Variational principles for relativistic smoothed particle hydrodynamics

Transcription

1 Mon. Not. R. Astron. So. 328, (2001) Vritionl priniples for reltivisti smoothed prtile hydrodynmis J. J. Monghn nd D. J. Prie P Epsilon Lortory, Deprtment of Mthemtis nd Sttistis, Monsh University, Clyton 3800, Austrli Aepted 2001 June 12. Reeived 2001 April 30 ABSTRACT In this pper we show how the equtions of motion for the smoothed prtile hydrodynmis (SPH) method my e derived from vritionl priniple for oth non-reltivisti nd reltivisti motion when there is no dissiption. Beuse the SPH density is funtion of the oordintes the derivtion of the equtions of motion through vritionl priniples is simpler thn in the ontinuum se where the density is defined through the ontinuity eqution. In prtiulr, the derivtion of the generl reltivisti equtions is more diret nd simpler thn tht of Fok. The symmetry properties of the Lgrngin led immeditely to the fmilir dditive onservtion lws of liner nd ngulr momentum nd energy. In ddition, we show tht there is n pproximtely onserved quntity whih, in the ontinuum limit, is the irultion. Key words: reltivity methods: numeril. 1 INTRODUCTION Approximte equtions of motion re etter if they inorporte importnt properties of the originl system (for disussion of this point for Hmiltonin systems see Slmon 1988). These importnt properties inlude the onservtion lws of momentum nd energy nd, for rytropi fluids with onservtive ody fores, the onservtion of irultion. Smoothed prtile hydrodynmis (SPH, for review see Monghn 1992) is n pproximtion to the ontinuum equtions of fluid dynmis tht n e written in form whih onserves liner nd ngulr momentum. If the externl fores re onservtive the energy is lso onserved. The SPH equtions therefore, lthough pproximte, retin these desirle fetures of the originl equtions. If the pressure is funtion of the density, nd there is no dissiption, then the fluid equtions hve nother invrint, the irultion. The irultion invrint is tully n infinite numer of invrints euse it onstrins the irultion round ny losed pth. In the fluid dynmis of either inompressile or rytropi fluids the irultion ples severe onstrint on the llowed motion. In strophysis the irultion is less importnt euse the pressure is usully not funtion of the density lone, nd dissiption results in hnges in the vortiity nd the irultion. However, there re exmples where the irultion is importnt. For exmple when the gs is isotherml or degenerte, or the motion is diti. It would therefore e desirle if form of SPH ould e found tht onserves irultion. However, euse SPH is prtile method it is not ler whether irultion theorem exists or to wht extent it is n pproximtion. For exmple, the fluid equtions re derived from lssil version of moleulr dynmis whih n in turn e derived from Lgrngin with the lssil dditive invrints of momentum nd energy. Where then is the irultion theorem hidden, or is it the result of the sttistil verging leding to the ontinuum equtions? In this pper we estlish the onservtion lws diretly from Lgrngin nd we show how the irultion theorem n e derived. The SPH Lgrngins for the reltivisti se n e esily estlished nd with them the equtions of motion nd the onservtion lws. In the se of generl reltivity the SPH Lgrngin leds to the equtions of motion in mnner whih is oth simpler nd more diret thn the lssi nlysis of Fok (1964). The SPH equtions of motion n lso e written in Hmiltonin form. Then, euse the phse spe of the SPH prtiles is finite the system stisfies Liouville s eqution nd the Poinré invrints. The extent to whih this Hmiltonin struture n e mde the sis of estimte of hos nd the sttistil equilirium of fluid system without dissiption is not known, ut it SPH would seem to provide trnsprent formlism for suh n investigtion. P E-mil: joe.monghn@si.monsh.edu.u (JJM); dprie@st.m..uk (DJP) q 2001 RAS

