives of the generators are also annulled by / . It is obvious that such a map annihilates every form in the ideal I. One immediately veri®es that / is the solution map of the set of equations Eq. (1.1) since the relations (2.9), in view of Eqs. (2.11a), (2.10b), (2.10c) and (2.11d), are none other than the relations (2.1), (2.2) and (1.1). 3. Isovector ®elds of the closed ideal I A vector ®eld V of the tangent space T(K) is an isovector ®eld of the ideal I if and only if 8a 2 I ) LV a 2 I;
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where LV represents the Lie derivative of a form a with respect to a vector ®eld V. The necessary and sucient conditions for V to be an isovector ®eld of I are that Lie derivatives of all generators should remain in the ideal [10]. It can be shown that isovector ®elds constitute a Lie subalgebra of T(K). Their orbits generate one-parameter Lie groups of continuous transformations [10]. Since I is a graded ideal a more systematic approach to determine isovector ®elds is plausible [10]. Moreover, we have to distinguish two cases. 3.1. The case N ¹ 1 A general vector ®eld V 2 T K is expressible in the following form with respect to the usual bases of the manifold K V Xi
o o o o o o o o U a a Vi a a S ai ai T a a Sjai ai Sai Sbaij aij b ai i ox ou ovi oR osj orb oR osb
Tia
o o o Tab a Tbai ai ; a oti osb otb
3:1
where all coecients (isovector components) are smooth functions of coordinates in the list (2.3). Let us ®rst consider the contact the ideal C ra ; Xai ; Xa I generated by 1-forms ra ; Xai ; Xa . If V is to be an isovector of C then the following conditions should be satis®ed LV ra kab rb Kbia Xbi Kba Xb ;
3:2a
b ai bj ai b LV Xai kai b r Lbj X Lb X ;
3:2b
LV Xa lab rb Mbia Xbi Mba Xb
3:2c
with appropriate coecients as smooth functions of K0 K. In order to evaluate the Lie derivative of an arbitrary exterior form we shall make use of the well-known relation LV a V yda d V ya:
3:3
Let us start with Eq. (3.2a) Since F a V yra U a ÿ vai X i ;
3:4a
V ydra ÿVi a dxi X i dvai ;
3:4b
we have LV ra ÿVi a dxi X i dvai dF a :
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Introducing the above relation into left-hand side and the relations (2.4) into the right-hand side of Eq. (3.2a) collecting similar terms and equating coecients of independent 1-forms in the resulting expression to zero we immediately obtain Kbia oF a osbi j
oF a ; oRbi
Kba oF a
0;
orbi c
oF a Vi i ox
a
dab X i
oF a ovbi
oF a ; oRb
0;
kab
oF a
oF a oF a ci oF a c r s ; oub oRci b oRc b
0;
osbij c
oF a otib
0;
oF a osbc
0;
oF a otcbi
oF a oF a cj oF a c b oF a bj oF a b r s v s b ti ; oub oRcj b oRc b i oRbj i oR
0;
3:5
oF a cji oF a ci s c tb 0 oR oRcj b
which implies obviously that F a F a xi ; ub ; vbi ; Rbi ; Rb : Since we have assumed that N 6 1 we can always make the choice a 6 b in the last expression of Eq. (3.5) so that we have oF a ovbi
oF a cji oF a ci s c tb 0; oR oRcj b
a 6 b
whence we deduce that oF a 0; oRcj
oF a 0; oRc
oF a ovbi
0;
a 6 b:
Hence the last expression of Eq. (3.5) reduces to dab X i
oF a ovbi
0:
If we introduce Eq. (3.4a) into the foregoing relation we obtain oU a ovbi
ÿ vaj
oX j ovbi
0:
By dierentiating the above expression with respect to vck and recalling the symmetry of mixed derivatives we conclude that
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dac
oX k ovbi
dab
oX i : ovck
3:6
By contracting on indices a and c in Eq. (3.6) we ®nd that N
oX k ovbi
oX i ovbk
:
Inserting this expression into the right-hand side of Eq. (3.6) and contracting this time on indices a and b we have N 2 ÿ 1
oX k 0; ovci
N 6 1
whence it follows that oX k 0 ovci
and
oU a ovbi
0:
We thus conclude that the isovector components X i ; U a and Vi a are to be given by X i X i xj ; ua ;
3:7a
U a U a xi ; ub ;
3:7b
F a U a xj ; ub ÿ vai X i xj ; ub ;
3:7c
Vi a
oU a oX j a oU a b oX j a b ÿ i vj b vi ÿ b vj vi oxi ox ou ou
3:7d
which are exactly of the same form as those corresponding to symmetry groups [11]. Similarly the left-hand side of Eq. (3.2b) becomes aij aij b b ai b b ai ai LV Xai ÿSjai dxj X j dsai j ÿ Sb du U drb ÿ Sb dvj Vj dsb dF ;
3:8
where we de®ne aij b j ai b F ai V yXai S ai ÿ sai j X ÿ rb U ÿ sb Vj :
Then it is relatively easy to see that Eq. (3.2b) yields now the following results: Lai bj
oF ai ; oRbj
Lai b
oF ai ; oRb
ai kai b kb ÿ Sb ; ai
3:9
E.S. Sß uhubi / International Journal of Engineering Science 37 (1999) 1901±1925 ai ai ai kai oF oF rcj oF sc ; b oub oRcj b oRc b
oF ai otjb
0;
oF ai osbc
0;
oF ai otcbj
1909
0;
ai b Sjai Sj ÿ Sai b vj ;
3:10
oF ai oF ai oF ai b ai ai kb vbj bk sbk tj ; Sj j j ox oR oRb Sbaij
oF ai ovbj
oF ai ckj oF ai cj s t oRc b oRck b
and dab dik X j
oF ai osbk j
0;
dac dij U b
oF ai 0; orcj b
dac dij Vkb
oF ai oscjk b
0:
3:11
It is clear that the functions F ai cannot depend on variables tia ; sab ; tbai . On the other hand X i ; U a and ai aij Vi a are independent of sai j ; rb ; sb . Hence Eq. (3.11) can easily be integrated to give aij b ai j bj b j ai b b b F ai ÿsai j X ÿ rb U ÿ sb Vj F x ; u ; vj ; R ; R :
3:12
On comparing Eq. (3.12) with Eq. (3.9) we immediately observe that S ai Fai xj ; ub ; vbj ; Rbj ; Rb :
3:13
In exactly the same fashion it follows from 3:23 that Ga V yXa T a ÿ tia X i ÿ sab U b ÿ tbai Vi b ; Mbia ab l oGa osbi j
oGa ; oRbi
Mba
oGa ; oRb
ab ÿ Tab ; lab l
oGa oGa ci oGa c r s ; oub oRci b oRc b
0; a
oGa orbi c
0;
Tia Ti ÿ Tab vbi ;
oGa osbij c
0;
3:14
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oGa oGa oGa b a ab vbi bj sbj ti ; Ti i l i ox oR oRb Tbai
oGa ovbi
oGa cji oGa ci s c tb oR oRcj b
and dab X i
oGa
0;
otib
dac U b
oGa 0; oscb
dac Vi b
oGa 0: otbci
3:15
aij ai i a a The functions Ga are free from the variables sai j ; rb ; sb : Since X ; U and Vi are also independent of tia ; sab ; tbai ; Eq. (3.15) can easily be integrated to a
Ga ÿtia X i ÿ sab U b ÿ tbai Vi b G xj ; ub ; vbj ; Rbj ; Rb
3:16
whence it follows that a
T a G xj ; ub ; vbj ; Rbj ; Rb :
3:17
Consequently, isovector ®elds of the contact ideal can now be characterised by the expression V Xi
o o o o o o ai o a o U a a Vi a a S ai ai T a a Sj ai Sbaij aij Ti a oxi ou ovi oR osj oti oR osb
Tbai
o V1 V2 ; otbai
3:18
where the vector ®elds V1 and V2 are de®ned as ! ! o o b o b o a ai ÿ vj ai ; V2 Tb ÿ vi a : V1 Sb orai osj osab oti b ai a The components X i ; U a ; Vi a ; S ai ; T a ; Sj ; Sbaij ; Ti and Tbai are now fully determined in terms of the a b ai functions X i xj ; ub ; U a xj ; ub ; F xj ; ub ; vj ; Rbj ; Rb and G xj ; ub ; vbj ; Rbj ; Rb in view of the relaa tions (3.7), (3.10), (3.12), (3.13), (3.14), (3.16) and (3.17). On the other hand Sai b and Tb are left as arbitrary functions of variables (2.3). However, it is straightforward to verify that
LV1 ra 0;
b LV1 Xai ÿSai br ;
LV1 Xa 0;
LV1 xa 0;
LV2 ra 0;
LV2 Xai 0;
LV2 Xa ÿTab rb ;
LV2 xa 0:
This result, in turn, implies that the vector ®elds V1 and V2 are isovector ®elds of the ideal I for a every choice of functions Sai b and Tb . Therefore they are trivial isovector ®elds of I and they will be discarded henceforth without loss of generality.
