A Group Theoretic Approach to Generalized Harmonic Vibrations in a One Dimensional Lattice

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1 Virginia Commonwealth Univerity VCU Scholar Compa Mathematic and Applied Mathematic Publication Dept. of Mathematic and Applied Mathematic 984 A Group Theoretic Approach to Generalized Harmonic Vibration in a One Dimenional Lattice J.. Boyd Virginia Commonwealth Univerity P.. Raychowdhury Virginia Commonwealth Univerity Follow thi and additional work at: Copyright 986. Thi i an open acce article ditributed under the Creative Common Attribution Licene, which permit unretricted ue, ditribution, and reproduction in any medium, provided the original work i properly cited. Downloaded from Thi Article i brought to you for free and open acce by the Dept. of Mathematic and Applied Mathematic at VCU Scholar Compa. It ha been accepted for incluion in Mathematic and Applied Mathematic Publication by an authorized adminitrator of VCU Scholar Compa. For more information, pleae contact libcompa@vcu.edu.

2 Internat. J. Math. & Math. Sci. Vol. 9 o. (986) -6 A GROUP THEORETIC APPROACH TO GEERALIZED HARMOIC VIBRATIOS I A OE DIMESIOAL LATTICE J.. BOYD and P.. RAYCHOWDHURY Department of Mathematical Science Virginia Commonwealth Univerity Richmond, Virginia 84 U.S.A. (Received July I0, 984) ABSTRACT. Beginning with a group theoretical implification of the equation of motion for harmonically coupled point mae moving on a fixed circle, we obtain the natural frequencie of motion for the array. By taking the number of vibrating point mae to be very large, we obtain the natural frequencie of vibration for any arbitrary, but ymmetric, harmonic coupling of the mae in a one dimenional lattice. The reult i a coine erie for the quare of the frequency, i j f, j {,, } and a() j --F a()coe where 0 B... depend upon =0 the attractive force contant between the j-th and (j+g)-th mae. Latly, we how that thee frequencie will be propagated by wave form in the lattice. KEY WORDS AD PHRASES. Harmonic coupling, frequencie of motion, wave form, lattice. AMS SUBJECT CLASSIFICATIO CODE. 0C5. i. itroductio. In thi paper, we uppoe that identical, point mae are ymmetrically and uniformly arranged around a fixed circle o that the mae are coupled with male, ideal pring. All motion are confined to the fixed circle. We write potential energy matrice and how how to diagonalize thee matrice when the ymmetry i that of the rotational group of order. Having begun with quite imple example, we next take to be very large to obtain the natural frequencie for a one dimenional crytal lattice in which both nearet neighboring point mae and next nearet neighbor are coupled by harmonic force. We then extend thi reult to include all ymmetric coupling and how that the lattice will upport wave diturbance of preciely the natural frequencie of the lattice. It hould be noted that our reult follow from the conideration of the interaction of all pair of particle around the circular lattice rather than from a Fourier expanion of the potential energy of interaction. Thi work repreent a further ue of group theorectic method which have been reported elewhere [,,,4,5].. POTETIAL EERGY MATRICES. Let u conider the ymmetric matrix V(,) (Vjk) defined in the following manner: _> ; v..j for j {,,,...,}; Vjk -I for k j -+ (mod ) where

3 5 J.. BOYD and P.. RAYCHOWDHIRY e {,,,..[l--!l]} and [l--!l] +/- the greatet integer le than or equal to --; and 0 for all other entrie of Vjk V(,). For example, -i 0 0 -I 0 -i 0 -i 0 V(,) - V(5,) and V(6,) i 0 0 -I - 0 -i I -i 0 -I 0 0 -I 0 -i 0 The matrix V(,) arie in writing the elatic potential energie of a ymmetric circular array of point mae interconnected with ideal pring [i]. If each point ma i coupled to both nearet neighbor but to no other mae, then i. If each point ma i coupled to it two next nearet neighbor but to no other, then. Conider the cae for 4, i. Four identical mae move on a fixed frictionle circle. They are connected by pring of force contant k a hown. -i x I x x 4 x FIGURE I. Let the coordinate, Xl,X,X,X4, denote very mall diplacement of the mae from their equilibrium poition. All motion i confined to the circle, and the elatic potential energy i i P.E. k [(Xl-X ) + (x-x ) + (x-x4) + (x4-xi) k V(4,)X where X x i the tranpoe of X, and V(4,) i a defined. x x 4 We can repreent the ytem with a graph on four vertice. The vertice repreent the point mae, and two vertice are connected by an edge if and only if the two correponding mae are coupled by pring. 4 FIGURE.

