Energy balance invariance for interacting particle systems

Transcription

1 Z. ngew. Mh. Phys. 60 (009) /09/ DOI 0.007/s c 008 Birkhäuser Verlg, Bsel Zeischrif für ngewne Mhemik un Physik ZAMP Energy blnce invrince for inercing pricle sysems Arsh Yvri n Jerrol E. Mrsen Absrc. This pper suies he principle of invrince of blnce of energy n is consequences for sysem of inercing pricles uner groups of rnsformions. Blnce of energy n is invrince is firs exmine in Euclien spce. Unlike he cse of coninuous mei, i is shown h conservion n blnce lws o no follow from he ssumpion of invrince of blnce of energy uner ime-epenen isomeries of he mbien spce. However, he posule of invrince of blnce of energy uner rbirry iffeomorphisms of he mbien (Euclien) spce, oes yiel he correc conservion n blnce lws. These ies re hen exene o he cse when he mbien spce is Riemnnin mnifol. Pirwise inercions in he cse of geoesiclly complee Riemnnin mbien mnifols re efine by ssuming h he inercion poenil explicily epens on he pirwise isnces of pricles. Posuling blnce of energy n is invrince uner rbirry ime-epenen spil iffeomorphisms yiels blnce of liner momenum. I is seen h pirwise forces re irece long ngens o geoesics heir en poins. One lso obins iscree version of he Doyle Ericksen formul, which reles he mgniue of inernl forces o he re of chnge of he ineromic energy wih respec o iscree meric h is rele o he bckgroun meric. Keywors. Coninuum mechnics, pricle mechnics, energy blnce, covrince.. Inroucion This pper is concerne wih blnce of energy for sysem of inercing pricles n fins connecion beween iscree blnce lws n invrince, somehing h hs been suie horoughly in he seing of coninuum mechnics (see [4, 9, 6] n references herein). Pricle mechnics is normlly formule in R 3 vi blnce lws such s blnce of momenum; however, he liner srucure of R 3 cn someimes obscure imporn geomeric informion; for exmple, blnce of liner momenum is no covrin noion in h i looks rher ifferen in curviliner coorines; see [9, 6]. Of course one wy o overcome his is o mke use of generlize Deice o he memory of Shhrm Kvinpour ( ). Reserch suppore by he Georgi Insiue of Technology. Reserch prilly suppore by he Cliforni Insiue of Technology n NSF-ITR Grn ACI

2 74 A. Yvri n J. E. Mrsen ZAMP coorines n Lgrngin mechnics. However, noher ineresing lernive pproch, h is lso use in geomeric coninuum mechnics, is o mke use of invrince properies of blnce of energy. A mjor gol of his pper is o revisi his issue o bring he heory more ino line wih wh one oes in geomeric coninuum mechnics. In geomeric coninuum mechnics, one usully works wih wo configurions reference configurion n curren configurion. The configurion spce is he mnifol of mps beween hese wo configurions [9]. The curren configurion hs cler physicl inerpreion; i is wh one cn see in he lborory. Equilibri of he curren configurion in he seing of coninuum mechnics correspon o locl minim of he corresponing pricle sysem in he cse of qusisic eformions. In he corresponing pricle sysem he resuln force on ech pricle is blnce by is inernl forces. In geomeric coninuum mechnics, he curren configurion evolves in Riemnninn mnifol (S, g) [9, 6]. The reference configurion on he oher hn hs less cler physicl inerpreion. In he riionl remens of elsiciy i is usully ssume h here is well-efine sress-free reference configurion. This is no lwys rue s ws noice by Eckr few eces go [, ]. For n elsic boy, in generl, one cn hve se of nurl configurions. So, sress-free reference configurion coul be ny of hese nurl configurions. In he corresponing omic sysem nurl configurion woul be locl minimum of he energy (or free energy) in he bsence of exernl forces. I shoul be noe h reference configurion is in some sense rbirry n one cn choose ifferen reference configurions for he sme problem (see [7] for some iscussions on his). We shoul emphsize h we re no criicizing he exising remens of pricle mechnics in Euclien spce. As mer of fc, his woul be he nurl wy of formuling, for exmple, moleculr sysems h re embee in Euclien spce. However, one shoul noe h even for he simple exmple for he clssicl rigi boy in roion bou is cener of mss, he configurion spce is mnifol, i.e. SO(3) n he kineic energy meric is non-euclien. The oher hing o noe is h suying mechnicl sysems geomericlly cn give nonrivil insigh ino mechnics of pricles in Euclien spce. A goo exmple of his is he reucion proceure for suying he ynmics of he shpe spce of molecule; h is, he spce obine when one elimines rnslions n roions. When his is one, one obins ynmics on non-euclien spce n geomeric mehos re criicl when unerking such suy. There re mny exmples of his in he lierure in he works of, for exmple, Lilejohn n Iwi. For concree exmple pplie o he nlysis of conformion chnges in Argon-6, see [5] n references herein. This pper is srucure s follows. In we briefly review he clssicl heorem of Green, Nghi n Rivlin n he covrince ies in elsiciy. In 3 we sr wih blnce of energy for sysem of inercing pricles embee in Euclien spce n suy he consequences of posuling is invrince uner ifferen