2 382 J. J. Monghn nd D. J. Prie 2 THE NON-RELATIVISTIC LAGRANGIAN 2.1 Equtions of motion The reder is ssumed to e fmilir with the si ides of SPH (see Monghn 1992 for review). The Lgrngin for non-reltivisti fluid dynmis, with self grvity n e sed on Ekrt s (1960) Lgrngin L ¼ r 1 2 v : v 2 ur; sþ dv; 2:1Þ where v is the veloity, r is the density, s is the entropy nd u(r, s) is the therml energy per unit mss. The integrtion is over the volume. In the SPH formlism the density n e written s funtion of the msses of the prtiles nd their oordintes. For prtile the density r is given y r ¼ mk Wjr 2 r k jþ; k 2:2Þ where m k is the mss of prtile k, r k is the oordinte vetor of prtile k nd W is smoothing kernel. The summtion is over ll prtiles lthough the kernel vnishes eyond speified distne nd only neighours ontriute. It is the ft tht the density n e defined s funtion of the oordintes, rther thn through the eqution of ontinuity, tht simplifies the derivtion of the equtions of motion from vritionl priniple. The SPH form of (2.1), generlized to inlude self grvity, is L ¼ m 2 v2 2 ur ; s Þ G 4 mk 5: 2:3Þ k jr 2 r k j with dr ¼ v : 2:4Þ Lgrnge s equtions of motion follow from vrying the tion keeping the entropy of eh prtile onstnt. Lgrnge s equtions for prtile re d L 2 L ¼ 0: 2:5Þ v The nonil momentum is p ¼ L v ¼ m v 2:6Þ nd L ¼ 2 m u r r 2 G s r 2 r Þ m jr 2 r j 3 : 2:7Þ From eqution (2.2) r ¼ W k mk d 2 d k Þ; 2:8Þ k nd from the first lw of thermodynmis u ¼ P r s r 2 ; where P is the pressure (whih n e lulted one the form of u(r, s) is given). Using these results Lgrnge s equtions for prtile n e written dv ¼ 2 m r 2 1 P r 2 r Þ r 2 7 W 2 G m jr 2 r j 3 ; 2:9Þ 2:10Þ where 7 denotes the grdient tken with respet to the oordintes of prtile. W denotes Wjr 2 r jþ. Eqution (2.10) is the SPH

3 equivlent of dv ¼ 2 P r 2 7r 2 7 P r Vritionl priniples for reltivisti SPH F; 2:11Þ ¼ 2 1 7P 2 7F; r 2:12Þ where F is the grvittionl potentil. These results show tht Lgrnge s equtions led to the stndrd non-dissiptive SPH equtions of fluid dynmis. If we hd hosen seprte resolution length h for eh prtile then the eqution of motion would hve een identil to tht ove exept tht the grdient of the kernel would hve een repled y 1 2 Wjr 2 r j; h Þ 1 Wjr 2 r j; h ÞÞ; 2:13Þ where the resolution lengths h nd h re shown expliitly. In prtie, the resolution length is required to hnge during the motion. For the present we ssume the resolution lengths re onstnt. The effet of hnges in h on the equtions of motion is usully smll. 2.2 Conservtion lws Additive integrls The symmetry of the Lgrngin leds immeditely to the onservtion lws. In prtiulr, in the present se where the entropy is onstnt, nd the summtion for the density is invrint to trnsltions nd rottions, liner nd ngulr momentum re onserved. For exmple, if eh prtile is given n ritrry infinitesiml trnsltion q, the hnge in L is L dl ¼ : q ¼ q : L ; 2:14Þ r r from whih, using Lgrnge s equtions, the totl liner momentum L ¼ m v ; v 2:15Þ is onserved. Other exmples re given y Lndu & Lifshitz (1976). The invrine of L to disrete shift in the time shows tht the energy E ¼ v : L 2 L 2:16Þ v 1 ¼ m 2 v2 1 u 1 F : 2:17Þ is onserved The irultion The prtile system is invrint to other trnsformtions. Consider, for exmple Fig. 1 whih shows set of prtiles eh with the sme mss Figure 1. A set of prtiles eh with the sme mss nd entropy nd mrked loop.