4 GROUP THEORETIC APPROACH TO GEERALIZED HARMOIC VIBRATIOS Suppoe that 5,. Thai i, each ma i coupled only to it two next nearet neighbor a indicated below. We emphaize that the pring lie along the circle and all motion i on the circle itelf. i -- FIGURE. If the pring have force contant k, we have P.E. k ] [(Xl-X ) + (x-x4) + (x-x5) + (x4-xi) + (x5-x ) - k V(5,)X.. GROUP REPRESETATIOS AD UITARY TRASFORMATIOS. A rotation by leave the ytem of point mae and pring unchanged in o far a kinetic and potential energie are concerned, we take the rotation group C() {R,R,...R} where R denote a rotation of the circle by to be the ymmetry group of the ytem. A the group i abelian, all irreducible, nonequivalent matrix repreentation of C() are one dimenional Thu the cyclic group under complex repreentation []. From thee repreentation we can contruct the unitary tranformation matrix U which will imultaneouly diagonalize -- - V(,) -- for all : "P-- "PT T 4i 8i i 4(-) t exp exp exp exp U 6i i 8i 6(-)_t exp -- Since much of what i to follow depend upon thi fact, the tatement hould be - jutified. Conider the tranformed matrix UV(,)U The j-th row of UV(,) i (exp exp - exp exp )V(,) (4 exp in 4 exp in rj /4 exp in j /4 exp jt in j - Recalling that U -I i imply the complex conjugate of the tranpoe of U, we can write the j, k-th entry of [UV(,)]U -I-" Thi entry i "4 in for k, (_4) in (j-k)i zj exp =i for j # k. The claim ha been jutified, and UV(,)U -I i diagonal for each. multiplication which have generator exp(-), {,, }, give all irreducible exp ---

5 4 J.. BOYD and P.. RAYCHOWDHURY 4. A APPLICATIO. Let u conider a circular arrangement of identical, uniformly paced particle of ma m. For very large, we have, in effect, applied the Born condition to tranform a linear, one dimenional crytal into a circular array []. Suppoe that nearet neighbor particle are coupled with ideal, male pring of force contant k() and that next nearet neighbor are coupled with ideal, male pring of force contant k() a indicated below. / \ FIGURE 4. Thi configuration would erve a a model for a phyical lattice in which harmonic force between next nearet neighbor were much larger than anharmonic force between nearet neighbor. In the event that the reader doe not wih to viualize uch a mechanical arrangement, he could eaily interpret the problem in term of identical LC circuit with capacitor coupled in The Lagrangian for the ytem i L [ where I i the identity matrix, the appropriate manner. k() C(,)X + k() (,)X] and X (XlX ). We can now tranform the Lagrangian to diagonalize V(,I) and V(,): e m =- -i + k() U (UV(,)U-I)ux] -- in in xn -i I -i (UIU-I)u[- [--k() iu (UV(,I)U I)UX /k() * - in " 0 in 0 n 0 in i! o o o 0 The column matrix UX give the new ymmetry coordinate in which all the potential energie are uncoupled The row matrix i the complex conjugate of the tranpoe of. The equation of motion are given by 0