3 Vol. 60 (009) Energy blnce invrince for inercing pricle sysems 75 groups of rnsformions of he mbien spce. 4 suies hese issues when he mbien spce is Riemnnin mnifol. Consequences of covrince of energy blnce re invesige in eil. Conclusions re given in 5.. Energy blnce, he Green Nghi Rivlin heorem n covrince in elsiciy In every mechnicl sysem here re blnce lws n some ssocie conserve quniies. For exmple, in elsiciy, one hs blnce of liner n ngulr momen n conservion of mss. One cn buil coninuum heory by posuling he relevn blnce lws. However, one cn lwys quesion he significnce of ech blnce lw n wheher hey hve ny inrinsic menings. Mos engineering heories, incluing he heory of elsiciy, re riionlly buil wih he implici ssumpion h he mbien spces re Euclien. I urns ou h he srucure of Euclien spce cn obscure he covrince of blnce lws since hese blnce lws look rher ifferen in curviliner coorines. A more nurl wy of builing fiel heories is o ssume h he mbien spces re mnifols. This wy, for exmple, one nurlly obins covrin heory n, in iion, one cn more esily sepre he meric epenen n inepenen relions. I hs long been known h here is eep connecion beween blnce lws n symmeries (see, for insnce, [0]). Noeher s heorem, for exmple, ienifies conserve quniy for ech locl symmery of he unerlying Lgrngin ensiy. In he cse of globl symmeries in coninuum mechnics Green n Rivlin [4] showe h posuling blnce of energy n is invrince uner spil isomeries of he Euclien mbien spce, one cn obin conservion of mss n blnces of liner n ngulr momen. Moive by his observion n he fc h blnce of energy cn lwys be inrinsiclly efine Mrsen n Hughes [9] evelope covrin heory of elsiciy by posuling blnce of energy n is spil covrince. This ssumpion resuls in conservion of mss, blnce of liner n ngulr momen n he Doyle Ericksen formul. Simo n Mrsen [4] erive meril version of Doyle Ericksen formul in erms of he roe sress ensor. Recenly Yvri e l. [6] suie he covrince conceps in elsiciy in some eil, possibiliy of meril covrince of energy blnce n he connecion beween covrince n Noeher s heorem. In he cse of linerize elsiciy, recenly i ws shown h covrince of linerize energy blnce cn give ll he fiel equions of linerize elsiciy [9]. See lso [8] for connecion beween energy blnce of iscreize soli n is blnce lws. To he bes of our knowlege, here is no sysemic suy of possible connecion beween invrince n blnce lws for pricle sysems. In his pper, we firs consier he cse in which he mbien spce is Euclien n show h here re some suble ifferences beween coninuum elsiciy n sysem of