4 384 J. J. Monghn nd D. J. Prie nd entropy nd mrked loop. Imgine eh prtile in the loop eing shifted to its neighour s position (in the sme sense round the loop) nd given its neighour s veloity. Sine the entropy is onstnt, nothing hs hnged, nd the Lgrngin is therefore invrint to this trnsformtion. The hnges in L n e pproximted y L dl ¼ : dr 1 L : dv ; 2:18Þ r v where denotes the lel of prtile on the loop. The hnge in position nd veloity re given y dr ¼ r 11 2 r ; 2:19Þ nd dv ¼ v 11 2 v : Using Lgrnge s equtions (2.5) we n rewrite (2.18) in the form m dv : r 11 2 r Þ 1 v : v 11 2 v Þ ¼ 0; 2:20Þ 2:21Þ nd relling tht the prtile msses re ssumed identil, we dedue tht d v : r 11 2 r Þ¼0: 2:22Þ so tht C ¼ v : r 11 2 r Þ; 2:23Þ is onserved to this pproximtion, for every loop. The onservtion is only pproximte euse the hnge to the Lgrngin is disrete, nd only pproximted y the first order terms. However, if the prtiles re suffiiently lose together (2.23) pproximtes the irultion theorem to ritrry ury. A relted rgument ws used y Feynmn (1957) to estlish from the invrine of the wve funtion tht irultion should e quntized in quntum fluids. These results re mirrored in Slmon s (1988) nlysis of Lgrngin nd Hmiltonin methods in fluid mehnis. Slmon (1988), following Bretherton s (1970) work, estlishes the onservtion lws y ppeling to the invrine to prtile interhnge. However, euse their nlysis is within the ontext of the ontinuum, it is more omplited thn the derivtion given ove. The system is lso invrint to the prtiles shifting round the loop in the opposite sense. This gives n pproximtion to the irultion with the opposite sign to tht ove. If these two re omined (tking ount of their signs so we sutrt one from the other) we get d v : r 11 2 r 21 Þ ¼ 0: 2 2:24Þ whih is etter pproximtion to the irultion of the ontinuous fluid. The ury of the pproximte irultion invrint n e estimted esily for simple systems of prtiles. For exmple, if there re no fores the veloity is onstnt nd the rte of hnge of irultion from (2.21) is v : v 11 2 v 21 Þ; 2:25Þ whih vnishes on summing round the loop. Another exmple is the rte of hnge of C for set of prtiles of equl mss on the sme irulr orit of rdius r out muh more mssive ojet of mss M. It is given y dc ¼ GMr 2 : r 11 2 r 21 Þ 1 v : v 11 2 v 21 Þ ; 2:26Þ r 3 nd this vnishes on summing round the orit. If the prtiles re on n ellipse out the entrl mssive ojet, then the rte of hnge of C does not vnish extly. It is esy to show, however, tht the error is seond order in the sping, nd pproximtely third order if the hnge in sping from one pir to the next is muh less thn the sping.