6 GROUP THEORETIC APPROACH TO GEERALIZED HARMOIC VIBRATIOS 5 where j i the j-th ymmetry coordinate and j 0 for j {i }. have mj + 4(k()in J_i + k()in )nj The natural frequencie are then fj =--I 4 /k()in j_ + k()in j i it complex conjugate. Thu we Thi expreion reduce to the well known reult for the linear lattice for k() > O, k() 0[]. 5. A EXTESIO. It hould be clear that if the point mae are coupled with force contant k() 0 and potential energy matrice V(,) for e{,, }, then the J-th frequency would become fj /i [k() in ]/m 7j=i In thi event, we can write k()in j k() (i co( )). [ a()cob where a(0) k() =i = =0 k() a() for e {i,, } and 0 < j B. The variable 8 will be called the wave number of the vibration. If we let and while auming abolute convergence for I we have a model in which we can take the harmonic =l vibration for all poible ymmetric coupling to approximate ome frequency ditribution obtained from an arbitrary potential energy of interaction 4m Suppoing that f i computed or found experimentally to be then we can obtain a() by noting that g() cob d8 a()cobd8 and a() 0 repect to j- 0 0 g()cosd. Since, the frequency function hould be ymmetric with our model will be conitent for g(8) ymmetric with repect to B 7, thu explaining the abence of ine term in the Fourier erie. 6. LOCALIZED DISTURBACES AD WAVE PROPAGATIO. The frequencie obtained are thoe for the lattice moving a a whole. That i, the frequencie are thoe of the normal mode of ocillation which involve motion of all contituent part of the lattice at once. We now return to the original coordinate in our circular array. Thee are xj for j e {,,.,} where x. i the diplacement of the j-th point ma from it equilibrium poition. For the moment, letu take "the force contant k() > 0 with k() 0 if # Suppoe that at t O, the j-th particle i diturbed. The diturbance will not be regitered at the ite of the (j + )-th particle until ome time later. If we potulate that a wave front will move. out from the ite of the j-th particle at t 0, then the time required for the front to reach the (J + )-th particle will be proportional to J + J

7 6 J.. BOYD and P.. RAYCHOWDHURY Taking the proportionality factor to be b, we expre thi Dhyical obervation by the introduction of a phae factor e bji/ to multiply the time dependent part of the diplacement of the j-th particle from it equilibrium poition. That i, we make the change of variable bji/ x.(t) e yj(t). Then xj+(t)/xj(t) bei/ e yj+e(t)/yj(t) implie a difference in phae equal to Ibl a required. From the Lagrangian in the original coordinate, L m XIX + k ()V(,)X, we obtain the equation of motion m. k(%)[-x + j xj+ + xj_]. If we now uppoe that a periodic wave move through the array, we write for each vbj i/ bj i/y fit j, e e x. yj(t) e where Y i a contant amplitude. Then the equation of motion become mj -m(f)ye bji/efit= k(g)yefit(e b(j+)i/ ej i/ + e b (j-)i/ ),or m(f) -k() (eb li/ + e-bi/) Therefore f k().b. k() b i /()(l-co in(--) (I co() or f j where Thu a longitudinal traveling wave having freuqency of any of the normal mode (k() > 0, k() 0 for ) will be upported by the lattice. Similarly, from the general Lagrangian in the coordinate., i e m I + E()V(,)X, we obtain m. k()[-xj + xj+ + =i xj_e]. all wave for ome fixed b and with the ummation taken over We have x. yeji/ fit [ m =i =m =0 a with a() and 8 a previouly defined, provided that b i an integer. implying the exitence of a traveling wave form contructed from the uperpoition of e where f k()in b i [ )cos We conclude with a change to a notation more familiar in olid tate phyic. Let u 0 be the equilibrium pacing between ucceive particle in our array, let ju 0 u, and let < u 0 Then x. x(u) ye<ui fit e Ye (<ui+=fit) and, for J (4f x x fixed b, x(u) atifie the wave equation -)---=----" u t the wave i / if _ 7 [ a()co < < =0 REFERECES Then the velocity of i. BOYD, J.. and RAYCHOWDHURY, P.., Repreentation Theory of Finite Abelian Group Applied to a Linear Diatomic Crytal, International Mathematic and Mathematical Science (980) BOYD, J.. and RAYCHOWDHURY, P.., An Application of Projection Operator to a One Dimenional Crytal, Bulletin of the Intitute of Mathematic, Academia Sinica 7 (979) BOYD, J.. and RAYCHOWDHURY, P.., A One Dimenional Crytal with earet eighbor Coupled Through Their Velocitie, ASME Sytem Meaurement and Control 0 (98) BOYD, J.. and RAYCHOWDHURY, P.., Group Repreentation in Lagrangian Mechanic, Phyica I4A(98) BOYD, J.. and RAYCHOWDHURY, P.., Two Dimenional Lattice Vibration from Direct Product Repreentation of Symmetry Group, International Mathematic and Mathematical Science, 6 (98) b

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