4 76 A. Yvri n J. E. Mrsen ZAMP inercing pricles. In priculr, we will show h he Green Nghi Rivlin (GNR) Theorem fils for pricle sysems. We hen efine pirwise inercions when he mbien spce is Riemnnin mnifol n suy he consequences of spil covrince of energy blnce. We will lso obin iscree version of he Doyle Ericksen formul. 3. Blnce of energy for pricle sysems in Euclien spce We firs consier sysem of inercing pricles in Euclien mbien spce n show h, unlike clssicl elsiciy, blnce lws o no follow from invrince of energy blnce uner ime-epenen isomeries of he Euclien mbien spce. In oher wors, he GNR Theorem fils. We hen show h moifie GNR rgumen using n enlrge group of rnsformions works n oes give he blnce lws for he pricle sysem. The filure of he GNR heorem Here, we look blnce of energy for n rbirry collecion of oms L n mke connecion beween invrince of blnce of energy for sysem of pricles n he clssicl Green Nghi Rivlin Theorem for coninuous mei. For he ske of simpliciy, we resric ourselves o pirwise inercions. Suppose we re given collecion L of pricles which hs he configurion {x i ()} R n ime. Blnce of energy for L cn be wrien s φ ij ( x i x j ) + m iẋ i ẋ i = F i ẋ i, (3.) where is he snr inner prouc of R n, m i is he mss of pricle i n F i is he exernl force on pricle i. Blnce of energy cn be simplifie o re f ij (v i v j ) + m i v i i = F i v i, (3.) where v i = ẋ i, i = ẍ i, f ij = φ ( ij x i x j ) x i x j x i x j x i x j, (3.3) n we hve use he snr frmework of clssicl mechnics in which he pricle msses re ssume o be ime inepenen. Noe h for pirwise inercions f ji = f ij. (3.4) Le us now posule h blnce of energy for L is invrin uner imeepenen rigi rnslion of he mbien spce S = R 3. Consier ξ : S S,

5 Vol. 60 (009) Energy blnce invrince for inercing pricle sysems 77 where x i = ξ (x i ) = x i + ( 0 )w, (3.5) for some rbirry w R 3. This implies h v i = v i + w for ll i L. (3.6) Also ime = 0 i is ssume h he boy forces F i rnsform s follows (see [9]): m i i F i = m i i F i for ll i L. (3.7) Invrince of blnce of energy mens h φ ( x i x j ) + m iv i v i = F i x i, (3.8) where he prime quniies re rele o he unprime ones hrough Crn s clssicl spceime heory. Simplifying (3.8) n evluing i ime = 0, we obin f ij (v i v j ) + m i (v i + w) i = F i (v i + w). (3.9) Subrcing (3.) from (3.9) yiels m i w i = F i w. (3.0) Becuse w is rbirry one cn conclue h m i i. (3.) F i = Eq. (3.) is nohing bu Newon s secon lw for he collecion of pricles. I is seen h he bove posule oes no give he known governing equions for ech pricle. Inse, i gives blnce of ol liner momenum for he whole collecion of pricles. Le us now look (3.) n ry o rewrie i for n rbirry subse M L. This is of course no lwys possible becuse in he collecion L, ech pricle inercs wih ll he oher pricles n blnce of energy for subcollecion cnno be wrien unmbiguously, in generl. Noe lso h energy my no even be conserve loclly []. In oher wors, for nonlocl sysems loclize blnce lw, in generl, involves so-clle resiul erm. In he cse of pirwise inercions, one my be empe o wrie he blnce

