Lagrangan Theory of Several-body Systems Ncholas Wheeler, Reed College Physcs Department November 995 Introducton. Let the N-tuple of 3-vectors {x (t) : =, 2,..., N} descrbe, relatve to an nertal frame, the confguraton of an N-partcle system at tme t. To descrbe the dynamcs of the system we would fnd t natural to ntroduce the Lagrangan L = 2 m ẋ ẋ U(x, x 2,..., x N ) () Suppose, however, we had elected to descrbe the partcles n terms of ther Cartesan relatonshp to a reference pont X(t) n arbtrarly prescrbed moton: x = X + r. We would then have L = 2 m (Ẋ + ṙ ) (Ẋ + ṙ ) U(X + r) = 2 MẊ Ẋ + Ẋ m ṙ + 2 m ṙ ṙ U(X + r) (2) n whch the dynamcal varables formerly {x, x 2,..., x N }, presently {r, r 2,..., r N } are stll (as before, and not at all surprsngly) N n number. The equatons of moton now read m r = m Ẍ U(X + r) ( =, 2,..., N) (3) n whch we nterpret the frst term on the rght to be a fcttous force term, an artfact of the crcumstance that the orgn of the X-centered r-frame s (except when Ẍ = 0) non-nertal. None of whch s n any respect problematc.
2 Lagrangan theory of several body systems Now wth an eye to the algebrac smplfcaton of (2) mpose upon the r-varables ths constrant: m r = 0 (4.) Equvalently, assocate X wth the center of mass of the N-partcle system: X = m x wth M = M m (4.2) Equaton (2) can now be notated L = 2 MẊ Ẋ + 2 m ṙ ṙ U(X + r) (5) (note the dsappearance of the cross-term), whch on ts face appears to refer to a system wth dynamcal varables {X, r, r 2,..., r N } more numerous that the varables of the system wth whch we started. And (5) gves rse to equatons of moton whch are not only more numerous than but also nconsstent wth equatons (3). What s gone wrong? In (4) we see that when we subjected the varables x to a cross-term-kllng constrant we effectvely promoted X to the status of a dynamcal varable; t s, accordng to (4.2), a varable a collectve varable whose t-dependence has now to be extracted from equatons of moton, and can no longer be sad to be arbtrarly prescrbed. But the Lagrangan (5) s heedless of ths crcumstance, and gves rse to equatons of moton the solutons of whch wll, n general, stand n volaton of (4.). To obtan correct results we mght, for example, ntroduce N N r N = m N m r and ṙ N = m N m ṙ nto (5) to obtan a Lagrangan of type L(Ẋ, ṙ,..., ṙ N, X, r,..., r N ), but such a procedure bears the formal blemsh of a dscrmnatory asymmetry not natural to the physcs of the stuaton. How to proceed more symmetrcally? In place of (5) wrte L = 2 MẊ Ẋ + 2 m ṙ ṙ U(X + r) g m r (6) where g s a Lagrange multpler whch wll be accorded the formal status of a supernumerary dynamcal varable. The resultng equatons of moton read MẌ = U (7.) m r = U gm ( =, 2,..., N) (7.2) 0 = m r (7.3)