E.S. Sß uhubi / International Journal of Engineering Science 37 (1999) 1901±1925
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Due to the commutation rule dLV LV d; valid for arbitrary vector ®elds and since the ideal is closed, the Lie derivatives of forms dra ; dXai ; dXa with respect to the vector ®eld given by Eq. (3.18) are already in the ideal I. Thus the last step to reach our goal is to impose certain conditions on (3.18) so that the forms LV xa remain in the ideal I. These conditions can be expressed as follows LV xa T a l ÿ X i dRa ^ li d S ai li ÿ X j dRai ^ lji Ra X i li T a l dS ai Ra dX i ^ li ÿ dX j ^ dRai ^lji mab xb rb ^ Aab drb ^ Bab Xbi ^ Cbia dXbi ^ Dabi Xb ^ Cba dXb ^ Dab ;
3:19
where we have employed Eq. (2.6). Unknown coecients are of course exterior forms of appropriate degrees, namely, mab 2 K0 K;
a Aab ; Cbi ; Cba 2 Knÿ1 K;
Bab ; Dabi ; Dab 2 Knÿ2 K:
To evaluate the expressions (3.19) we ®rst have to calculate the dierentials dX i and dS ai appearing there taking into account relations (3.7a) and (3.13). We then insert the de®nitions (2.8) into Eq. (3.19), replace the forms dua ; dRai and dRa by ra ; Xai and Xa , respectively, employ the identities (2.7) in the resulting expressions and ®nally collect the similar forms on both sides. We see then at once that we simply have to take Dabi 0;
Dab 0;
whereas the remaining terms can be arranged as follows h i bi bj a b ai ai a bi c b a a ci a a i C ÿ mb R si rc vi l r ^ Cb li ÿ Ab ÿ mc rb li X ^ Cbj li ÿ Cbj ÿ mb dj li
oS ai oX j ai oX j ai b oX i aik b a b c ÿ C l X ^ r ^ l r r ^ r ^ l ÿ s r ^ dvck ^ lji X ^ i ji ji b b b b c b c ou ou ou oR a cij dvbj ^ li dvbi ^ dxi ^ Bab 0 Caij 3:20 b ÿ mc sb b
aij ai where the functions Ca ; Cai b ; Cbj and Cb are de®ned by the relations: ai ai i i oS ai oS oS oX k a i oX i a b a a oX a oX a bj c C T i R R vi k db dj ÿ j db sbj i rc v i bj i b b ox ox ou ou ox ox oR
oS ai b oX j ai c b b c ai ai c b t s v si rai i i c c vi vj ÿ sj rc vj vi ; b b ou oR
3:21a
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oS ai oX i b Ra b ou ou
Cai b
oS ai oX k a i oX i a cj oS ai c oX j ai c dd ÿ d r s ÿ s rai c vj oRcj oxk c j oxj c b oRc b oub j
oX i aj oX j ai c oX i aj c aj c s r v r v ÿ r v; c j oub j ouc b j ouc b j
oS ai oX k a i oX i a oX k c a i oX i c a d d ÿ d vk db dj ÿ c vj db ; ou oRbj oxk b j oxj b ouc
Cai bj Caij b
oS ai ovbj
oS ai oX l a i oX i a ckj oS ai cj oX k aij c oX i akj c dd ÿ d s c tb c sb vk ÿ c sb vk : oR ou ou oRck oxl c k oxk c b
3:21b
3:21c
3:21d
Comparing both sides of Eq. (3.20) we immediately deduce the expressions bi c a mab Rb sbi i rc m i C ;
Cba
oS ai li ; oRb
a a j Cbia Caj bi ÿ mb di lj db
a ci Aab Cai b ÿ mc rb li
oX j c r ^ lji ; ouc
oX j ai c oX j aik c r r ^ l ÿ s dvk ^ lji : ji ouc b oub c
On the other hand if we choose Bab Bajk b ljk ;
akj 0 Bajk b ÿBb 2 K K;
3:22
we notice that on using Eq. (2.7b) we obtaina aji aij b b b i aik dvbi ^ dxi ^ Bab Bajk b dvi ^ dx ^ ljk dvi ^ Bb lk ÿ Bb lj ÿ2Bb dvj ^ li :
Inserting this expression into Eq. (3.20) we ®nally get aij aij mac scij b Cb 2Bb :
The antisymmetric part of the foregoing expression determines Baij b since it is antisymmetric in indices i and j while its symmetric part leads to the relations c ij
mac sb
a ij
Cb :
Therefore the equations to be satis®ed by the generating functions for isovector components are reduced to bi c a mab Rb sbi i rc v i C
3:23a
E.S. Sß uhubi / International Journal of Engineering Science 37 (1999) 1901±1925 cji aij aji mac scij b sb Cb Cb
1913
3:23b
These equations involve the unknown functions mab in addition to the functions a X i xj ,ub ; Fai xj ; ub ; vbj ; Rbj ; Rb ; G xj ; ub ; vbj ; Rbj ; Rb and some of their ®rst order derivatives. It is straightforward to check that the number of unknowns is n N N 2 Nn while the number of equations are N 12 n 1nN 2 for N > 1. Therefore the maximal number of compatibility conditions which might be imposed are n ÿ 1 n 2N 2 ÿ n N 1: 2 If they admit nontrivial solutions, Eqs. (3.23) determine all independent isovector ®elds associated with the equivalence groups of balance equations (1.1). After having obtained all linearly independent isovector ®elds we can determine an equivalence group corresponding to a particular isovector ®eld by integrating the following system of ordinary dierential equations in the group parameter dxi X i xj ; ub ; d
d ua U a xj ; ub ; d
dR bj ; R b ; Fai xj ; ub ; vbj ; R d ai
d vai Vi a xj ; ub ; vbj ; d
dR ai bj ; R b ; G xj ; ub ; vbj ; R d a
under the initial conditions xi 0 xi ;
ua 0 ua ;
vai 0 vai ;
ai 0 Rai ; R
a 0 Ra : R
3.2. The case N 1 If there is a single dierential equation of the balance form then all Greek indices take just the value one and they will be omitted henceforth to simplify the notation. With this notational artifact in mind we can copy all equations related to the contact ideal until to the end of Eqs. (3.5). However, the reasoning which had worked quite well there fails in N 1 and from then on we have to pursue the analysis on a some what dierent path. The relations which we could borrow from Eqs. (3.4a), (3.4b) and (3.5) are summarised below Xi ÿ
oF oF oF i ÿ j sji ÿ t; ovi oR oR
oF Vi i ox
U F vi X i ;
F F xi ; u; vi ; Ri ; R;
oF oF i oF oF oF ir s vi j sji ti : ou oR oR oR oR
3:24
One must note here that sji , sji and ti , ti represent entirely dierent quantities and they should not be mistaken as associated tensor components. Similarly we have almost verbatim copies of the
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E.S. Sß uhubi / International Journal of Engineering Science 37 (1999) 1901±1925
relations (3.10), (3.11) and (3.14), (3.15) with F i being independent of ti , s and ti and G being independent of sji , ri and sji . However these properties imply through the relations (3.11) and (3.15) that X i ; U and Vi given by Eq. (3.24) cannot depend on sji , ti and sii , ti . Hence it follows that oF 0; oRi
oF 0 ) F F xi ; u; vi oR
3:25
and Eq. (3.24) reduces to Xi ÿ
oF ; ovi
U F ÿ vi
oF ; ovi
Vi
oF oF vi : oxi ou
3:26
In the same fashion Eqs. (3.11) and (3.15) yield F i ÿsij X j ÿ ri U ÿ sij Vj Fi xj ; u; vj ; Rj ; R; G ÿti X i ÿ sU ÿ ti Vi G xj ; u; vj ; Rj ; R; whereas the relations Eqs. (3.10) and (3.14) reduce now to S i Fi xj ; u; vj ; Rj ; R; i Sji Sj ÿ Si vj ; i
oF i Sj j ox S ij
oF i oF i k oF i oF i oF i kr s vj k skj tj ; ou oR oR oR oR
oF i oF i kj oF i j s t; ovj oRk oR
T G xj ; u; vj ; Rj ; R; Tj Ti ÿ Tvi ; oG Ti i ox Ti
oG oG j oG oG oG jr s vi j sji ti ; ou oR oR oR oR
oG oG ji oG i s t: ovi oRj oR
Hence an isovector ®eld of the contact ideal is expressible as
3:27
E.S. Sß uhubi / International Journal of Engineering Science 37 (1999) 1901±1925
o o o o o o o o i o U Vi Si i T Sj i S ij ij Ti T i i V1 V2 ; i ox ou ovi oR osj os oti ot oR
V Xi
1915
3:28
where the vector ®elds V1 and V2 are de®ned now as i
V1 S
! o o ÿ vj i ; ori osj
o o V2 T : ÿ vi os oti
Again it is straightforward to see V1 and V2 are trivial isovector ®elds which can be discarded without loss of generality. The components of (3.28) are entirely determined in terms of three functions F xi ; u; vi , Fi xj ; u; vj ; Rj ; R, G xj ; u; vj ; Rj ; R via the relations (3.26) and (3.27). In order that the ideal I x; r; Xi ; X; dr; dXi ; dX remain invariant under the vector ®eld (3.28) we require that the following relation, which can be deduced directly from Eq. (3.19), should also be satis®ed LV x T l ÿ X i dR ^ li d S i li ÿ X j dRi ^ lji RX i li T l dS i RdX i ^ li ÿ dX j ^ dRi ^ lji mx r ^ A dr ^ B Xi ^ Ci X ^ C; where m and A; Ci ; C and B are arbitrary forms of degrees 0 and n ÿ 2 and n ÿ 1, respectively. The forms Di and D which should appear in the foregoing expressions as coecients of 2-forms dXi and dX are taken as zero from the start without loss of generality. The functions X i depend, however, also on vj . Proceeding in exactly the same fashion as we have done previously we obtain the following expression C ÿ m R sii ri vi l r ^ Ci li ÿ A ÿ mri li Xj ^ Cij li ÿ Cj ÿ ldij li i j oS oX j i oX i oX j ik X^ r ÿ l ÿC X ^ r ^ lji s r ^ dvk ^ lji oR i ou ovk ou
oX j i X ^ dvk ^ lji Cij ÿ msij dvj ^ li dvi ^ dxi ^ B Cijkl dvl ^ dvk ^ lji 0; ovk
3:29
where the functions C, Ci ; Cij and Cij are now de®ned by the relations oS i oX i CT i R i ox ox
i oS i oX i oS oX k i oX i oS i j j s R vi si svi d ÿ r v i i ou ou oR oRj oxk j oxj
oX j i s vj ÿ sij vi ; ou i
oS i oX i R C ou ou i
oS i oX k i oX i j oS i oX j i oX i j r s ÿ s s; d ÿ oR ou j ou j oRj oxk j oxj
3:30
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E.S. Sß uhubi / International Journal of Engineering Science 37 (1999) 1901±1925
Cij
oS i oX i jÿ j ox oR
oX k oX k oX i i d ÿ v vj ; k j oxk ou ou
i oS i oX i oS oX l i oX i kj oS i j oX k i oX i k i s R d ÿ ÿ s r v s rk vk t C k ovj ovj oR ovj k ovj k oRk oxl k oxk k oX ij oX i kj s ÿ s vk ; ou ou ij
C
ijkl
ÿC
jikl
ÿC
ijlk
j i i 1 il oX j ik oX jk oX jl oX ÿs : s ÿs s ovk ovl ovl ovk 4
It is clear that these expressions cannot be obtained directly from (3.21) since X i depends also on vj . We further take B Bjk ljk Bjklm dvm ^ ljkl ;
Bjk ÿBkj ;
Bjklm Bjklm 2 K0 K
from which we simply obtain ÿdvi ^ dxi ^ B 2Bij dvj ^ li 3Bijlk dvl ^ dvk ^ lji : On comparing both sides of Eq. (3.29) we get j oX i oX j ik i i A C ÿ mr li dvk ^ lji ; r ÿ s ovk ou C
oS i l oR i
Ci Cji ÿ mdji lj
oX j oX j r ^ lji dvk ^ lji ou ovk
and m R sii ri vi C;
3:31a
ms ij C ij
3:31b
) m sij sji Cij Cji ;
Ci jkl 0 ) Cijkl Cikjl 0:
3:31c
Eqs. (3.31) are determining equations for the function m and the functions F ; Fi and G through which the components of isovector ®elds of equivalence associated with the balance equation
E.S. Sß uhubi / International Journal of Engineering Science 37 (1999) 1901±1925
oRi R 0; oxi
fRi ; R Ri ; R xj ; u; vj g;
i 1; 2; . . . ; n;
vi u;i ;
1917
3:32
may be found. Eqs. (3.23b) and (3.31b) generate chains of equations to determine the desired function since these functions depend on restricted sets of variables of the underlying manifold K. The forms of these equations are heavily aected by the numbers n and N. It seems that it is plausible to ®nd the general solutions of equations (3.23) or (3.31). This will be the subject of a subsequent paper. If some of the arguments of Rai , Ra or Ra , R are missing, isovector components corresponding to these arguments should consequently vanish. Hence additional conditions should be imposed on the general solutions which might change completely their structure. Even if the explicit general solutions cannot be provided, we believe that the work presented here builds quite an ecient shortcut to reach directly to the determining equations for isovector ®elds. These equations may easily be generated by a personal computer and sometimes even solutions may be handled by a computer employing, for instance, Mathematica. Since the cumbersome auxiliary calculations to be carried out in the classical approach to ®nd the determining equations are entirely removed in the present scheme memory requirement for computer is relaxed to a great extent and computation time may be shortened a great deal.