6 78 A. Yvri n J. E. Mrsen ZAMP of energy for subcollecion M L s follows. { φ ij ( x i x j ) + i,j M j L\M + φ ij( x i x j ) } m iẋ i ẋ i = F i ẋ i, (3.) where we hve ssume h he energy of he bon beween he pricles i n j is eqully shre by hem in he cse of pirwise inercions. Noe lso h unlike clssicl coninuum mechnics energy ensiy is no one-poin funcion, i.e., for given pricle energy hs conribuions from ll he oher pricles in he collecion, in generl. Blnce of energy (3.) cn be simplifie o re i,j M f ij (v i v j ) + j L\M f ij (v i v j ) + m i v i i = F i ẋ i. (3.3) Now le us posule h his blnce lw is invrin uner n rbirry imeepenen rigi rnslion of he mbien spce, i.e. i,j M f ij (v i v j )+ j L\M f ij (v i v j )+ m iv i i = F i x i. (3.4) Subrcing (3.3) from (3.4) evlue = 0 yiels f ij (w i w j )+ f ij (w i w j )+ m i w i i = F i w i, i,j M j L\M (3.5) where w i = ξ (x i ). Now if ξ is rigi rnslion, hen w i = c, for ll i L n hence m i c i = F i c. (3.6) Becuse c is rbirry, his hen implies h F i = m i i. (3.7) M is rbirry so we cn choose M = {i} n hence F i = m i i. (3.8) Noe h φij ( x i x j ) is he energy of he pir (i, j). There is fcor in he firs erm of he lef-hn sie of (3.) becuse in he sum boh (i, j) n (j, i) re coune. On he oher hn, he fcor ppers in he secon erm becuse i is ssume h he energy φ ij( x i x j ) is shre eqully beween he oms i n j.

7 Vol. 60 (009) Energy blnce invrince for inercing pricle sysems 79 This mens h he ineromic forces re self-equilibre, i.e. f ij = 0, (3.9) j L which is no lwys rue! This shows h he version of energy blnce given by (3.) is no invrin uner ime-epenen rigi rnslions, in generl. This goes bck o he nonlocl nure of inercions in pricle sysem n lso he presence of self inercions in given subse. In oher wors, blnce of energy for he whole sysem shoul be consiere. See [5] n [6] for similr iscussions. In summry, he GNR Theorem fils for pricle sysem. Remrk. In nonlocl sysem, one cn unmbiguously wrie globl blnce of energy, e.g. (3.) for our pricle sysem, bu pssge from globl energy blnce o locl energy blnce or o blnce of energy for n rbirry meril subse my no be unique [, 3]. In wh follows from here on we only wrie globl blnce of energy n suy is invrince (covrince) properies. The success of moifie GNR heorem Le us now consier lrger group of rnsformions for energy blnce invrince. Inse of consiering blnce of energy for M L n is invrince uner rigi rnslions, le us look blnce of energy for L n consier n rbirry C iffeomorphism ξ : S S. Thus, inse of (3.0) we hve where Noe h Hence f ij (w i w j ) + m i w i i = F i w i, (3.0) w i = ξ (x i ). (3.) f ij (w i w j ) = f ij w i. (3.) j L f ij w i = (F i m i i ) w i. (3.3) j L This cn be rewrien s [ F i j L f ij + m i i ] w i = 0. (3.4)

8 730 A. Yvri n J. E. Mrsen ZAMP Now ssuming h ξ is such h he w j = 0, j i, one obins F i + j L f ij = m i i, (3.5) which is wh one expecs. Le us now consier ime-epenen rigi roion in he mbien spce, i.e. ξ (x i ) = e ( 0)Ω x i (3.6) for some skew-symmeric mrix Ω. This mens h w i = Ωx i for ll i L. Obviously, posuling invrince uner rigi roions of he mbien spce oes no give ny new blnce lws s rigi roion is jus specil cse of iffeomorphisms consiere bove in (3.0). In oher wors, blnce of ngulr momenum is rivilly sisfie for collecion of inercing pricles. In summry, we hve prove he following proposiion. Proposiion 3.. For collecion of pricles wih pirwise inercions, posuling blnce of energy n is invrince uner isomeries of he mbien Euclien spce is no enough o fin ll he blnce lws (equions of moion). If, however, one posules blnce of energy n is invrince uner rbirry iffeomorphisms of Euclien spce, hen hese equions re ll obine. Remrk. In his pper, we o no consier inepenen roions, n in generl microsrucure, for pricles. However, he rgumens presene here cn be exene o more complice pricle sysems, e.g. pricle sysems wih nisoropic inercions, ec. 4. Energy blnce for pricle sysems on Riemnnin mnifols To pu he previous resuls in more generl frmework n lso for he ske of clriy, compleeness, n inellecul sisfcion, le us ssume h he pricles move on Riemnnin mnifol. Alhough his my seem oo bsrc firs glnce, i is nurl o sk how he blnce lws look when he mbien spce is Riemnnin. This is lso moive by he previous covrin formulions of coninuum elsiciy. We shoul lso menion h here hve been recen effors in formuling mechnics on non-euclien spces, e.g. [, 3]. Suppose he inercing pricles lie in Riemnnnin mnifol (S, g), which we ssume is geoesiclly complee. For he ske of simpliciy, le us ssume h only wo-boy inercions re presen. Consier wo pricles i, j L, which in he curren configurion lie in S, i.e., x i, x j S. We ssume h he poenil energy of hese wo pricles explicily epens on heir relive isnce in he Riemnin mnifol. Of course, here my be oher possibiliies bu his ssumpion woul be goo sring poin. As physicl sysem represene by his moel, le us consier finie number of pricles lying on wo-mnifol M R 3 connece o ech oher by some