A smple example: the one-dmensonal A 2 molecule 3 where =. These are N + 2 equatons of moton n as many varables. The last of these the g th Lagrange equaton of moton (7.3) s smply the constrant relaton (4.), and entals m r = 0. Addng equatons (7.2) together, and subtractng the result from (7.), we obtan The equatons of moton (7) reduce therefore to g = Ẍ (8) MẌ = U (9.) m { r + Ẍ} = U ( =, 2,..., N) (9.2) whch are attractvely symmetrc (no r has been dscrmnated aganst), but redundant: addng equatons (9.2) together gves back (9.). Equatons (9.2) are consstent wth (3), of whch they are a partcularzed nstance. It s nstructve to note also that the ntroducton of (8) nto (6) yelds a Lagrangan whch s dstnct from but (by Ẍ m r = d dt [Ẋ m r ] Ẋ m ṙ ) gauge equvalent to the Lagrangan of (2). A smple example: the one-dmensonal A 2 molecule. We look now to the Lagrangan theory of what mght be called a one-dmensonal A 2 molecule. The consttuent atoms resde at x and x 2 > x, and both have mass m. The molecule tself therefore has mass M = 2m. We assume the molecule to be bound together by a sprng of natural length a and strength k, and to move n an ambent potental U(x). In natural varables the Lagrangan reads L = 2 m(ẋ2 + ẋ 2 2) U(x ) U(x 2 ) 2 k[(x 2 x ) a] 2 (0) The varables ntutvely most natural to ths smple system are the external coordnate X = M (mx + mx 2 ) = 2 (x + x 2 ) and the nternal coordnate 2s = x 2 x, whch descrbes the nstantaneous length of the molecule. Immedately } x = X s x 2 = X + s () gvng L = m(ẋ2 + ṡ 2 ) U(X s) U(X + s) 2 k(2s a)2 (3) It becomes analytcally advantageous at ths pont to ntroduce the varable q = 2s a whch descrbes molecular length relatve to the rest length of the molecule;
4 Lagrangan theory of several body systems then s = 2 (a + q) and we have where K = 2k and We are led thus to wrte L = mẋ2 + 2 { 2 m q2 2 Kq2 } U(X, q) U(X, q) = U(X + s) + U(X s) = {U(X) + U (X)s + 2 U (X)s 2 +...} + {U(X) U (X)s + 2 U (X)s 2 +...} = 2U(X) + U (X)s 2 +... L = {mẋ2 2U(X)} + 2 { 2 m q2 2 Kq2 } U nteracton (X, q) (4) wth U nteracton (X, q) = 4 U (X)(a 2 + 2aq + q 2 ) +... = 2 au (X) q +... The operatve assumpton here s that the ambent potental changes lttle over the dmenson of the molecule. The strkng absence of a U (X)-term s an artfact of our assumpton that m = m 2 = m, and means that the nteracton s, n leadng approxmaton, tdal. In ths respect the physcs of A 2 molecules s dstnct from the physcs of AB molecules. Lookng now to the equatons of moton mẍ = U (X) 2 qau (X) (5.) m q + Kq = au (X) (5.2) we fnd t natural on physcal grounds to abandon the 2 nd term on the rght sde of (5.). Returnng wth X(t) a soluton of the equaton thus obtaned to (5.2), we have m q + Kq = F(t) wth F(t) = au (X(t)) (6) Evdently the tdal term on the rght sde of (5.2) serves n effect to force the nternal oscllaton of the molecule. The reader who was awatng the entry of relatve varables r and r 2 nto the preceedng dscusson wll have been struck by ther absence. Ther non-appearance can be attrbuted to the crcumstance that the asymmetry problem whch motvated our ntal dscusson does not arse n the case N = 2; t s non-dscrmnatory to speak of s = 2 (r 2 r ). In ths respect the physcs of A n molecules (n > 2) s margnally more nterestng. The physcs even of A 2 molecules becomes markedly more nterestng when we gve up the one-dmensonalty or our problem, for then the molecule can be expected to experence torques, and to tumble n ways responsve to the dervatve