4. Applications 4.1. One-dimensional nonlinear wave equation To illustrate the advantage of the general approach we have developed so far we shall try ®rst to derive the determining equations for the equivalence groups associated with the following single nonlinear wave equation utt f x; t; u; ux ; ut x g x; t; u; ux ; ut :
4:1
To simplify the notation we de®ne x1 x; R1 f ;
x2 t;
R2 ÿv;
s11 fx f1 ; s21 0;
r2 0;
r1 fu f3 ; s21 0;
t2 gt g2 ;
X2 T;
v2 ut v;
R g;
s12 ft f2 ;
s22 0;
t1 gx g1 ; X1 X;
v1 ux p;
S1 F ;
s11 fp f4 ;
s12 fv f5 ;
s22 ÿ1;
s gu g3 ;
t 1 gp g4 ;
S 2 ÿV2 ÿV ;
T G:
t2 gv g5 ;
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E.S. Sß uhubi / International Journal of Engineering Science 37 (1999) 1901±1925
It then follows readily from (3.30) that C G Fx ÿ Vt g Xx Tt pXu vTu pFu ÿ vVu Ff Tt vTu f1 p Ff Tt ÿ vTx f3 ÿ Tx pTu f2 Fg g1 g3 p; C11 Fp gXp Ff Tt vTu f4 Fg g1 ÿ Tp f2 f3 v; C22 ÿXx ÿ Vv ÿ pXu Tv g f1 f3 p ÿ Tx pTu f5 ;
4:2
C12 Xt gXv vXu Fv ÿ Tv f2 f3 v Ff Tt vTu f5 Fg g5 ; C21 ÿVp Tp g f1 f3 p ÿ Tx pTu f4 : Thus (3.31) take simply the following forms: g f1 f3 pm ÿ C 0;
4:3a
g f1 f3 pC11 ÿ f4 C 0;
4:3b
g f1 f3 pC22 C 0;
4:3c
g f1 f3 p C12 C21 ÿ f5 C 0;
4:3d
f5 Tp Xp ÿ f4 Tv 0
4:3e
If we eliminate C between Eqs. (4.3c) and (4.3d) we obtain on noting that g f1 f3 p 6 0 C12 C21 f5 C22 0:
4:4
When we employ (4.2) in (4.4) and arrange the resulting expression we get Xt gXv vXu gTp ÿ Vp Fv Ff Tt ÿ Xx ÿ Vv vTu gTv ÿ pXu f5 Fg g5 ÿ Tv f2 Tp f1 pTp ÿ vTv f3 ÿ Tx pTu f4 Tv f1 f3 pf5 ÿ Tx pTu f52 0 which leads to the following set of equations Fg 0;
Tv 0;
Tp 0;
Xt gXv vXu ÿ Vp Fv 0;
Tx 0;
Tu 0;
4:5a 4:5b
E.S. Sß uhubi / International Journal of Engineering Science 37 (1999) 1901±1925
Ff Tt ÿ Xx ÿ Vv ÿ pXu 0:
1919
4:5c
Using these results in Eq. (4.2) we reduce those expressions to C11 Fp gXp Ff Tt f4 ;
C11 ÿXx ÿ Vv ÿ pXu ;
C G Fx ÿ Vt g Xx Tt pXu pFu ÿ vVu Ff Tt f1 f3 p: Then Eqs. (4.3b) and (4.3c) yield Fp gXp Ff Tt ÿ Xx ÿ Vv ÿ pXu f4 0; f1 f3 p Ff Tt ÿ Xx ÿ Vv ÿ pXu G Fx ÿ Vt pFu ÿ vVu g Tt ÿ Vv 0: Recalling that F and X are independent of g and noting Eq. (4.5c) we ®nally end up with Xp 0;
Fp 0;
G ÿFx Vt ÿ pFu vVu g Vv ÿ Tt :
4:6
Eqs. (4.5a), (4.5b), (4.5c), (4.6) are determining equations of the equivalence groups associated with Eq. (4.1). They are exactly the same as those found earlier [5]. Their solutions were provided in Ref. [5]. 4.2. Homogeneous hyperelastictiy As a second application to the general theory corresponding to a system involving more than one dependent variable we consider a homogeneous hyperelastic solid whose motion is described by the functions xk xk XK ; t, k 1,2,3; K 1,2,3 where XK , xk and t represent, respectively, material coordinates, spatial coordinates and the time, FkK X; t oxk =oXK are deformation ~ gradients, vk X; t oxk =ot are velocity components, R R F R C is the stress potential T (strain energy function), C X; t F F is the Green deformation tensor. The well-known ®eld equations of hyperelasticity in material description without body forces are given below in a nondimensional form o oR ovk ÿ 0: 4:7 ot oXK oFkK We replace for convenience our usual variables xi i 1; 2; 3; 4 by fXK K 1; 2; 3; tg and ua a 1; 2; 3 by xk k 1; 2; 3. We thus de®ne RkK
oR The first Piola-Kirchhoff stress tensor; oFkK
Rk4 ÿvk ;
Ra 0;
vkK FkK ;
vk4 vk ;
skKlL
oRkK ; oFlL
sk4l4 ÿ
ovk ÿdkl : ovl
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E.S. Sß uhubi / International Journal of Engineering Science 37 (1999) 1901±1925
Coordinate cover of the extended manifold K is reduced now to fXK ; t; xk ; FkK ; vk ; RkK ; skKlL g. Isovector ®eld in the tangent space of K may be conveniently expressed as V ÿUK
o o o o o o o ÿ w Uk VkK Vk SkK SkKlL : oXK ot oxk oFkK ovk oRkK oskKlL
It is straightforward to see that relations (3.7) yield in the present case UK UK X; t; x; Vk
w w X; t; x;
Uk Uk X; t; x;
4:8a
oUk oUK ow oUk oUK ow vl FkK vl vk vl ; FkK vk ot ot oxl oxl ot oxl
VkK
oUk oUL ow oUk oUL ow FkL vk FlK FkL FlK vk FlK : oXK oXK oxl oxl oXK oxl
4:8b
4:8c
We can easily observe that we should now replace F ai by fGkK ; 0g. When then get GkK ÿskKlL VlL HkK XL ; t; xl ; FlL ; vl ; RlL ;
SkK HkK ;
Sk4 ÿVk :
ai However the vanishing of Sj and SkKl4 implies, respectively
oGkK 0; oxl
oGkK 0 oXL
oGkK 0 ot
and
oGkK 0: ovl
Hence we conclude that VkK VkK FlL ;
SkK HkK FlL ; RlL :
Thus the determining equations become: oVk oUK oUK oVk oVk ÿ FmK dkl ; vl 0; mkl ovl oXK oxm ot oxl
4:9
4:10a
oVk ow ow oUK oUK ÿ skLlK FmL vm dkl 0; oFlK ot oxm oXL oxm
4:10b
CkKlL CkLlK ÿ vkm smKlL smLlK 0;
4:10c
where we de®ne CkKlL
oHkK oHkK oUK oUK ÿ smMIL FnM dkm oFlL oRmM oXM oxn oUN oUN ow ow ÿ FnN vn dkm dKM : oXN oxn ot oxn
E.S. Sß uhubi / International Journal of Engineering Science 37 (1999) 1901±1925
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Inserting Eq. (4.8b) into Eq. (4.10b) we obtain w w t and oUK oUK vm dkl 0: 2 ot oxm Hence UK UK X. In the same fashion (4.10a)2 yields o2 Uk o 2 Uk o 2 Uk 2 v wd 2 vl vm 0; kl l ot2 otoxl oxl oxm whence we deduce that o2 Uk 0; ot2
2
o2 Uk 0; oxl oxm
o2 Uk 1 ÿ wd kl : otoxl 2
0 or The last expression above, in view of the ®rst one, gives w w a1 t2 2a2 t a3 : Therefore the function Uk are found as Uk X; t; x ÿ a1 t a2 xk Kkl Xxl xk Xt ck X: We thus obtain VkK Kkl;K xl xk;K t ck;K UL;K FkL ÿ a1 t a2 FkK Kkl FlK : On the other hand Eqs. (4.9) require that a1 0;
xk a k ;
Kkl akl ;
ck AkK XK Ak ;
UK BKL XL BK ;
where all coecients are constants. Finally Eq. (4.10c) leads to the relations oHkK oHkL smMlN RkKmM dLN RkLmM dKN 0; oFlL oFlK
4:11
where RkKmM
oHkK BKM dkm ÿ 3a2 dKM dkm ÿ dKM akm : oRmM
Since Eq. (4.11) depends linearly on smMlN , one immediately observes that it yields RkKmM 0 which can be integrated to HkK ÿBKM RkM 3a2 RkK akm RmK rmK F; where the functions rkK are to satisfy orkK orkL 0; oFlL oFlK
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E.S. Sß uhubi / International Journal of Engineering Science 37 (1999) 1901±1925
whose solution can be found as [12] 1 rkK eKLM elmn ckn FlL FmM eKLM cLkl FlM ckK ; 2 where e's are permutation symbols and c's are constants. Let us ®nally summarise the results concerning relevant components of the isovector ®eld: w 2a2 t a3 ;
UK BKL XL BK ;
Uk akl ÿ a2 dkl xl ak t AkK XK Ak ; VkK AkK ÿ a2 FkK BLK FkL akl FlK ; SkK ÿBKM RkM 3a2 RkK akm RmK rkK : Equivalence transformations are then obtained through equations dX x dF dt ÿw t; d t; x ; ÿU X; U X; V F d d d d dR F; S R; d
R RkK ;
V VkK ;
S SkK ;
under usual initial conditions. Transformation of the stress potential is governed by dFkK dR oR R kK : VkK F d oF kK d However, the stress potential is actually a function of the deformation tensor C so that the One can readily see that this is feasible C. function R C should be transformed to a function R only if we choose AkK 0;
akl a0 dkl ;
ckl c0 dkl ;
cLkl cL dkl ;
ckK 0:
5. Symmetry groups It might be quite instructive to verify that the determining equations for isovector ®elds of symmetry groups associated with balance equations are directly deducible from those corresponding to equivalence groups. To this end we should consider that Rai and Ra and their derivatives with respect to their arguments are no longer independent variables. Therefore the only surviving isovector components will be X i ; U a and Vi a in (3.1). However, we have to note that the base vectors o=oxi ; o=oua and o=ovai in (3.1) were evaluated by keeping Rai and Ra constant. Hence
E.S. Sß uhubi / International Journal of Engineering Science 37 (1999) 1901±1925
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they should now be replaced by total derivatives by taking into account the functional forms of Rai and Ra . It is quite clear that we have now to write o o oRaj o oRa o ÿ ÿ oxi oRaj oxi oRa oxi Rai ;Ra oxi and similar expressions for the derivatives with respect to ua and vai . Consequently the isovector ®eld is now expressible as o oRaj i oRbj a oRbj a o a o a o bj V X iU Vi S ÿ i X ÿ a U ÿ a Vi a a ox ou ovi ox ou ovi oRbj oRb i oRb a oRb a o b T ÿ i X ÿ a U ÿ a Vi : ox ou ovi oRa i
Since the components of the isovector ®eld on the base vectors o=oRbj and o=oRb should now vanish we conclude that S ai V Rai ;
T a V Ra ;
5:1
where for a smooth function f xi ; ua ; vai we de®ne of i of a of a X a U a Vi : oxi ou ovi
V f
Inserting (5.1) and (2.2) into 3:211 and 3:214 and the resulting expressions into (3.23) we obtain: mab
mac
oRbi oRbi R i c vci ox ou b
oRci ovbj
oRcj
!