9 Vol. 60 (009) Energy blnce invrince for inercing pricle sysems 73 springs h re consrine o lie in M. Energy of his sysem is obviously n explici funcion of lenghs of he springs. Given ny wo pricles, energy of he corresponing spring is minimize when he spring coincies wih he geoesic connecing he wo poins on M. Wh we fin covrinly in he following is blnce of forces projece on ngen spces of M pricle posiions. As noher moivion for suying his moel, we shoul menion he recen work by Koni n Sun [7]. Koni n Sun [7] consier weighe finie grph X = (V, E), where V n E re he verex n ege ses, respecively, n consier piecewise smooh mp Φ from X o Riemnnin mnifol (Y, g). They enoe he resricion of his mp o e E by Φ e (), [0, ]. Then, hey efine energy of his mp s E(Φ) = e E m E (e) 0 Φe, Φ e, (4.) where m E is weigh funcion efine on E. They show h mp Φ is criicl mp for his energy if n only if Φ e is geoesic for every e E. For such mp, energy is n explici funcion of geoesic lenghs. Remrk. Noe h, for generic Riemnnin mnifol n wo rbirry poins on he mnifol, here my be more hn one n even infiniely mny isnce minimizing geoesics connecing he wo poins. An exmple woul be he norh n souh pole of sphere. Here, we ssume h in given configurion of he pricle sysem here re no pirs of pricles lying on such pir of poins. Le us enoe he geoesic connecing x i n x j by l ij g, where i is cler from his noion h he geoesic explicily epens on he meric g. This curve hs prmerizion l : [, b] S n is lengh is efine s L g (l ij g ) = s lij g (s), s lij g (s) g s, (4.) where.,. g is he inner prouc inuce from he Riemnnin meric g. Therefore, for x i, x j S ( φ ij = φ ij L(l ij g ) ). (4.3) The ol inercion energy is efine s e ( {x i }, g ) = j L φ ij ( Lg (l ij g ) ). (4.4) Noe h we cn hink of g ij := L g (l ij g ) s iscree meric for he collecion {x i } S. Of course, his iscree meric is n explici funcion of he bckgroun Riemnnin meric g. Now blnce of energy cn be wrien s φ ij ( Lg (l ij g ) ) + m i ẋi, ẋ i g = F i, ẋ i g. (4.5)

10 73 A. Yvri n J. E. Mrsen ZAMP Noe h F i T x is is he exernl force on om i. Figure. A spil chnge of frme for sysem of inercing pricles. Le us ssume h uner spil chnge of frme ξ : S S blnce of energy is invrin, i.e. φ ij ( L g (l g ij ) ) + ẋ i m i, ẋ i g = F i, ẋ i. (4.6) g Noe h g = ξ g (see Fig. ). We ssume h he pirwise poenil rnsforms onsorilly, i.e. ( x i, x j, g ) ( = φ ij x i, x j, ξ g ). (4.7) φ ij Equivlenly φ ij ( Lg (l ij g ) ) = φ ij ( L ξ g(l ij ξ g) ). (4.8)