A less smple example: the one-dmensonal A 3 molecule 5 structure of U(X); I shall, however, resst the temptaton to enter nto an mmedate dscusson of the detals. A less smple example: the one-dmensonal A 3 molecule. We now assume partcles of dentcal mass m to resde at x, x 2 > x and x 3 > x 2 and to be bound by sprngs dentcal to those encountered n the prevous example. In natural varables the Lagrangan (compare (0)) reads L = 2 m(ẋ2 + ẋ 2 2 + ẋ 2 3) U(x ) U(x 2 ) U(x 3 ) 2 k{[(x 2 x ) a] 2 + [(x 3 x 2 ) a] 2 } (7) Drect appropraton of (6) gves rse for such a system to L = 2 MẊ2 + 2 m(ṙ2 + ṙ 2 2 + ṙ 2 3) U(X + r ) U(X + r 2 ) U(X + r 3 ) 2 k[(r 2 r ) a] 2 2 k[(r 3 r 2 ) a] 2 g m(r + r 2 + r 3 ) (8) where M = 3m s the mass of the A 3 molecule, and where X = 3 (x + x 2 + x 3 ) serves to locate ts center of mass. Equatons (9) acqure therefore ths partcularzed meanng: MẌ = U (X + r ) U (X + r 2 ) U (X + r 3 ) (9.) m( r + Ẍ) = U (X + r ) + k[(r 2 r ) a] m( r 2 + Ẍ) = U (X + r 2 ) k[(r 2 r ) a] + k[(r 3 r 2 ) a] (9.2) m( r 3 + Ẍ) = U (X + r 3 ) k[(r 3 r 2 ) a] Lookng frst to (9.), we observe that RHS of (9.) = 3U (X) m U (X) 3 mr 2m U (X) 3 mr 2 +... (20) The second term on the rght vanshes by defnton of the center of mass. It s nterestng n ths lght to notce that the sum encountered n the thrd term on the rght serves to defne the nstantaneous moment of nerta relatve to the center of mass (.e., the centered second moment of the molecular mass dstrbuton), and that the sums encountered n hgher-order terms defne nameless hgher moments of the mass dstrbuton. Upon the abandonment of all such neglgble terms, (9.) reduces to mẍ = U (X): the center of mass moves as a sngle atom would move n the ambent potental. Lookng now to (9.2), we are motvated by the structure of the sprng terms to ntroduce } q = (r 2 r ) a (2) q 2 = (r 3 r 2 ) a Inverson of q + a = r + r 2 q 2 + a = r 2 + r 3 0 = +r + r 2 + r 3
6 Lagrangan theory of several body systems gves r = 3 (2q + q 2 ) a r 2 = + 3 ( q q 2 ) r 3 = + 3 ( q + 2q 2 ) + a (22) and when we return wth (2) and (22) to (8) we by computaton obtan (compare (4)) L = 3{ 2 mẋ2 U(X)} + { 3 m( q2 + q q 2 + q 2 2) 2 k(q2 + q 2 2)} U nteracton (X, q, q 2 ) (23) where U nteracton (X, q, q 2 ) = 2m U (X) m [ 2 9 a2 + 2 9 a(q 2 q ) = 9 au (X) (q 2 q ) +... + 2 3 (q2 + q q 2 + q 2 2) ] +... To better emphasze the essentals of the stuaton as t now stands, I wrte L = L center of mass (Ẋ, X) + L nternal( q, q) + L nteracton (X, q) and assert (on the physcal grounds to whch I have already alluded, and to a more careful dscusson of whch I promse to return) that L nteracton (X, q) contrbutes essentally to the moton of the nternal varables q, but nessentally to the moton of X. We are led thus to the equatons of moton mẍ = U (X) (24.) { d } L = 0 ( =, 2) (24.2) dt q q where L = L nternal ( q, q) + L nteracton (X(t), q). I have refraned from wrtng out the detaled mplcatons of (24.2) because those equatons are not welladapted to analytcal treatment. To obtan more workable equatons we very The tedum of the computaton s much reduced f one wrtes r r 2 = G q q 2 and ṙ ṙ 2 = G q 2 q 2 wth G = 3 + 0 r 3 a ṙ 3 0 2 and notces that G T G = 6 3 9 3 6 2 for then (r 2 + r 2 2 + r 2 3) and (ṙ 2 + ṙ 2 2 + ṙ 2 3) become qute easy to evaluate.