ovbi
aij b c b Aa Aai b vi Abc vi vj ;
aji akij akji Aaij A vck ; A ÿ 2 A b b bc bc
5:2
where we de®ned: Aa V Ra Aai b 2Aaij bc
i oV Rai oX k oRai oX i oRaj a oX R ÿ j ; oxi oxk oxi ox oxi oxi
i oV Rai oX k oRai oX i oRaj oX i oRaj oX j oRai a oX R ÿ j ÿ b ; oub oxk oub ox oub oub oxj ou oxj oub
oX i oRai oX j oRai ÿ ; ouc oub oub ouc
5:3
1924
E.S. Sß uhubi / International Journal of Engineering Science 37 (1999) 1901±1925
Aaij b 2Aakij b
oV Rai oX k oRai oX i oRak k ÿ k ; ox ovbj ox ovbj ovbj oX i oRak oX k oRai ÿ c : ouc ovbj ou ovbj
The relations (5.2) and (5.3) are identical with those given in Ref. [11]. In the same fashion we can handle the case N 1 and ®nd that the relations (3.31) are cast into the form: oRi oRi vi A Ai vi ; m R i ox ou
oRi oRj m ovj ovi
ÿ Aij Aji ÿ 2 Bkij Bkji vk ;
5:4
C ijkl C ikjl 0 where A V R
oV Ri oX i oX j oRi oX i oRi R j ÿ ; oxi ox oxi oxj oxi oxi
Ai
oV Ri oX i oX j oRi oX i oRj oX i oRi oX j oRi j ÿ j ÿ ; R ou ox ou ox ou ou oxj ou oxj ou
Aij
oV Ri oX i oX k oRi oX i oRk oX i oRk oX k oRi R k ÿ ÿ ; ovj ox ovj oxk ovj ovj oxk ovj oxk ovj
2Bkij
oX i oRk oX k oRi oX k oRi oX i oRk ÿ ÿ ; ou ovj ou ovj ovj ou ovj ou
4C ijkl
oRi oX j oRi oX j oRj oX i oRj oX i ÿ ÿ : ovl ovk ovk ovl ovk ovl ovl ovk
5:5
For a function f xi ; u; vi the function V f is de®ned as V f
of i of of X U Vi : oxi ou ovi
One easily veri®es that Eqs. (5.4) and (5.5) agree completely with those given in Ref. [11]. Acknowledgements The author gratefully acknowledges the partial support provided by Turkish Academy of Sciences.
E.S. Sß uhubi / International Journal of Engineering Science 37 (1999) 1901±1925
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References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12]
L.V. Ovsiannikov, Group Analysis of Dierential Equations, Academic Press, New York, 1982. N.H. Ibragimov, M. Torrisi, A. Valenti, J. Math Phys. 32 (1991) 2988. N.H. Ibragimov, M. Torrisi, J. Math Phys. 33 (1992) 3931. N.H. Ibragimov, M. Torrisi, J. Math Anal. Appls. 184 (1994) 441. E.S. S ß uhubi, ARI (Formerly Bull Tech. Univ. Istanbul) 50 (1997) 151. N.H. Ibragimov (Ed.), CRC handbook of Lie group analysis of dierential equations, vol. I, Symmetries, Exact Solutions and Conservation Laws. CRC Press, Boca Raton, 1994. B.K. Harrison, F.B. Estabrook, J. Math Phys. 12 (1971) 653. E. Cartan, Les systemes dierentiel exterieurs et leurs applications geometriques. Hermann, Paris, 1945. D.G.B. Edelen, Isovector Methods for Equations of Balance. Sijtho Noordho, Alphen aan den Rijn, 1980. D.G.B. Edelen, Applied Exterior Calculus. Wiley, New York, 1985. E.S. S ß uhubi, Int. J. Engng. Sci. 29 (1991) 133. E.S. S ß uhubi, Int. J. Engng. Sci. 27 (1989) 441.
www.elsevier.com/locate/ijengsci
Equivalence groups for second order balance equations Erdo gan S. S ß uhubi
*
Department of Engineering Sciences, Faculty of Science and Letters, Istanbul Technical University, Maslak 80626, Istanbul, Turkey Received 28 October 1998; accepted 12 November 1998
Abstract The groups of equivalence transformations for a family of second order balance equations involving arbitrary number of independent and dependent variables are investigated. Equivalence groups are much more general than symmetry groups in the sense that they map equations containing arbitrary functions or parameters onto equations of the same structure but with dierent functions or parameters. Our approach to attack this problem is based on exterior calculus. The analysis is reduced to determine isovector ®elds of an ideal of the exterior algebra over an appropriate dierentiable manifold dictated by the structure of the dierential equations. The isovector ®elds induce point transformations, which are none other than the desired equivalence transformations, via their orbits which leave that particular ideal invariant. The general scheme is applied to a one-dimensional nonlinear wave equation and hyperelasticity. It is shown that symmetry transformations can be deduced directly from equivalence transformations. Ó 1999 Elsevier Science Ltd. All rights reserved.
1. Introduction If a given set of equations contains some arbitrary functions or parameters we have in fact a family of sets of equations of the same structure. Almost all ®eld equations of classical continuum physics possess this property since they describe certain common or similar behaviours of diverse materials. The equivalence groups are de®ned as groups of continuous transformations which leave a given family of equations invariant. In other words, they map an arbitrary member of the family onto another member of the same family and they transform a solution of a set of equations onto a solution of another set in the same family. Although the concept of equivalence transformations is well-known in the theory of dierential equations, both ordinary and partial,
*
Tel.: +0090 212 285 3434; fax: +00 90 212 285 6386. E-mail address: [email protected] (E.S. S ß uhubi)
0020-7225/99/$ ± see front matter Ó 1999 Elsevier Science Ltd. All rights reserved. PII: S 0 0 2 0 - 7 2 2 5 ( 9 9 ) 0 0 0 1 2 - 9
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E.S. Sß uhubi / International Journal of Engineering Science 37 (1999) 1901±1925
we owe their ®rst systematic treatment to Ovsiannikov [1] who observed that the usual Lie in®nitesimal invariance approach could be applied to construct equivalence groups as well. However, to obtain all equivalence transformation for a given family of equations is quite a complicated task. Apart from some rather trivial cases we can refer to [2] for special one-dimensional wave equations, to [3] for equations modelling detonation, to [4] simple equations of balance form and to [5] for general one-dimensional nonlinear wave equations of balance form. Some rather special cases can also be found in [6]. The aim of the present work is to study equivalence groups associated with second order system of balance equations given in the following form oRai Ra 0; oxi
i 1; 2; . . . ; n;
a 1; 2; . . . ; N ;
1:1
where Rai and Ra are smooth functions of independent variables xi , dependent variables ua and derivatives ua;i . Throughout the analysis the summation convention on repeated indices is adopted. Subscript comma indicates partial dierentiation. A great number of ®eld equations of continuum physics fall into this category of equations with n 4. Since the solution of Eq. (1.1) is in the form u u x, we have in fact a quasilinear system of second order equations as follows oRai oub;j
ub;ij
oRai b oRai u i Ra 0: oub ;i ox
For such equations symmetry transformation x x x; u;uu x; u imply u oRai x; u; $u oRai x;u;$ a u 0 R x; u; $u 0 ! Ra x;u;$ i i ox ox and they transform a solution u u x of an equation to another solution uu x of the same equation whereas equivalence transformations x x x; u; uu x; u imply u ai oRai x; u; $u oR x;u;$ a u 0 a R x; u; $u 0 ! R x;u;$ i i ox ox and a solution u u x of a given equation is transformed to a solution uu x of another equation of the same family. In order to determine groups of equivalence transformations we employ the geometrical approach which was ®rst formulated in the seminal work of Harrison and Estabrook [7]. This approach is based on Cartan's geometrical formulation of partial dierential equations in terms of exterior dierential forms [8]. Within the framework of exterior calculus, in®nitesimal generators of symmetry groups associated with a given set of partial dierential equations are identi®ed as components of isovector ®elds of an ideal of the exterior algebra over an appropriate dierentiable manifold. Isovector ®elds are certain special vector ®elds in the tangent space of the manifold whose orbits generate transformations which leave the ideal invariant. This scheme has been explored in studying symmetry groups in Refs. [9] and [10] with special emphasis on second
E.S. Sß uhubi / International Journal of Engineering Science 37 (1999) 1901±1925
1903
order balance equations whereas a general approach for arbitrary system of balance equations of ®nite order was presented in [11]. In order to employ the geometrical approach in studying the equivalence groups of Eq. (1.1), we have ®rst to extend largely the manifold over which symmetry transformations were used to be determined by taking into account Rai , Ra and their ®rst order derivative with respect to their arguments as additional independent variables. Although this extended manifold will usually acquire a huge dimension we see that it would be possible to carry out a large part of intermediary calculations and to obtain quite a reduced number of equations to determine components of isovector ®elds. The appropriate dierential manifold and the fundamental contact and balance forms which generate a closed ideal which is annihilated by solutions of Eq. (1.1) are introduced in Section 2. Section 3 is devoted to the determination of the isovector ®elds of that ideal. The general structure of the isovector components and quite a reduced number of relations to determine them are separately provided for the cases N 6 1 and N 1. As applications of the general scheme a onedimensional nonlinear wave equation and ®eld equations of homogeneous hyperelasticity are considered in Section 4. Section 5 is concerned with deducing determining equations for isovector components corresponding to symmetry transformations from those of equivalence transformations.