11 Vol. 60 (009) Energy blnce invrince for inercing pricle sysems 733 Blnce of energy for he new frming = 0 res m ẋi i + w i, i + ( ) φ g ij L ξ g(l ij ξ g) = F i, ẋ i + w i, =0 g (4.9) where we ssume h F i m i i = ξ (F i m i i ) [9]. Now he nonrivil sk is o simplify he secon erm on he lef hn sie of Eq. (4.9). Noe h given wo poins in Riemnnin mnifol he geoesic n is lengh boh explicily epen on he meric g. Noe lso h ( ) φ ij L ξ g(l ij ξ g) =0 = φ ij g ij ( ) L ξ g(l ij ξ g). (4.0) =0 Before proceeing ny furher, le us firs simplify he blnce of energy for he originl frme, i.e., he lef-hn sie of Eq. (4.5) φ ij ( Lg (l ij g ) ) = φ ij g ij L g(l ij g ). (4.) Consier he geoesic joining he poins x i (), x j () S. As hese poins re ime epenen, he geoesic joining hem woul be ime epenen s well, i.e., l = l(, s) := l ij g (, s), where s is he curve prmeer n is ime. Noe h L g(l ij g ) = l s, l s s, (4.) where l s = sl(, s) is he velociy of he prmeerize geoesic. We ssume h he curve l is prmeerize by rc lengh, i.e., i hs uni spee everywhere. Thus l s, l s s = l s, l s s = l s D l s, l s s, (4.3) where D is he covrin erivive long he geoesic n for vecor fiel long he curve l i is efine s D V = l Ṽ, (4.4) where Ṽ is n exension of V o S, is he Riemnnin connecion corresponing o he meric g (see [8] for more eils), n l = l(, s). I is esy o show h [8] D l s = D s l (4.5) s in locl coorine chr {x } hey boh hve he following represenion ( x c D l s = D s l = s + x x b ) s γc b c, (4.6)

12 734 A. Yvri n J. E. Mrsen ZAMP where γb c re Chrisoffel coefficiens of he Riemnnin connecion. Using his propery n he fc h l s = we obin b ( ) L g(l ij g ) = D s l, l s s = s l, l s l, D s l s s. (4.7) Bu becuse l is geoesic we hve D s l s = 0 n hence b L g(l ij g ) = s l, l s s = l, l s b (4.8). Noe h l (, ) = ẋ i, l (b, ) = ẋ j (4.9) n le us enoe he velociy vecors of he geoesic poins x i n x j by l s (, ) = ˆ ij T x is, l s (b, ) = ˆ ji T x j S. (4.0) Therefore L g(l ij g ) = ẋ j,ˆ ji ẋ i,ˆ ij. (4.) Now blnce of energy cn be wrien s φ ij ( ẋj,ˆ ji ẋ i,ˆ ij ) + m i i, ẋ i = F i, ẋ i. (4.) g ij Noe h he pricle ccelerion i is he covrin ime erivive of ẋ i. Le us now look blnce of energy for chnge of frme ξ : S S such h ξ =0 = I. Consier fmily of geoesics l(, s) joining he poins x i () = ξ (x i ()), x j () = ξ (x j ()) S. Moive by (4.8), for fixe, l(, s) is he geoesic wih respec o he meric g = ξ g joining hese wo poins. Assume h he geoesic l(s) = l( 0, s) is prmeerize by rc lengh, i.e., i hs uni spee everywhere. Noe h D s ls (, s) = 0 is sisfie for ech geoesic, where D s is he covrin erivive long he geoesic l(, s) wih respec o he meric g. The ime re of chnge of he lengh of his fmily is simplifie o re L g ( l ij g ) = ls, l s s = l s (, s) D ls, l s s, (4.3) where D is he covrin ime erivive long he curve l(, s) wih respec o he meric g. Noe h sill we hve he relion D ls = D s l s in he locl represenion (4.6) he only ifference woul be he -epennce of he Chrisoffel symbols. Thus L g ( l ij g ) = = l s (, s) l s (, s) D s l, l s ( s l, l s s ) l, D s ls s. (4.4)