5 Vritionl priniples for reltivisti SPH Liouville s theorem nd Poinré invrints The equtions of motion n esily e written in Hmiltonin form with the Hmiltonin p 2 H ¼ m 1 u ; 2:27Þ 2m 2 with the nonil momentum p defined y (2.6). If there re n SPH prtiles then the phse spe hs nonil oordintes r 1 ; r 2 ; ; r n nd nonil moment p 1 ; p 2 ; ; p n. Louiville s theorem then shows tht dr 1 dr 2 dr n dp 1 dp 2 dp n 2:28Þ is invrint. In (2.28) dr denotes dx dy dz nd dp denotes dp x dp y dp z for three-dimensionl Crtesin oordinte system. The Poinré invrints involving integrls on su-mnifolds, e.g. the integrl over mnifold of two dimensions dq dp ; 2:29Þ is invrint. These integrl invrints pply to n ensemle of systems. Aordingly, if we set up dense set of repli SPH systems, the volume in phse spe ssoited with the repli systems will remin invrint. This ide is used s sis for sttistil mehnis nd rises the interesting question of the equilirium stte of non-dissiptive fluid, nd how tht stte might e relted to the dditive invrints nd the pproximte irultion invrint. The existene of the Hmiltonin lso suggests tht the powerful Hmiltonin methods for nlysing dynmil systems might e pplied to non-dissiptive fluids. 2.3 The prtile energy eqution The rte of hnge of totl energy per unit mss ê n e found esily from the expression for the totl energy y writing E s P m ^e where ^e ¼ 1 2 v 2 1 u 1 F: 2:30Þ Thus de ¼ d ^e m ¼ m dv v 1 u dr 1 df : 2:31Þ s Using the elertion eqution, the rte of hnge of the density, nd the expression for the grvittionl potentil energy we find d ^e m ¼ 2 m m r 2 v 1 P Gv : r Þ r 2 v 7 W 2 m m : 2:32Þ We n then identify d ^e ¼ 2 m r 2 v 1 P Gv : r Þ r 2 v 7 W 2 m : 2:33Þ r 3 r 3 This expression is the SPH equivlent of d ^e ¼ 2 P r 2 7 : rvþ 2 v : 7 P 1 df r ¼ 2 1 r 7 : PvÞ 1 df ; whih is the usul energy eqution. 2:34Þ 2:35Þ 3 THE SPECIAL RELATIVITY LAGRANGIAN 3.1 Equtions of motion The SPH speil reltivity equtions hve een dedued from the ontinuum reltivisti hydrodynmi equtions y Chow & Monghn (1997) using SPH pproximtions to the sptil derivtives. It is onvenient to onsider the fluid s omposed of ryons eh with the sme rest mss m 0 nd then sle the energy with m 0 2 nd

6 386 J. J. Monghn nd D. J. Prie the veloity with the speed of light. We require the reltivisti tion to e Lorentz invrint nd this requires the Lgrngin L to e the integrl of Lorentz invrint quntity over volume. It is esy to guess tht the Lgrngin is L ¼ 2 T mn U m U n dv; 3:1Þ where U m is the four-veloity, nd T mn is the energy momentum tensor defined y T mn ¼n 1 nun; sþ 1 PÞU m U n 1 Ph mn ; 3:2Þ where n is the ryon numer density, u(n, s) is the therml energy per ryon nd P is the pressure. These quntities re defined in the rest frme of the element of fluid eing onsidered. The metri tensor h mn hs signture (21,1,1,1). The Lgrngin n e simplified to L ¼ 2 n½1 1 un; sþš dv: 3:3Þ The SPH formlism will e set up in seleted frme whih we ll the omputing frme. In this frme the ryon numer density is N nd it is relted to n ording to N ¼ nu 0 n ¼ ng ¼ p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; 3:4Þ 1 2 v 2 Þ where v is the veloity of the fluid reltive to the omputing frme nd g is the usul Lorentz ftor. The SPH interpoltion in the omputing frme is sed on the integrl interpolnt A I rþ ¼ Ar 0 ÞWjr 2 r 0 jþ dv 0 ; where Wjr 2 r 0 jþ dv 0 ¼ 1: 3:5Þ 3:6Þ The integrl interpolnt (3.5) n e