A less smple example: the one-dmensonal A 3 molecule 7 much n the sprt of standard small oscllaton theory subject L nternal ( q, q) to some preparatory massagng. Wrtng T L nternal ( q, q) = 6 m q M q 2 wth M = ( q q 2 ) T 2 k q K q 2 2 2 ( q K = q 2 ) 0 0 we ntroduce new varables (wrte q = RQ wth R a rotaton matrx: R T R = I) to obtan T T Q L nternal ( Q, Q) = 6 m Q R Q T M R 2 Q 2 k Q R T K R 2 Q 2 ( Q Q 2 ) (25) and look to the smultaneous dagonalzaton of R T M R and R T K R. Ths, of course, s standard theory of small oscllatons methodology; the only unusual crcumstance s that here t s not be sprng matrx K but the mass matrx M whch comes to us n ntally non-dagonal form. Some prelmnary observatons: from det(m λi) = λ 2 4λ + 3 we conclude that the egenvalues of M can be descrbed λ = 2 ±, and that our assgnment, therefore, s to dscover the matrx cos ϕ sn ϕ R = sn ϕ cos ϕ such that R T M R = 0 0 3 Our assgnment (to say the same thng another way) s to dscover the ϕ such that ( 2 cos = 2 ϕ + 3 sn 2 ) ϕ 2 sn ϕ cos ϕ 2 2 sn ϕ cos ϕ sn 2 ϕ + 3 cos 2 ϕ Immedately ϕ = 45, gvng R = 2 (26) In any event qute apart from the detals of the argument whch led us to (26) we have only to nsert (26) nto (25) to obtan L nternal ( Q, Q) = 6 m( Q 2 + 3 Q 2 2) 2 k(q2 + Q 2 ) The modal moton of the free A 3 molecule can therefore be descrbed Q + ω 2 Q = 0 (27)
8 Lagrangan theory of several body systems wth ω = 3k/m ω 2 = k/m (28) For such a molecule (.e., for a one-dmensonal A 3 molecule n the total absence of an ambent potental) we have X(t) = X 0 + V 0 t q (t) = A cos(ω t + δ ) + A 2 cos(ω 2 t + δ 2 ) q 2 (t) = A cos(ω t + δ ) + A 2 cos(ω 2 t + δ 2 ) so (recall (22)) n natural varables we have the followng explct descrpton of the free moton latent n the Lagrangan (7): x (t) = X(t) 3 A cos(ω t + δ ) A 2 cos(ω 2 t + δ 2 ) x 2 (t) = X(t) + 2 3 A cos(ω t + δ ) x 3 (t) = X(t) 3 A cos(ω t + δ ) + A 2 cos(ω 2 t + δ 2 ) (29) These equatons make clear the sense n whch the fast mode (the mode under the control of A ) s a hp-swngng mode, and the slow mode (controlled by A 2 ) s a breather mode a dance n whch m 2 does not partcpate. So much for the free moton of the system. Now renstate the ambent potental U(x). The moton X(t) of the center of mass s now no longer unform, but accelerated as descrbed (n leadng approxmaton) by (24.). Of more partcular nterest s the fact that L nteracton = 9 au (X(t)) (q q 2 ) = F(t) Q where F(t) = 2 9 au (X(t)). The mplcaton s that tdal forces couple (n leadng approxmaton) only to the fast mode. Ths I fnd somewhat counterntutve, snce t s the slow breather mode of the A 3 molecule whch most resembles the soltary mode of the A 2 molecule a breather mode whch, as we know from prevous work, does respond to tdal forces. From L molecular ( Q, Q, t) = 6 m( Q 2 + 3 Q 2 2) 2 k(q2 + Q 2 ) + F(t) Q we obtan Q + ω 2 Q = 3 m F(t) Q 2 + ω 2 2Q 2 = 0 (30) whch gve back (27) n the absence of tdal forces. Moton of a struck A 3 molecule. In place of (7) we now have L = 2 m(ẋ2 + ẋ 2 2 + ẋ 2 3) 2 k{[(x 2 x ) a] 2 + [(x 3 x 2 ) a] 2 } x F (t) the nessental assumpton here beng that (snce objects are most commonly struck on ther exposed surfaces) t s the end-partcle m whch has been struck.