13 Vol. 60 (009) Energy blnce invrince for inercing pricle sysems 735 Using he geoesic equion n evluing he bove relion = 0, we obin L g ( l ij g ) = l, l b s. (4.5) =0 Noe h l ( 0, ) = ẋ i + w i, l ( 0, b) = ẋ j + w j, (4.6) where w = ξ. Also l s ( 0, ) = ˆ ij T x is, ls ( 0, b) = ˆ ji T x j S. (4.7) Thus L g ( l ij g ) = ẋ j + w j,ˆ ji ẋ i + w i,ˆ ij. (4.8) =0 Blnce of energy for he new frming ime = 0 res φ ij ( ẋj + w j,ˆ ji ẋ i + w i,ˆ ij ) + m i i, ẋ i + w i g ij = F i, ẋ i + w i. (4.9) Subrcing (4.) from he bove blnce equion n noing h he vecor fiel w is rbirry, we obin he following blnce lw j L φ ij g ij ˆ ij + F i = m i i for ll i L. (4.30) I is seen h ssuming h pricles re embee in Riemnnin mnifol n posuling energy blnce n is invrince uner rbirry spil iffeomorphisms resuls in blnce of liner momenum. In erms of he number of blnce lws, hving Riemnnin mbien spce oes no give us ny new relions. However, efining more generl pirwise inercion we see h inercion force i L ue o he pricle j L is irece long he ngen o he geoesic joining x i, x j S he poin x i n is mgniue is equl o he re of chnge of he poenil energy of i, j L wih respec o he iscree meric g ij. Thus, we cn hink of f ij = φ ij = e (4.3) g ij g ij s iscree Doyle Ericksen formul. Noe h becuse ˆ ij n ˆ ji lie in wo ifferen ngen spces, he relion f ji = f ij is meningless, in generl. However, f ij = f ji. In summry, we hve prove he following proposiion. Proposiion 4.. Assuming h blnce of energy for sysem of pirwisely inercing pricles in n mbien Riemnnin mnifol is spilly covrin is equivlen o blnce of liner momenum.

14 736 A. Yvri n J. E. Mrsen ZAMP The proof of converse of his proposiion is similr o h of he coninuum version [6]. Remrks. () From Eq. (4.30) i is seen h he bckgroun meric eners he blnce of liner momenum only hrough he iscree meric g ij. (b) Noe h Eq. (4.30) is he ynmic version of Eq. () in [7]. Exmple. We show in his exmple h wh we jus erive for generl Riemnnin mnifol is reuce o he clssicl resuls when he mbien spce is Euclien. In his cse he geoesic joining x i, x j S = R n hs he following prmerizion l(s) = bxi x j b + xj x i s s [, b]. (4.3) b Becuse he curve is prmeerize by he rc lengh we mus hve b = x j x i. Thus ˆ ij = xj x i x j x i. (4.33) Also in his cse g ij = r ij := x j x i. Hence he following clssicl blnce of liner momenum is recovere j L φ r ij x i x j x j x i + Fi = m i i for ll i L. (4.34) Noe h in his cse he relion f ji = f ij hols s ngen spce o R n ny poin cn be ienifie wih R n. Remrk. Blnce of liner momenum for pricle sysem on Riemnnin mnifol cn of course lso be obine using Hmilon s principle of les cion which is noher covrin pproch vi Lgrngin mechnics. 5. Concluing remrks This pper suie he connecion beween blnce lws n energy blnce invrince for sysem of inercing pricles. I ws shown h, unlike clssicl elsiciy, posuling invrince of energy blnce uner isomeries of he (Euclien) mbien spce is no enough o obin he blnce lws. Inse, if one posules invrince of energy blnce uner rbirry iffeomorphisms, hen one recovers ll he blnce lws. This shows funmenl ifference beween coninuum n sysem of inercing pricles n cn be ssocie wih he nonlocl nure of inercions in sysem of pricles. Blnce of energy for sysem of pricles embee in Riemnnin mnifol ws lso invesige vi generlize form of pirwise inercions by ssuming