pproximted y sudividing the spe into smll volumes suh tht the smll volume DV ontins n ¼ N DV ryons. Repling the integrl y summtion over the elements of volume we get the summtion interpolnt n ArÞ ¼ A Wjr 2 r jþ; N 3:7Þ As n exmple the numer density is given y NrÞ ¼ n Wjr 2 r jþ; nd from (3.6) NrÞ dv ¼ n ; 3:8Þ 3:9Þ whih shows tht the totl numer of ryons is onserved. All integrls over volume n e repled y summtions over the SPH prtiles. The Lgrngin (3.3) n then e pproximted y L ¼ 2 n n 1 1 u Þ; 3:10Þ N ¼ 2 n qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 2 v 2 Þ 1 1 u Þ: The reltivisti SPH equtions n now e otined from Lgrnge s equtions. We first need the prtil derivtives of L with respet to the veloity nd to the oordintes. The veloity ours oth in the squre root ftor in L nd in u through the dependene of the therml energy on n ¼ N/g. We find for prtile qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi L ¼ n g v 1 1 u Þ 2 n 1 2 v v 2 Þ u v 3:11Þ : 3:12Þ s; r Mking use of the thermodynmi reltions t onstnt entropy we n write u ¼ n : 3:13Þ v v n 2

7 Vritionl priniples for reltivisti SPH 387 Using the reltion etween n nd N we find n ¼ 2N g v ; 3:14Þ v so tht the nonil momentum of SPH prtile is p ¼ L ¼ n 1 1 u 1 P g v : 3:15Þ v n For prtile the sptil derivtive is L ¼ 2 n g u ; 3:16Þ nd from the thermodynmi reltions u ¼ n n 2 ¼ P N g n 2 : 3:17Þ Anlogously to the non-reltivisti se we find L ¼ 2n n N 2 1 P N 2 7 W : 3:18Þ The Lgrngin eqution of motion of prtile is therefore dp ¼ 2n n N 2 1 P N 2 7 W : 3:19Þ If p is repled y the momentum per ryon S ¼ p /n this eqution n e written ds ¼ 2 n N 2 1 P N 2 7 W ; 3:20Þ whih is the sme s the reltivisti SPH eqution derived from the ontinuum equtions y Chow & Monghn (1997) nd similr in form to the fluid prt of the non-reltivisti elertion eqution (2.10). 3.2 Speil reltivisti onservtion lws Additive integrls As in the non-reltivisti se the invrine of the Lgrngin to n ritrry infinitesiml trnsltion of the system shows tht the totl momentum p ¼ n 1 1 u 1 P g v ; 3:21Þ n is onstnt. The invrine to rottions shows tht the ngulr momentum r p ¼ n 1 1 u 1 P g r v ; n 3:22Þ is onserved. From the sene of ny expliit time dependene the energy E given y L E ¼ : v 2 L 3:23Þ v ¼ n S : v u Þ 3:24Þ g ¼ n g 1 1 u 1 P 2 P ; 3:25Þ n N is onserved.

8 388 J. J. Monghn nd D. J. Prie The irultion The rgument leding to the pproximte irultion theorem follows s efore. Assuming eh SPH prtile ontins the sme numer of ryons we find (using the definition of the momentum per ryon S given erlier) tht C ¼ S : r 11 2 r Þ; is invrint. The ontinuum limit of (3.26) is the speil reltivisti irultion þ C ¼ 1 1 u 1 P gv : dr: n 3:26Þ 3:27Þ Liouville s theorem nd Poinré invrints Hving identified the nonil momentum the Hmiltonin n e written down nd the equtions of motion n e written in Hmiltonin form. The Hmiltonin is just the energy with g written in terms of the nonil momentum. From (3.15) we find p 2 g 2 ¼ 1 1 n 1 1 u 1 P 2 : 3:28Þ n As in the non reltivisti se Liouville s theorem nd the Poinré invrints n e used to disuss the sttistil ehviour of the system. 3.3 The prtile energy eqution P In order to dedue the rte of hnge of energy of eh SPH prtile we write E s n ^e where ê the energy per ryon of SPH prtile is ^e ¼ g 1 1 u 1 P 2 : 3:29Þ n N The time derivtive n then e found noting first tht d 1 1 u Þ dv ¼ 2S 1 dn g N 2 ; 3:30Þ nd dn ¼ n v 2 v Þ : 7 W ; 3:31Þ so tht de ¼ de^ n ¼ 2 n n N 2 v 1 P N 2 v : 7 W ; 3:32Þ from whih we n dedue tht d ^e ¼ 2 n N 2 v 1 P N 2 v : 7 W ; 3:33Þ whih grees with the eqution used y Chow & Monghn (1997) derived from the energy momentum tensor nd using SPH pproximtions of the sptil derivtives. Beuse of the symmetry of the grdient of the kernel this eqution leds diretly to the onservtion of energy. The reder will note tht (5.10) is the SPH equivlent of d ^e ¼ 2 1 N 7 : vpþ; whih is the usul reltivisti energy eqution for non-dissiptive fluid. 3:34Þ 4 THE GENERAL RELATIVITY LAGRANGIAN 4.1 Equtions of motion For onveniene we ssume tht the metri is speified funtion of the oordintes, nd the tsk is to determine the Lgrngin for fluid

9 moving in tht metri. The SPH equtions for this se hve een given y other uthors (Lgun, Miller & Zurek 1993; Siegler & Riffert 2000). Their derivtions strt with the ontinuum equtions whih re then pproximted y using SPH interpoltion to dedue the SPH equtions. As efore, our im is to show tht these equtions, or their equivlent, n e otined y using Lgrngin (or Hmiltonin). In the following Greek indies re summed over (0,1,2,3) while Ltin indies re summed over (1,2,3). The susripts,, nd re reserved for prtile lels. The Lgrngin for the fluid is (Fok 1964) L ¼ 2 T mn pffiffiffiffiffiffi U n U m 2g dv; 4:1Þ where the four-veloity U m is defined y Vritionl priniples for reltivisti SPH 389 U m ¼ dx m ; where t is the proper time. We note ¼2g nmv m v n Þ 1 2 : From the previous reltions we n write dx m ¼ v m ¼ U m U 0 ; where U 0 ¼ ¼2g nmv m v n Þ 21 2 : For perfet fluid the Lgrngin is pffiffiffiffiffiffiffi L ¼ 2 ½n 1 nun; sþš 2g dv: 4:2Þ 4:3Þ 4:4Þ 4:5Þ 4:6Þ As efore we prefer to work with the numer density n of ryons with rest mss m 0, nd we sle the energy with m 0 2 nd sle the veloity with the speed of light. The reltivisti numer onservtion eqution (the ontinuity eqution) is 1 pffiffiffiffiffiffi 2g x n pffiffiffiffiffiffiffi 2g nu n Þ; 4:7Þ whih suggests using the trnsformed numer density p N* ¼ ffiffiffiffiffiffiffi 2g nu 0 : 4:8Þ N* is the numer density equivlent of the vrile D* of Siegler & Riffert (2000). The numer onservtion eqution then eomes N* 1 t x i N*v i Þ¼0: 4:9Þ This eqution shows tht the totl numer of ryons N*dV ¼ nu 0 pffiffiffiffiffiffiffi 2g dv; 4:10Þ is onserved. As suggested y Siegler & Riffert, we n interpolte ording to ArÞ ¼ Ar 0 ÞWjr 2 r 0 jþ dv; where Wjr 2 r 0 jþ dv ¼ 1: 4:11Þ 4:12Þ This normliztion is over flt spe so tht we n use the sme kernels s in the non-reltivisti lultions. The summtion interpolnt is A ArÞ ¼ n Wr 2 r Þ; 4:13Þ N*

10 390 J. J. Monghn nd D. J. Prie where n ¼ N * DV ¼ n U 0 pffiffiffiffiffiffiffiffi 2g DV ; pffiffiffiffiffiffiffi is the numer of ryons in the volume 2g DV. We n now write the Lgrngin s N* L ¼ un; sþþ dv; U 0 4:15Þ nd the SPH Lgrngin is then 4:14Þ L ¼ 2 n 1 1 u Þ 1=2 ; 4:16Þ where ¼2g mn v m v n Þ : 4:17Þ The vritionl priniple involves the vrition of the trjetory of prtiles with onstnt entropy. The prtil derivtive of L with respet to v i is strightforwrd to lulte. Noting v i ¼22g im v m Þ ; 4:18Þ nd the thermodynmis reltion u v i ¼ n n 2 v i ; 4:19Þ with n ¼ p N* ffiffiffiffiffiffiffi 1=2 : 4:20Þ 2g we find L v i ¼ n 1=2 1 1 u 1 P n g im v m Þ ; from whih we n identify the nonil momentum per ryon of prtile s S iþ ¼ 1 1=2 1 1 u 1 P g im v m Þ n : 4:22Þ in greement, in the flt spe limit, with the speil reltivisti nonil momentum dedued erlier. This expression is similr to tht of Siegler & Riffert (2000) who use the 311 formlism nd inlude n rtifiil dissiption term. To omplete Lgrnge s equtions we need the sptil derivtive of L. The derivtive of the therml energy is u x i ¼ P n n 2 x i ¼ P 1 N * n N * x i 2 p ffiffiffiffiffiffiffiffi 2g x i 4:21Þ 4:23Þ! x i : 4:24Þ Writing N * in SPH interpolnt form, nd noting tht p ffiffiffiffiffiffiffiffi 2g x i ¼ 1 2 g mn g mn x i d ; 4:25Þ together with x i ¼ 2v m v n g mn x i ¼ 2U m U n g mn x i ; 4:26Þ the sptil derivtive of L n e written L pffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffi! pffiffiffiffiffiffiffiffi 2g 2g P W 2g x i ¼ 2n n 1 x i 1 n T mn g mn 2N* x i : 4:27Þ

11 Lgrnge s eqution of motion then eomes ds iþ pffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffi! pffiffiffiffiffiffiffiffi 2g 2g P W 2g ¼ 2 n 1 x i 1 T mn g mn 2N* x i : 4:28Þ This eqution is similr in form to eqution (41) of Siegler & Riffert (2000). It is not identil euse they inlude n rtifiil dissiption, pffiffiffiffiffiffiffi nd there is ftor 2g outside the summtion. The fluid prt of our eqution of motion onserves momentum extly. This is not the se for the Siegler & Riffert eqution euse their fluid term hs not een symmetrized. With the Lgrngin formultion this symmetriztion ours nturlly. 4.2 Generl reltivisti onservtion lws Additive integrls In generl, with n ritrry metri, momentum is not onserved. It is onserved when the field equtions re solved s prt of the lultion. If the metri terms hve rottionl symmetry then the ngulr momentum out the xis of symmetry is onserved. Provided the metri does not hve expliit time dependene the energy E ¼ v i L v i 2 L 4:29Þ ¼ ½v i S iþ u Þ 1=2 Š 4:30Þ ¼ n 1=2 1 1 u 1 P g im v m v i u Þ n is onserved. P Introduing n energy per ryon ê, the totl energy n e written s n e^, where ^e ¼ 1 1=2 1 1 u 1 P g im v m v i u Þ 4:32Þ n The energy per ryon is equivlent to the expression E 2 i S i Þ introdued y Siegler & Riffert (2000). The energy eqution per prtile n e otined s efore y tking the rte of hnge of E with time nd identifying the rte of hnge of ê in terms of the other physil quntities. The first step gives de ¼ d ^e n ¼ n v i ds iþ We n write the lst term s d 1=2 1 1 uþþ ¼ 2 1=2 2n T mn dg mn dv i 1 S iþ 1 d 1=2 ½ 1 1 u ÞŠ 1 P p ffiffiffiffiffiffiffi 2g 4:31Þ : 4:33Þ dn* ; 4:34Þ nd the rte of hnge of N* n e found from the SPH interpolnt for N*. Comining these results we find d ^e pffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffi! pffiffiffiffiffiffiffiffi 2g ¼ 2 n v i 1 2g P v i W 2g x i 2 T mn g mn t 2 Vritionl priniples for reltivisti SPH 391 : 4:35Þ This expression for the energy is similr to tht of Siegler & Riffert (2000, eqution 43). It differs from their result euse of the symmetrized form of the pressure terms The irultion The rgument leding to the pproximte irultion onservtion follows s efore. We now find tht the SPH equivlent is tht C ¼ S : r 11 2 r Þ; 4:36Þ where S : r denotes S i x i, is pproximtely onstnt. In the ontinuum limit this result grees with Tu (1978) Liouville s theorem nd Poinré invrints Beuse we hve prtile Lgrnin nd ssoited nonil momentum it is possile to write the equtions of motion in Hmiltonin form. In the phse spe defined for the oordintes nd moment of the SPH prtiles, Liouville s theorem nd the Poinré invrints hold.

12 392 J. J. Monghn nd D. J. Prie 5 CONCLUSION AND DISCUSSION In this pper we hve shown tht the SPH equtions for n idel fluid n e dedued from Lgrngin. The derivtion is strightforwrd nd hinges on the ft tht in the SPH method the density is funtion of the oordintes. The existene of the Lgrngin enles us to dedue the onservtion lws inluding n pproximtion to the irultion whih eomes the usul irultion in the ontinuum limit. These results suggest tht muh of the suess of SPH n e ttriuted to the ft tht it preserves mny of the properties of n idel fluid. Of ourse, in mny strophysil pplitions, dissiption ours nd the equtions n no longer e derived from Lgrngin. However, this does not mke it ny less desirle to hve equtions for numeril work whih, in the sene of dissiption, preserve importnt properties of the originl equtions. In ddition to the use of SPH for numeril work, the formlism llows us to write down Hmiltonin equtions in prtiulrly simple wy. This leds to questions regrding the hoti motion nd sttistil equilirium of the system. Cn we predit when it will e hoti nd, given the dditive integrls of the motion nd the pproximte irultion invrint, n we predit the equilirium stte of the nondissiptive fluid? In our disussion we hve not inluded the field equtions in the Lgrngin exept in the non-reltivisti se where we hve ssumed the field is due to self grvity. In tht se the potentil is known s funtion of the prtile oordintes. We ould hve inluded term whih, when vried gve n SPH pproximtion to Poisson s eqution for the grvittionl potentil ut tht is not the preferred form for omputtions. For tht reson we hve not inluded it. In the non-reltivisti se the strophysilly signifint field is usully the grvittionl field nd, if reltivisti speeds re generted, the fields must e omputed from the full GR field equtions. In our disussion we hve ssumed the grvittionl field is known nd not ffeted y the motion of the fluid, vlid pproximtion for fluid of reltively low totl mss oriting eyond the event horizon of lk hole. A disussion of the other signifint field, the mgneti field, will e onsidered elsewhere. The Lgrngin for the reltivisti grvittionl field is well known nd it my e dded to the fluid Lgrngin we hve onsidered nd the vrition with respet to the metri oeffiients then gives the usul Einstein field equtions. There is nothing new in this nd for tht reson we hve not inluded these terms in our disussion of vritionl priniples. REFERENCES Bretherton F. P., 1970, J. Fluid Meh., 44, 19 Chow E., Monghn J. J., 1997, J. Computt. Phys., 134, 296 Ekrt C., 1960, Phys. Fluids, 3, 421 Feynmn R. P., 1957, Prog. Low Temp. Phys., 1, 17 Fok V., 1964, The Theory of Spe, Time nd Grvittion. 2nd edn. Pergmon Press, Oxford Lndu L. D., Lifshitz E. M., 1976, Mehnis, Course of Theoretil Physis. 3rd edn. Vol. 1, Pergmon Press, Oxford Lgun P., Miller W. A., Zurek W. H., 1993, ApJ, 404, 678 Monghn J. J., 1992, ARA&A, 30, 543 Slmon R., 1988, Ann. Rev. Fluid Meh., 20, 225 Siegler S., Riffert H., 2000, ApJ, 531, 1053 Tu A., 1978, Ann. Rev. Fluid Meh., 10, 301 This pper hs een typeset from TE/LATE file prepred y the uthor.