2. Fundamental equations We ®rst reduce Eq. (1.1) to a ®rst order system of dierential equations by introducing the variables vai ua;i :
2:1
To construct equivalence groups we have to consider Rai and Ra as independent variables. To recognise their functional dependence we further de®ne the following variables sai j
oRai ; oxj
rai b
oRai ; oub
saij b
oRai ovbj
;
tia
oRa ; oxi
sab
oRa ; oub
tbai
oRa ovbi
:
2:2
Let M Rn denote the space of independent variables with the cartesian coordinate cover fxi g and G Rn RN the graph space with a coordinate cover fxi ; ua g. We call K the extended manifold which has the coordinate cover n o aij ai a a ai xi ; ua ; vai ; Rai ; Ra ; sai 2:3 ; r ; s ; t ; s ; t b j b i b b : One can easily observe that the dimension of K is n n 1 n 2N n 12 N 2 which is quite a large number even for rather simple systems. Let K K denote the exterior algebra over the manifold K. We next de®ne contact 1-forms
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E.S. Sß uhubi / International Journal of Engineering Science 37 (1999) 1901±1925
ra dua ÿ vai dxi 2 K1 K; aij b 1 j ai b Xai dRai ÿ sai j dx ÿ rb du ÿ sb dvj 2 K K;
2:4
Xa dRa ÿ tia dxi ÿ sab dub ÿ tbai dvbi 2 K1 K and balance n-forms xa dRai ^ li Ra l Kn K;
2:5
where the symbol ^ expresses the exterior product and l; li and lji ; which will be utilised later, are bases of form spaces Kn M; Knÿ1 M and Knÿ2 M, respectively, and they are given by Edelen [10] l dx1 ^ dx2 ^ ^ dxn ; li
o yl; oxi
lji
o o o yl y yl ÿlij ; oxj i oxj oxi
2:6
lijk
o yl ; oxi jk
where y represents the interior product of a vector with an exterior form. They satisfy the following identities dxi ^ lj dij l; dxk ^ lji dkj li ÿ dki lj ;
2:7a dxm ^ lijk dmi ljk dmj lki dmk lij :
2:7b
Let us now consider the ideal I of K K generated by the forms xa ; ra ; Xai ; Xa ; dra ; dXai and dXa where the operator d represents the exterior dierentiation. We have dra ÿdvai ^ dxi 2 K2 K; aij b 2 j ai b dXai ÿdsai j ^ dx ÿ drb ^ du ÿ dsb ^ dvj 2 K K;
2:8
dXa ÿdtia ^ dxi ÿ dsab ^ dub ÿ dtbai ^ dvbi 2 K2 K: The ideal I is closed. In fact, it is a simple exercise to show that dxa dRa ^ l Xa ^ l sab rb ^ l ÿ tbai drb ^ li 2 I: Next we consider a map / : M ! K which induces a map / : K K ! K M annihilating the generators of the ideal I, i.e.,
E.S. Sß uhubi / International Journal of Engineering Science 37 (1999) 1901±1925
/ ra 0;
/ Xai 0;
/ Xa 0;
/ xa 0:
1905
2:9
In fact, the solution map / : M ! K is a composite map given by / /2 /1 where /1 : M ! H , /2 : H ! K such that the induced maps transforms the following exterior algebras: / K K ! K M;
/ /1 /2 ;
/2 K K ! K H ; /1 K H ! K M; where the space H is de®ned by the coordinate cover xi ; ua ; vai . We choose /2 in such a way that /2 ra ra ; /2 Xai
/2 Xa
/2 xa
ai oRai oR ai j ai ÿ sj dx ÿ rb dub oxj oub
a oRa oR a i a ÿ ti dx ÿ sb dub oxi oub
oRai ovbj oRa ovbi
! ÿ saij dvbj 0; b
2:10a
! ÿ tbai dvbi 0;
oRai oRai b oRai b a R l du ^ l dvj ^ li : i oxi oub ovbj
2:10b
2:10c
By applying then /1 we require: / ra /1 /2 ra ua;i ÿ vai dxi 0;
2:11a
/ Xai /1 /2 Xai 0;
2:11b
/ Xa /1 /2 Xa 0; / wa /1 /2 wa
oRai oub;j
!
ub;ij
oRai b oRai u i Ra l 0: oub ;i ox
2:11c 2:11d
Due to the well-known commutation rule / d d/ ; the exterior derivatives of the generators are also annulled by / . It is obvious that such a map annihilates every form in the ideal I. One immediately veri®es that / is the solution map of the set of equations Eq. (1.1) since the relations (2.9), in view of Eqs. (2.11a), (2.10b), (2.10c) and (2.11d), are none other than the relations (2.1), (2.2) and (1.1). 3. Isovector ®elds of the closed ideal I A vector ®eld V of the tangent space T(K) is an isovector ®eld of the ideal I if and only if 8a 2 I ) LV a 2 I;
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E.S. Sß uhubi / International Journal of Engineering Science 37 (1999) 1901±1925
where LV represents the Lie derivative of a form a with respect to a vector ®eld V. The necessary and sucient conditions for V to be an isovector ®eld of I are that Lie derivatives of all generators should remain in the ideal [10]. It can be shown that isovector ®elds constitute a Lie subalgebra of T(K). Their orbits generate one-parameter Lie groups of continuous transformations [10]. Since I is a graded ideal a more systematic approach to determine isovector ®elds is plausible [10]. Moreover, we have to distinguish two cases. 3.1. The case N ¹ 1 A general vector ®eld V 2 T K is expressible in the following form with respect to the usual bases of the manifold K V Xi
o o o o o o o o U a a Vi a a S ai ai T a a Sjai ai Sai Sbaij aij b ai i ox ou ovi oR osj orb oR osb
Tia
o o o Tab a Tbai ai ; a oti osb otb
3:1
where all coecients (isovector components) are smooth functions of coordinates in the list (2.3). Let us ®rst consider the contact the ideal C ra ; Xai ; Xa I generated by 1-forms ra ; Xai ; Xa . If V is to be an isovector of C then the following conditions should be satis®ed LV ra kab rb Kbia Xbi Kba Xb ;
3:2a
b ai bj ai b LV Xai kai b r Lbj X Lb X ;
3:2b
LV Xa lab rb Mbia Xbi Mba Xb
3:2c
with appropriate coecients as smooth functions of K0 K. In order to evaluate the Lie derivative of an arbitrary exterior form we shall make use of the well-known relation LV a V yda d V ya:
3:3
Let us start with Eq. (3.2a) Since F a V yra U a ÿ vai X i ;
3:4a
V ydra ÿVi a dxi X i dvai ;
3:4b
we have LV ra ÿVi a dxi X i dvai dF a :
E.S. Sß uhubi / International Journal of Engineering Science 37 (1999) 1901±1925
1907
Introducing the above relation into left-hand side and the relations (2.4) into the right-hand side of Eq. (3.2a) collecting similar terms and equating coecients of independent 1-forms in the resulting expression to zero we immediately obtain Kbia oF a osbi j
oF a ; oRbi
Kba oF a
0;
orbi c
oF a Vi i ox
a
dab X i
oF a ovbi
oF a ; oRb
0;
kab
oF a
oF a oF a ci oF a c r s ; oub oRci b oRc b
0;
osbij c
oF a otib
0;
oF a osbc
0;
oF a otcbi
oF a oF a cj oF a c b oF a bj oF a b r s v s b ti ; oub oRcj b oRc b i oRbj i oR
0;
3:5
oF a cji oF a ci s c tb 0 oR oRcj b
which implies obviously that F a F a xi ; ub ; vbi ; Rbi ; Rb : Since we have assumed that N 6 1 we can always make the choice a 6 b in the last expression of Eq. (3.5) so that we have oF a ovbi
oF a cji oF a ci s c tb 0; oR oRcj b
a 6 b
whence we deduce that oF a 0; oRcj
oF a 0; oRc
oF a ovbi
0;
a 6 b:
Hence the last expression of Eq. (3.5) reduces to dab X i
oF a ovbi
0:
If we introduce Eq. (3.4a) into the foregoing relation we obtain oU a ovbi
ÿ vaj
oX j ovbi
0:
By dierentiating the above expression with respect to vck and recalling the symmetry of mixed derivatives we conclude that
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E.S. Sß uhubi / International Journal of Engineering Science 37 (1999) 1901±1925
dac
oX k ovbi
dab
oX i : ovck
3:6
By contracting on indices a and c in Eq. (3.6) we ®nd that N
oX k ovbi
oX i ovbk
:
Inserting this expression into the right-hand side of Eq. (3.6) and contracting this time on indices a and b we have N 2 ÿ 1
oX k 0; ovci
N 6 1
whence it follows that oX k 0 ovci
and
oU a ovbi
0:
We thus conclude that the isovector components X i ; U a and Vi a are to be given by X i X i xj ; ua ;
3:7a
U a U a xi ; ub ;
3:7b
F a U a xj ; ub ÿ vai X i xj ; ub ;
3:7c
Vi a
oU a oX j a oU a b oX j a b ÿ i vj b vi ÿ b vj vi oxi ox ou ou
3:7d
which are exactly of the same form as those corresponding to symmetry groups [11]. Similarly the left-hand side of Eq. (3.2b) becomes aij aij b b ai b b ai ai LV Xai ÿSjai dxj X j dsai j ÿ Sb du U drb ÿ Sb dvj Vj dsb dF ;
3:8
where we de®ne aij b j ai b F ai V yXai S ai ÿ sai j X ÿ rb U ÿ sb Vj :
Then it is relatively easy to see that Eq. (3.2b) yields now the following results: Lai bj
oF ai ; oRbj
Lai b
oF ai ; oRb
ai kai b kb ÿ Sb ; ai
3:9
E.S. Sß uhubi / International Journal of Engineering Science 37 (1999) 1901±1925 ai ai ai kai oF oF rcj oF sc ; b oub oRcj b oRc b
oF ai otjb
0;
oF ai osbc
0;
oF ai otcbj
1909
0;
ai b Sjai Sj ÿ Sai b vj ;
3:10
oF ai oF ai oF ai b ai ai kb vbj bk sbk tj ; Sj j j ox oR oRb Sbaij
oF ai ovbj
oF ai ckj oF ai cj s t oRc b oRck b
and dab dik X j
oF ai osbk j
0;
dac dij U b
oF ai 0; orcj b
dac dij Vkb
oF ai oscjk b
0:
3:11
It is clear that the functions F ai cannot depend on variables tia ; sab ; tbai . On the other hand X i ; U a and ai aij Vi a are independent of sai j ; rb ; sb . Hence Eq. (3.11) can easily be integrated to give aij b ai j bj b j ai b b b F ai ÿsai j X ÿ rb U ÿ sb Vj F x ; u ; vj ; R ; R :
3:12
On comparing Eq. (3.12) with Eq. (3.9) we immediately observe that S ai Fai xj ; ub ; vbj ; Rbj ; Rb :
3:13
In exactly the same fashion it follows from 3:23 that Ga V yXa T a ÿ tia X i ÿ sab U b ÿ tbai Vi b ; Mbia ab l oGa osbi j
oGa ; oRbi
Mba
oGa ; oRb
ab ÿ Tab ; lab l
oGa oGa ci oGa c r s ; oub oRci b oRc b
0; a
oGa orbi c
0;
Tia Ti ÿ Tab vbi ;
oGa osbij c
0;
3:14
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E.S. Sß uhubi / International Journal of Engineering Science 37 (1999) 1901±1925
oGa oGa oGa b a ab vbi bj sbj ti ; Ti i l i ox oR oRb Tbai
oGa ovbi
oGa cji oGa ci s c tb oR oRcj b
and dab X i
oGa
0;
otib
dac U b
oGa 0; oscb
dac Vi b
oGa 0: otbci
3:15
aij ai i a a The functions Ga are free from the variables sai j ; rb ; sb : Since X ; U and Vi are also independent of tia ; sab ; tbai ; Eq. (3.15) can easily be integrated to a
Ga ÿtia X i ÿ sab U b ÿ tbai Vi b G xj ; ub ; vbj ; Rbj ; Rb
3:16
whence it follows that a
T a G xj ; ub ; vbj ; Rbj ; Rb :
3:17
Consequently, isovector ®elds of the contact ideal can now be characterised by the expression V Xi
o o o o o o ai o a o U a a Vi a a S ai ai T a a Sj ai Sbaij aij Ti a oxi ou ovi oR osj oti oR osb
Tbai
o V1 V2 ; otbai
3:18
where the vector ®elds V1 and V2 are de®ned as ! ! o o b o b o a ai ÿ vj ai ; V2 Tb ÿ vi a : V1 Sb orai osj osab oti b ai a The components X i ; U a ; Vi a ; S ai ; T a ; Sj ; Sbaij ; Ti and Tbai are now fully determined in terms of the a b ai functions X i xj ; ub ; U a xj ; ub ; F xj ; ub ; vj ; Rbj ; Rb and G xj ; ub ; vbj ; Rbj ; Rb in view of the relaa tions (3.7), (3.10), (3.12), (3.13), (3.14), (3.16) and (3.17). On the other hand Sai b and Tb are left as arbitrary functions of variables (2.3). However, it is straightforward to verify that
LV1 ra 0;
b LV1 Xai ÿSai br ;
LV1 Xa 0;
LV1 xa 0;
LV2 ra 0;
LV2 Xai 0;
LV2 Xa ÿTab rb ;
LV2 xa 0:
This result, in turn, implies that the vector ®elds V1 and V2 are isovector ®elds of the ideal I for a every choice of functions Sai b and Tb . Therefore they are trivial isovector ®elds of I and they will be discarded henceforth without loss of generality.
E.S. Sß uhubi / International Journal of Engineering Science 37 (1999) 1901±1925
1911
Due to the commutation rule dLV LV d; valid for arbitrary vector ®elds and since the ideal is closed, the Lie derivatives of forms dra ; dXai ; dXa with respect to the vector ®eld given by Eq. (3.18) are already in the ideal I. Thus the last step to reach our goal is to impose certain conditions on (3.18) so that the forms LV xa remain in the ideal I. These conditions can be expressed as follows LV xa T a l ÿ X i dRa ^ li d S ai li ÿ X j dRai ^ lji Ra X i li T a l dS ai Ra dX i ^ li ÿ dX j ^ dRai ^lji mab xb rb ^ Aab drb ^ Bab Xbi ^ Cbia dXbi ^ Dabi Xb ^ Cba dXb ^ Dab ;
3:19
where we have employed Eq. (2.6). Unknown coecients are of course exterior forms of appropriate degrees, namely, mab 2 K0 K;
a Aab ; Cbi ; Cba 2 Knÿ1 K;
Bab ; Dabi ; Dab 2 Knÿ2 K:
To evaluate the expressions (3.19) we ®rst have to calculate the dierentials dX i and dS ai appearing there taking into account relations (3.7a) and (3.13). We then insert the de®nitions (2.8) into Eq. (3.19), replace the forms dua ; dRai and dRa by ra ; Xai and Xa , respectively, employ the identities (2.7) in the resulting expressions and ®nally collect the similar forms on both sides. We see then at once that we simply have to take Dabi 0;
Dab 0;
whereas the remaining terms can be arranged as follows h i bi bj a b ai ai a bi c b a a ci a a i C ÿ mb R si rc vi l r ^ Cb li ÿ Ab ÿ mc rb li X ^ Cbj li ÿ Cbj ÿ mb dj li
oS ai oX j ai oX j ai b oX i aik b a b c ÿ C l X ^ r ^ l r r ^ r ^ l ÿ s r ^ dvck ^ lji X ^ i ji ji b b b b c b c ou ou ou oR a cij dvbj ^ li dvbi ^ dxi ^ Bab 0 Caij 3:20 b ÿ mc sb b
aij ai where the functions Ca ; Cai b ; Cbj and Cb are de®ned by the relations: ai ai i i oS ai oS oS oX k a i oX i a b a a oX a oX a bj c C T i R R vi k db dj ÿ j db sbj i rc v i bj i b b ox ox ou ou ox ox oR
oS ai b oX j ai c b b c ai ai c b t s v si rai i i c c vi vj ÿ sj rc vj vi ; b b ou oR
3:21a
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E.S. Sß uhubi / International Journal of Engineering Science 37 (1999) 1901±1925
oS ai oX i b Ra b ou ou
Cai b
oS ai oX k a i oX i a cj oS ai c oX j ai c dd ÿ d r s ÿ s rai c vj oRcj oxk c j oxj c b oRc b oub j
oX i aj oX j ai c oX i aj c aj c s r v r v ÿ r v; c j oub j ouc b j ouc b j
oS ai oX k a i oX i a oX k c a i oX i c a d d ÿ d vk db dj ÿ c vj db ; ou oRbj oxk b j oxj b ouc
Cai bj Caij b
oS ai ovbj
oS ai oX l a i oX i a ckj oS ai cj oX k aij c oX i akj c dd ÿ d s c tb c sb vk ÿ c sb vk : oR ou ou oRck oxl c k oxk c b
3:21b
3:21c
3:21d
Comparing both sides of Eq. (3.20) we immediately deduce the expressions bi c a mab Rb sbi i rc m i C ;
Cba
oS ai li ; oRb
a a j Cbia Caj bi ÿ mb di lj db
a ci Aab Cai b ÿ mc rb li
oX j c r ^ lji ; ouc
oX j ai c oX j aik c r r ^ l ÿ s dvk ^ lji : ji ouc b oub c
On the other hand if we choose Bab Bajk b ljk ;
akj 0 Bajk b ÿBb 2 K K;
3:22
we notice that on using Eq. (2.7b) we obtaina aji aij b b b i aik dvbi ^ dxi ^ Bab Bajk b dvi ^ dx ^ ljk dvi ^ Bb lk ÿ Bb lj ÿ2Bb dvj ^ li :
Inserting this expression into Eq. (3.20) we ®nally get aij aij mac scij b Cb 2Bb :
The antisymmetric part of the foregoing expression determines Baij b since it is antisymmetric in indices i and j while its symmetric part leads to the relations c ij
mac sb
a ij
Cb :
Therefore the equations to be satis®ed by the generating functions for isovector components are reduced to bi c a mab Rb sbi i rc v i C
3:23a
E.S. Sß uhubi / International Journal of Engineering Science 37 (1999) 1901±1925 cji aij aji mac scij b sb Cb Cb
1913
3:23b
These equations involve the unknown functions mab in addition to the functions a X i xj ,ub ; Fai xj ; ub ; vbj ; Rbj ; Rb ; G xj ; ub ; vbj ; Rbj ; Rb and some of their ®rst order derivatives. It is straightforward to check that the number of unknowns is n N N 2 Nn while the number of equations are N 12 n 1nN 2 for N > 1. Therefore the maximal number of compatibility conditions which might be imposed are n ÿ 1 n 2N 2 ÿ n N 1: 2 If they admit nontrivial solutions, Eqs. (3.23) determine all independent isovector ®elds associated with the equivalence groups of balance equations (1.1). After having obtained all linearly independent isovector ®elds we can determine an equivalence group corresponding to a particular isovector ®eld by integrating the following system of ordinary dierential equations in the group parameter dxi X i xj ; ub ; d
d ua U a xj ; ub ; d
dR bj ; R b ; Fai xj ; ub ; vbj ; R d ai
d vai Vi a xj ; ub ; vbj ; d
dR ai bj ; R b ; G xj ; ub ; vbj ; R d a
under the initial conditions xi 0 xi ;
ua 0 ua ;
vai 0 vai ;
ai 0 Rai ; R
a 0 Ra : R
3.2. The case N 1 If there is a single dierential equation of the balance form then all Greek indices take just the value one and they will be omitted henceforth to simplify the notation. With this notational artifact in mind we can copy all equations related to the contact ideal until to the end of Eqs. (3.5). However, the reasoning which had worked quite well there fails in N 1 and from then on we have to pursue the analysis on a some what dierent path. The relations which we could borrow from Eqs. (3.4a), (3.4b) and (3.5) are summarised below Xi ÿ
oF oF oF i ÿ j sji ÿ t; ovi oR oR
oF Vi i ox
U F vi X i ;
F F xi ; u; vi ; Ri ; R;
oF oF i oF oF oF ir s vi j sji ti : ou oR oR oR oR
3:24
One must note here that sji , sji and ti , ti represent entirely dierent quantities and they should not be mistaken as associated tensor components. Similarly we have almost verbatim copies of the
1914
E.S. Sß uhubi / International Journal of Engineering Science 37 (1999) 1901±1925
relations (3.10), (3.11) and (3.14), (3.15) with F i being independent of ti , s and ti and G being independent of sji , ri and sji . However these properties imply through the relations (3.11) and (3.15) that X i ; U and Vi given by Eq. (3.24) cannot depend on sji , ti and sii , ti . Hence it follows that oF 0; oRi
oF 0 ) F F xi ; u; vi oR
3:25
and Eq. (3.24) reduces to Xi ÿ
oF ; ovi
U F ÿ vi
oF ; ovi
Vi
oF oF vi : oxi ou
3:26
In the same fashion Eqs. (3.11) and (3.15) yield F i ÿsij X j ÿ ri U ÿ sij Vj Fi xj ; u; vj ; Rj ; R; G ÿti X i ÿ sU ÿ ti Vi G xj ; u; vj ; Rj ; R; whereas the relations Eqs. (3.10) and (3.14) reduce now to S i Fi xj ; u; vj ; Rj ; R; i Sji Sj ÿ Si vj ; i
oF i Sj j ox S ij
oF i oF i k oF i oF i oF i kr s vj k skj tj ; ou oR oR oR oR
oF i oF i kj oF i j s t; ovj oRk oR
T G xj ; u; vj ; Rj ; R; Tj Ti ÿ Tvi ; oG Ti i ox Ti
oG oG j oG oG oG jr s vi j sji ti ; ou oR oR oR oR
oG oG ji oG i s t: ovi oRj oR
Hence an isovector ®eld of the contact ideal is expressible as
3:27
E.S. Sß uhubi / International Journal of Engineering Science 37 (1999) 1901±1925
o o o o o o o o i o U Vi Si i T Sj i S ij ij Ti T i i V1 V2 ; i ox ou ovi oR osj os oti ot oR
V Xi
1915
3:28
where the vector ®elds V1 and V2 are de®ned now as i
V1 S
! o o ÿ vj i ; ori osj
o o V2 T : ÿ vi os oti
Again it is straightforward to see V1 and V2 are trivial isovector ®elds which can be discarded without loss of generality. The components of (3.28) are entirely determined in terms of three functions F xi ; u; vi , Fi xj ; u; vj ; Rj ; R, G xj ; u; vj ; Rj ; R via the relations (3.26) and (3.27). In order that the ideal I x; r; Xi ; X; dr; dXi ; dX remain invariant under the vector ®eld (3.28) we require that the following relation, which can be deduced directly from Eq. (3.19), should also be satis®ed LV x T l ÿ X i dR ^ li d S i li ÿ X j dRi ^ lji RX i li T l dS i RdX i ^ li ÿ dX j ^ dRi ^ lji mx r ^ A dr ^ B Xi ^ Ci X ^ C; where m and A; Ci ; C and B are arbitrary forms of degrees 0 and n ÿ 2 and n ÿ 1, respectively. The forms Di and D which should appear in the foregoing expressions as coecients of 2-forms dXi and dX are taken as zero from the start without loss of generality. The functions X i depend, however, also on vj . Proceeding in exactly the same fashion as we have done previously we obtain the following expression C ÿ m R sii ri vi l r ^ Ci li ÿ A ÿ mri li Xj ^ Cij li ÿ Cj ÿ ldij li i j oS oX j i oX i oX j ik X^ r ÿ l ÿC X ^ r ^ lji s r ^ dvk ^ lji oR i ou ovk ou
oX j i X ^ dvk ^ lji Cij ÿ msij dvj ^ li dvi ^ dxi ^ B Cijkl dvl ^ dvk ^ lji 0; ovk
3:29
where the functions C, Ci ; Cij and Cij are now de®ned by the relations oS i oX i CT i R i ox ox
i oS i oX i oS oX k i oX i oS i j j s R vi si svi d ÿ r v i i ou ou oR oRj oxk j oxj
oX j i s vj ÿ sij vi ; ou i
oS i oX i R C ou ou i
oS i oX k i oX i j oS i oX j i oX i j r s ÿ s s; d ÿ oR ou j ou j oRj oxk j oxj
3:30
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E.S. Sß uhubi / International Journal of Engineering Science 37 (1999) 1901±1925
Cij
oS i oX i jÿ j ox oR
oX k oX k oX i i d ÿ v vj ; k j oxk ou ou
i oS i oX i oS oX l i oX i kj oS i j oX k i oX i k i s R d ÿ ÿ s r v s rk vk t C k ovj ovj oR ovj k ovj k oRk oxl k oxk k oX ij oX i kj s ÿ s vk ; ou ou ij
C
ijkl
ÿC
jikl
ÿC
ijlk
j i i 1 il oX j ik oX jk oX jl oX ÿs : s ÿs s ovk ovl ovl ovk 4
It is clear that these expressions cannot be obtained directly from (3.21) since X i depends also on vj . We further take B Bjk ljk Bjklm dvm ^ ljkl ;
Bjk ÿBkj ;
Bjklm Bjklm 2 K0 K
from which we simply obtain ÿdvi ^ dxi ^ B 2Bij dvj ^ li 3Bijlk dvl ^ dvk ^ lji : On comparing both sides of Eq. (3.29) we get j oX i oX j ik i i A C ÿ mr li dvk ^ lji ; r ÿ s ovk ou C
oS i l oR i
Ci Cji ÿ mdji lj
oX j oX j r ^ lji dvk ^ lji ou ovk
and m R sii ri vi C;
3:31a
ms ij C ij
3:31b
) m sij sji Cij Cji ;
Ci jkl 0 ) Cijkl Cikjl 0:
3:31c
Eqs. (3.31) are determining equations for the function m and the functions F ; Fi and G through which the components of isovector ®elds of equivalence associated with the balance equation
E.S. Sß uhubi / International Journal of Engineering Science 37 (1999) 1901±1925
oRi R 0; oxi
fRi ; R Ri ; R xj ; u; vj g;
i 1; 2; . . . ; n;
vi u;i ;
1917
3:32
may be found. Eqs. (3.23b) and (3.31b) generate chains of equations to determine the desired function since these functions depend on restricted sets of variables of the underlying manifold K. The forms of these equations are heavily aected by the numbers n and N. It seems that it is plausible to ®nd the general solutions of equations (3.23) or (3.31). This will be the subject of a subsequent paper. If some of the arguments of Rai , Ra or Ra , R are missing, isovector components corresponding to these arguments should consequently vanish. Hence additional conditions should be imposed on the general solutions which might change completely their structure. Even if the explicit general solutions cannot be provided, we believe that the work presented here builds quite an ecient shortcut to reach directly to the determining equations for isovector ®elds. These equations may easily be generated by a personal computer and sometimes even solutions may be handled by a computer employing, for instance, Mathematica. Since the cumbersome auxiliary calculations to be carried out in the classical approach to ®nd the determining equations are entirely removed in the present scheme memory requirement for computer is relaxed to a great extent and computation time may be shortened a great deal.
4. Applications 4.1. One-dimensional nonlinear wave equation To illustrate the advantage of the general approach we have developed so far we shall try ®rst to derive the determining equations for the equivalence groups associated with the following single nonlinear wave equation utt f x; t; u; ux ; ut x g x; t; u; ux ; ut :
4:1
To simplify the notation we de®ne x1 x; R1 f ;
x2 t;
R2 ÿv;
s11 fx f1 ; s21 0;
r2 0;
r1 fu f3 ; s21 0;
t2 gt g2 ;
X2 T;
v2 ut v;
R g;
s12 ft f2 ;
s22 0;
t1 gx g1 ; X1 X;
v1 ux p;
S1 F ;
s11 fp f4 ;
s12 fv f5 ;
s22 ÿ1;
s gu g3 ;
t 1 gp g4 ;
S 2 ÿV2 ÿV ;
T G:
t2 gv g5 ;
1918
E.S. Sß uhubi / International Journal of Engineering Science 37 (1999) 1901±1925
It then follows readily from (3.30) that C G Fx ÿ Vt g Xx Tt pXu vTu pFu ÿ vVu Ff Tt vTu f1 p Ff Tt ÿ vTx f3 ÿ Tx pTu f2 Fg g1 g3 p; C11 Fp gXp Ff Tt vTu f4 Fg g1 ÿ Tp f2 f3 v; C22 ÿXx ÿ Vv ÿ pXu Tv g f1 f3 p ÿ Tx pTu f5 ;
4:2
C12 Xt gXv vXu Fv ÿ Tv f2 f3 v Ff Tt vTu f5 Fg g5 ; C21 ÿVp Tp g f1 f3 p ÿ Tx pTu f4 : Thus (3.31) take simply the following forms: g f1 f3 pm ÿ C 0;
4:3a
g f1 f3 pC11 ÿ f4 C 0;
4:3b
g f1 f3 pC22 C 0;
4:3c
g f1 f3 p C12 C21 ÿ f5 C 0;
4:3d
f5 Tp Xp ÿ f4 Tv 0
4:3e
If we eliminate C between Eqs. (4.3c) and (4.3d) we obtain on noting that g f1 f3 p 6 0 C12 C21 f5 C22 0:
4:4
When we employ (4.2) in (4.4) and arrange the resulting expression we get Xt gXv vXu gTp ÿ Vp Fv Ff Tt ÿ Xx ÿ Vv vTu gTv ÿ pXu f5 Fg g5 ÿ Tv f2 Tp f1 pTp ÿ vTv f3 ÿ Tx pTu f4 Tv f1 f3 pf5 ÿ Tx pTu f52 0 which leads to the following set of equations Fg 0;
Tv 0;
Tp 0;
Xt gXv vXu ÿ Vp Fv 0;
Tx 0;
Tu 0;
4:5a 4:5b
E.S. Sß uhubi / International Journal of Engineering Science 37 (1999) 1901±1925
Ff Tt ÿ Xx ÿ Vv ÿ pXu 0:
1919
4:5c
Using these results in Eq. (4.2) we reduce those expressions to C11 Fp gXp Ff Tt f4 ;
C11 ÿXx ÿ Vv ÿ pXu ;
C G Fx ÿ Vt g Xx Tt pXu pFu ÿ vVu Ff Tt f1 f3 p: Then Eqs. (4.3b) and (4.3c) yield Fp gXp Ff Tt ÿ Xx ÿ Vv ÿ pXu f4 0; f1 f3 p Ff Tt ÿ Xx ÿ Vv ÿ pXu G Fx ÿ Vt pFu ÿ vVu g Tt ÿ Vv 0: Recalling that F and X are independent of g and noting Eq. (4.5c) we ®nally end up with Xp 0;
Fp 0;
G ÿFx Vt ÿ pFu vVu g Vv ÿ Tt :
4:6
Eqs. (4.5a), (4.5b), (4.5c), (4.6) are determining equations of the equivalence groups associated with Eq. (4.1). They are exactly the same as those found earlier [5]. Their solutions were provided in Ref. [5]. 4.2. Homogeneous hyperelastictiy As a second application to the general theory corresponding to a system involving more than one dependent variable we consider a homogeneous hyperelastic solid whose motion is described by the functions xk xk XK ; t, k 1,2,3; K 1,2,3 where XK , xk and t represent, respectively, material coordinates, spatial coordinates and the time, FkK X; t oxk =oXK are deformation ~ gradients, vk X; t oxk =ot are velocity components, R R F R C is the stress potential T (strain energy function), C X; t F F is the Green deformation tensor. The well-known ®eld equations of hyperelasticity in material description without body forces are given below in a nondimensional form o oR ovk ÿ 0: 4:7 ot oXK oFkK We replace for convenience our usual variables xi i 1; 2; 3; 4 by fXK K 1; 2; 3; tg and ua a 1; 2; 3 by xk k 1; 2; 3. We thus de®ne RkK
oR The first Piola-Kirchhoff stress tensor; oFkK
Rk4 ÿvk ;
Ra 0;
vkK FkK ;
vk4 vk ;
skKlL
oRkK ; oFlL
sk4l4 ÿ
ovk ÿdkl : ovl
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E.S. Sß uhubi / International Journal of Engineering Science 37 (1999) 1901±1925
Coordinate cover of the extended manifold K is reduced now to fXK ; t; xk ; FkK ; vk ; RkK ; skKlL g. Isovector ®eld in the tangent space of K may be conveniently expressed as V ÿUK
o o o o o o o ÿ w Uk VkK Vk SkK SkKlL : oXK ot oxk oFkK ovk oRkK oskKlL
It is straightforward to see that relations (3.7) yield in the present case UK UK X; t; x; Vk
w w X; t; x;
Uk Uk X; t; x;
4:8a
oUk oUK ow oUk oUK ow vl FkK vl vk vl ; FkK vk ot ot oxl oxl ot oxl
VkK
oUk oUL ow oUk oUL ow FkL vk FlK FkL FlK vk FlK : oXK oXK oxl oxl oXK oxl
4:8b
4:8c
We can easily observe that we should now replace F ai by fGkK ; 0g. When then get GkK ÿskKlL VlL HkK XL ; t; xl ; FlL ; vl ; RlL ;
SkK HkK ;
Sk4 ÿVk :
ai However the vanishing of Sj and SkKl4 implies, respectively
oGkK 0; oxl
oGkK 0 oXL
oGkK 0 ot
and
oGkK 0: ovl
Hence we conclude that VkK VkK FlL ;
SkK HkK FlL ; RlL :
Thus the determining equations become: oVk oUK oUK oVk oVk ÿ FmK dkl ; vl 0; mkl ovl oXK oxm ot oxl
4:9
4:10a
oVk ow ow oUK oUK ÿ skLlK FmL vm dkl 0; oFlK ot oxm oXL oxm
4:10b
CkKlL CkLlK ÿ vkm smKlL smLlK 0;
4:10c
where we de®ne CkKlL
oHkK oHkK oUK oUK ÿ smMIL FnM dkm oFlL oRmM oXM oxn oUN oUN ow ow ÿ FnN vn dkm dKM : oXN oxn ot oxn
E.S. Sß uhubi / International Journal of Engineering Science 37 (1999) 1901±1925
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Inserting Eq. (4.8b) into Eq. (4.10b) we obtain w w t and oUK oUK vm dkl 0: 2 ot oxm Hence UK UK X. In the same fashion (4.10a)2 yields o2 Uk o 2 Uk o 2 Uk 2 v wd 2 vl vm 0; kl l ot2 otoxl oxl oxm whence we deduce that o2 Uk 0; ot2
2
o2 Uk 0; oxl oxm
o2 Uk 1 ÿ wd kl : otoxl 2
0 or The last expression above, in view of the ®rst one, gives w w a1 t2 2a2 t a3 : Therefore the function Uk are found as Uk X; t; x ÿ a1 t a2 xk Kkl Xxl xk Xt ck X: We thus obtain VkK Kkl;K xl xk;K t ck;K UL;K FkL ÿ a1 t a2 FkK Kkl FlK : On the other hand Eqs. (4.9) require that a1 0;
xk a k ;
Kkl akl ;
ck AkK XK Ak ;
UK BKL XL BK ;
where all coecients are constants. Finally Eq. (4.10c) leads to the relations oHkK oHkL smMlN RkKmM dLN RkLmM dKN 0; oFlL oFlK
4:11
where RkKmM
oHkK BKM dkm ÿ 3a2 dKM dkm ÿ dKM akm : oRmM
Since Eq. (4.11) depends linearly on smMlN , one immediately observes that it yields RkKmM 0 which can be integrated to HkK ÿBKM RkM 3a2 RkK akm RmK rmK F; where the functions rkK are to satisfy orkK orkL 0; oFlL oFlK
1922
E.S. Sß uhubi / International Journal of Engineering Science 37 (1999) 1901±1925
whose solution can be found as [12] 1 rkK eKLM elmn ckn FlL FmM eKLM cLkl FlM ckK ; 2 where e's are permutation symbols and c's are constants. Let us ®nally summarise the results concerning relevant components of the isovector ®eld: w 2a2 t a3 ;
UK BKL XL BK ;
Uk akl ÿ a2 dkl xl ak t AkK XK Ak ; VkK AkK ÿ a2 FkK BLK FkL akl FlK ; SkK ÿBKM RkM 3a2 RkK akm RmK rkK : Equivalence transformations are then obtained through equations dX x dF dt ÿw t; d t; x ; ÿU X; U X; V F d d d d dR F; S R; d
R RkK ;
V VkK ;
S SkK ;
under usual initial conditions. Transformation of the stress potential is governed by dFkK dR oR R kK : VkK F d oF kK d However, the stress potential is actually a function of the deformation tensor C so that the One can readily see that this is feasible C. function R C should be transformed to a function R only if we choose AkK 0;
akl a0 dkl ;
ckl c0 dkl ;
cLkl cL dkl ;
ckK 0:
5. Symmetry groups It might be quite instructive to verify that the determining equations for isovector ®elds of symmetry groups associated with balance equations are directly deducible from those corresponding to equivalence groups. To this end we should consider that Rai and Ra and their derivatives with respect to their arguments are no longer independent variables. Therefore the only surviving isovector components will be X i ; U a and Vi a in (3.1). However, we have to note that the base vectors o=oxi ; o=oua and o=ovai in (3.1) were evaluated by keeping Rai and Ra constant. Hence
E.S. Sß uhubi / International Journal of Engineering Science 37 (1999) 1901±1925
1923
they should now be replaced by total derivatives by taking into account the functional forms of Rai and Ra . It is quite clear that we have now to write o o oRaj o oRa o ÿ ÿ oxi oRaj oxi oRa oxi Rai ;Ra oxi and similar expressions for the derivatives with respect to ua and vai . Consequently the isovector ®eld is now expressible as o oRaj i oRbj a oRbj a o a o a o bj V X iU Vi S ÿ i X ÿ a U ÿ a Vi a a ox ou ovi ox ou ovi oRbj oRb i oRb a oRb a o b T ÿ i X ÿ a U ÿ a Vi : ox ou ovi oRa i
Since the components of the isovector ®eld on the base vectors o=oRbj and o=oRb should now vanish we conclude that S ai V Rai ;
T a V Ra ;
5:1
where for a smooth function f xi ; ua ; vai we de®ne of i of a of a X a U a Vi : oxi ou ovi
V f
Inserting (5.1) and (2.2) into 3:211 and 3:214 and the resulting expressions into (3.23) we obtain: mab
mac
oRbi oRbi R i c vci ox ou b
oRci ovbj
oRcj
!
ovbi
aij b c b Aa Aai b vi Abc vi vj ;
aji akij akji Aaij A vck ; A ÿ 2 A b b bc bc
5:2
where we de®ned: Aa V Ra Aai b 2Aaij bc
i oV Rai oX k oRai oX i oRaj a oX R ÿ j ; oxi oxk oxi ox oxi oxi
i oV Rai oX k oRai oX i oRaj oX i oRaj oX j oRai a oX R ÿ j ÿ b ; oub oxk oub ox oub oub oxj ou oxj oub
oX i oRai oX j oRai ÿ ; ouc oub oub ouc
5:3
1924
E.S. Sß uhubi / International Journal of Engineering Science 37 (1999) 1901±1925
Aaij b 2Aakij b
oV Rai oX k oRai oX i oRak k ÿ k ; ox ovbj ox ovbj ovbj oX i oRak oX k oRai ÿ c : ouc ovbj ou ovbj
The relations (5.2) and (5.3) are identical with those given in Ref. [11]. In the same fashion we can handle the case N 1 and ®nd that the relations (3.31) are cast into the form: oRi oRi vi A Ai vi ; m R i ox ou
oRi oRj m ovj ovi
ÿ Aij Aji ÿ 2 Bkij Bkji vk ;
5:4
C ijkl C ikjl 0 where A V R
oV Ri oX i oX j oRi oX i oRi R j ÿ ; oxi ox oxi oxj oxi oxi
Ai
oV Ri oX i oX j oRi oX i oRj oX i oRi oX j oRi j ÿ j ÿ ; R ou ox ou ox ou ou oxj ou oxj ou
Aij
oV Ri oX i oX k oRi oX i oRk oX i oRk oX k oRi R k ÿ ÿ ; ovj ox ovj oxk ovj ovj oxk ovj oxk ovj
2Bkij
oX i oRk oX k oRi oX k oRi oX i oRk ÿ ÿ ; ou ovj ou ovj ovj ou ovj ou
4C ijkl
oRi oX j oRi oX j oRj oX i oRj oX i ÿ ÿ : ovl ovk ovk ovl ovk ovl ovl ovk
5:5
For a function f xi ; u; vi the function V f is de®ned as V f
of i of of X U Vi : oxi ou ovi
One easily veri®es that Eqs. (5.4) and (5.5) agree completely with those given in Ref. [11]. Acknowledgements The author gratefully acknowledges the partial support provided by Turkish Academy of Sciences.
E.S. Sß uhubi / International Journal of Engineering Science 37 (1999) 1901±1925
1925
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12]
L.V. Ovsiannikov, Group Analysis of Dierential Equations, Academic Press, New York, 1982. N.H. Ibragimov, M. Torrisi, A. Valenti, J. Math Phys. 32 (1991) 2988. N.H. Ibragimov, M. Torrisi, J. Math Phys. 33 (1992) 3931. N.H. Ibragimov, M. Torrisi, J. Math Anal. Appls. 184 (1994) 441. E.S. S ß uhubi, ARI (Formerly Bull Tech. Univ. Istanbul) 50 (1997) 151. N.H. Ibragimov (Ed.), CRC handbook of Lie group analysis of dierential equations, vol. I, Symmetries, Exact Solutions and Conservation Laws. CRC Press, Boca Raton, 1994. B.K. Harrison, F.B. Estabrook, J. Math Phys. 12 (1971) 653. E. Cartan, Les systemes dierentiel exterieurs et leurs applications geometriques. Hermann, Paris, 1945. D.G.B. Edelen, Isovector Methods for Equations of Balance. Sijtho Noordho, Alphen aan den Rijn, 1980. D.G.B. Edelen, Applied Exterior Calculus. Wiley, New York, 1985. E.S. S ß uhubi, Int. J. Engng. Sci. 29 (1991) 133. E.S. S ß uhubi, Int. J. Engng. Sci. 27 (1989) 441.