15 Vol. 60 (009) Energy blnce invrince for inercing pricle sysems 737 h he pirwise poenil energy of pricles explicily epens on heir pirwise isnces he lenghs of he geoesics joining hem. This efiniion nurlly reuces o he clssicl noion of pirwise inercions in Euclien spce. Posuling blnce of energy n is spil covrince, shows h one cn obin geomeric version of blnce of liner momenum. For pricle i L, blnce of liner momenum is wrien in he ngen spce of S x i S. I ws observe h in his generl seing, relion like f ji = f ij woul be meningless bu inse one hs he meningful relion f ji = f ij. Defining iscree meric s he pirwise isnces, we showe h f ij = e g ij, (5.) which cn be hough of s iscree version of he Doyle Ericksen formul. References [] Eckr, C., Remrks concerning forces on line efecs, Physicl Review 73 (948), [] Eelen, D. G. B. n Lws, N., Thermoynmics of sysems wih nonlocliy, Archive for Rionl Mechnics n Anlysis 43() (97), [3] Eringen, A. C. n Eelen, D. G. B., Nonlocl elsiciy, Inernionl Journl of Engineering Science 0 (97), [4] Green, A. E. n Rivlin, R. S., On Cuchy s equions of moion, ZAMP 5 (964), [5] Gurin, M. E. n Willims, W. O., On he firs lw of hermoynmics, Archive for Rionl Mechnics n Anlysis 4 (97), [6] Gurin, M. E. n Willims, W. O., On coninuum hermoynmics wih muul boy forces n inernl relion, ZAMP (97), [7] Koni, M. n Sun, T., Snr relizions of crysl lices vi hrmonic mps, Trnscions of he Americn Mhemicl Sociey 353 (000), 0. [8] Lee, J. M., Riemnnin Mnifol An Inroucion o Curvure, Springer-Verlg, New York 997. [9] Mrsen, J. E. n T. J. R. Hughes, Mhemicl Founions of Elsiciy, Dover, New York 983. [0] Mrsen, J. E. n T. Riu, Inroucion o Mechnics n Symmery, Springer, New York 003. [] Rjgopl, K. R. n A. R. Srinivs, On he hermomechnics of merils h hve muliple nurl configurions Pr II: Twinning n soli o soli phse rnsformion, ZAMP 55(6) (984), [] Shchepeilov, A. V., Reucion of he wo-boy problem wih cenrl inercion on simply connece spces of consn secionl curvure. Journl of Physics A-Mhemicl n Generl 3 (998), [3] Shchepeilov, A. V., Clculus n Mechnics on Two-Poin Homogenous Riemnnin Spces (Lecure Noes in Physics), Springer, New York 983. [4] Simo, J. C. n J. E. Mrsen, On he roe sress ensor n he meril version of he Doyle Ericksen formul, Archive for Rionl Mechnics n Anlysis 86 (984), 3 3. [5] Yno, T., W.-S. Koon, J.E. Mrsen, n I. Kevrekiis, Gyrion-rius ynmics in srucurl rnsiions of omic clusers, Journl of Chemicl Physics 6 (007), 7. [6] Yvri, A., J. E. Mrsen n M. Oriz, On he spil n meril covrin blnce lws in elsiciy, Journl of Mhemicl Physics 47 (006), 85.

16 738 A. Yvri n J. E. Mrsen ZAMP [7] Yvri, A., M. Oriz n K. Bhchry, A heory of nhrmonic lice sics for nlysis of efecive crysls, Journl of Elsiciy 86 (007), [8] Yvri, A., On geomeric iscreizion of elsiciy, Journl of Mhemicl Physics 49 (008), 36. [9] Yvri, A. n A. Ozkin, Covrince in linerize elsiciy, Zeischrif für Angewne Mhemik un Physik (ZAMP) (008), DOI 0.007/s Arsh Yvri School of Civil n Environmenl Engineering Georgi Insiue of Technology Aln, GA 3033 USA e-mil: rsh.yvri@ce.gech.eu Jerrol E. Mrsen Conrol n Dynmicl Sysems Cliforni Insiue of Technology Psen, CA 95 USA (Receive: My 9, 008) Publishe Online Firs: Ocober 0, 008 To ccess his journl online: