PHYS 705: Classical Mechanics. Small Oscillations

Transcription

1 PHYS 705: Classical Mechanics Small Oscillations

2 Fomulation of the Poblem Assumptions: V q - A consevative system with depending on position only - The tansfomation equation defining does not dep on time explicitly q Now, conside the situation when a system is nea an equilibium point q 0 whee the genealized foces acting on the is ZERO, i.e., q 0 V Q q0 0, q q 0 (consevative foce) Ou goal is to descibe the motion of the system when it is slightly deviated fom this equilibium point: q q 0

3 3 Fomulation of the Poblem Now, we expand the Lagangian aound :. The potential enegy tem: V q Fo small displacement, we can expand V in a Taylo expansion: V V n 0 0n k q q 0 q k 0,,, V q q V q q q 0 (E s sum ule applies hee and coefficients evaluated at.) Note: - can be set at zeo V q V q,, q n since is an equilibium 0 0 q 0 q 0

4 4 Fomulation of the Poblem So, nea an equilibium, can be well appoximated by a quadatic fom: q q 0 V q V V k k whee V k q V q k 0 (E s sum ule applies to all epeated indices) Note: - is ust a numbe (V evaluated at ) V k - Since V is assumed to be a smooth function and the ode of the q 0 V k Vk Vk patial deivatives can be switch, is symmetic:

5 5 Fomulation of the Poblem. The kinetic enegy tem: Recall that we can in geneal wite the KE as (Ch. ): i i T T TT0 m q q i k q q k i k i i i mi q m i q t t i i T T T 0 - quadatic in - linea in q - independent of q q With the assumption that tansfomation equation does not dependent on time explicitly, i.e., i t 0 we will only have the quadatic tem emained q,,, q t i i n

6 6 Fomulation of the Poblem This gives, Note: m T m k q qk whee, - in geneal can be a function of k q l - but we can also Taylo expand it aound,, m q,, q m q q m k ql q 0l m k k n k 0 0n l ql 0 i i i mi q qk - keeping only lowest ode tem (const tem) in, we then have T T k k whee, T m q,, q k k 0 0n q q (ecall & ) 0 q

7 7 Fomulation of the Poblem: Quadatic Fom of L q 0 With V and T given nea, we can now fom the Lagangian: LT V T V k k k k (E s sum ule) - the deviation fom the equilibium is ou genealized coods T n n - and evaluated at ae constant squae matices k V k q 0 - dynamic nea ANY equilibium quadatic foms in T and V - coupling infomation among diff dofs ae encoded in the coss tems - GOAL fo the following analysis is to To find a coodinate tansfomation so that in the new coods q 0 Tk V and diagonalize simultaneously k the poblem decoupled!

8 8 Fomulation of the Poblem: EOM nea Eq. LT V T V ik i k ik i k Now we use the EL equation to get the EOM fo : (E s sum ule) L V V k k i i V k ae symmetic L V k k (pick out only i=) Similaly, we have: L (pick out only k=) T k k d dt L T k k Then, EL eq gives EOM: T (sum ule ove k) V k k k k 0 Thee ae of these eqs and they ae all coupled.

9 9 Fomulation of the Poblem: EOM nea Eq. T V k k k k 0 (sum ove k) The EOM fo the ae a set of coupled nd ode ODEs with constant coefficients and the solution has the following geneal fom: t i t Ca e Ca - is the complex amplitude fo the genealized coodinate a with being eal. (We have chosen the fom of the pe-facto Ca fo convenience late.) - Obviously, only the eal pat of contibutes to the actual motion

10 0 Fomulation of the Poblem: Condition on Solution Plugging the oscillatoy solution into the nd ode ODE, we have the following matix equation fo the amplitudes : Fo nontivial solutions, we need the deteminant of the coefficients to vanish: V T a k k k a 0 a 0 (sum ove k) det Vk Tk 0 o V T V T n n V T V T n n nn nn 0

11 Fomulation of the Poblem: Eigenvalue Equation One can think of this as the chaacteistic equation fo the following genealized eigenvalue poblem: - ae the eigenvalues o esonant fequencies of the poblem a - is the eigenvecto which will detemine the elative elations among the genealized coods det V T 0 Va Ta in the diffeent eigenmodes (nomal modes) V and T ae eal and symmetic a is eal is eal and othogonal

12 Fomulation of the Poblem: Eigenvalue Equation Since V and T ae squae matices, thee ae in geneal n distinct eigenvalues and eigenvectos. (We will conside degeneacy late.) Labeling the n eigenmodes by the index, we can wite the matix equation fo the th eigenmode as: Va n n Ta The geneal solution fo will in geneal be a supeposition (linea combination) of all of these eigenmodes: i t it t a C e C e whee a is the th compoent of a (sum ove ),, T a a n

13 3 Fomulation of the Poblem: Real Physical Solution As we mentioned peviously, the actual physical motion is given by the eal pat of the complex solution, f whee and ae eal paametes and they will be detemined by initial conditions: and t Re t f a cos t f a cos a ReC 0 f a sin a ImC * C C C t (Hee, we have taken the choice that fo.)

14 4 Fomulation of the Poblem: Pops of Eigenmodes To show that ae eal, we will take the conugate tanspose of the equation: Then, Va Ta avat * V and T ae eal and symmetic a Va Ta * s s s T - av a a s * 0 s ata s (*)

15 5 Fomulation of the Poblem: Pops of Eigenmodes Conside the case when = s, we have * 0 ata Fo nontivial solutions, we have 0 and a ata 0 Then, the above condition gives * 0 ae eal

16 6 Fomulation of the Poblem: Pops of Eigenmodes Note: a eal eigenvalue can be associated with a complex eigenvecto. Let call it, then staightfowadly gives Convesely, this means that we can always find a eal eigenvecto Va a α iβ Ta fo if is eal! Vα Tα and ivβ itβ α and β Thus, ae also two eal eigenvecto fo the eal eigenvalue Va Ta Fom now on, we will only conside eal fo eal a

17 7 Fomulation of the Poblem: Pops of Eigenmodes Lastly, to show the othogonality of the eigenvectos, We look back at ou Eq. (*) (eigen-system condition) with and being eal: ata 0 s s s With distinct eigenvalues fo, we then have s (**) a s ata 0 s s With, the matix poduct in Eq. (**) is indeteminate. We will now show that it is nonzeo and in fact positive definite fom a physical agument, i.e., ata 0

18 8 Fomulation of the Poblem: Pops of Eigenmodes To do that, we will exploit the fact that the Kinetic enegy is a positive definite quantity, i.e., T m q q k k T 0 q q Now, we put in the expansion fo in tems of its eigenmodes: 0 T k k T T k sin sin 0 fa t s fsaks sts sum ove sum ove s fa cos t sum ove and k

19 9 Fomulation of the Poblem: Pops of Eigenmodes s We have ust shown that fo, we have the othogonality condition: ata 0 o s Tkaaks 0 So, the pevious Q-tiple sum fo the kinetic enegy T collapses with only tems suviving: sum ove,, and k s T Tkaa k f sin t 0 Notice that the [] tem ae all squaes and it will geneically be positive T a a The matix poduct o k k when we ae consideing nontivial solutions. 0 ata 0 a 0

20 0 Fomulation of the Poblem: Pops of Eigenmodes So, we have the following othogonality condition fo the matix poduct: Recall that the eigenvalue equation,,detemines the diection of the eigenvecto and we do have a feedom in choosing its magnitude. ata s a We now impose the nomalization of the eigenvectos such that: 0 0 Va s s Ta a ata s 0 s s T a a o k ks s

21 Nomal Modes Now, we will define a new set of coodinates (nomal modes) so that the oscillations decoupled! Recall we have the geneal expansion fo ou genealized coods: i * t it t a C e C e (sum ove ) We will define the nomal coodinates as the tem inside the ( ), * t C e C e a i t it,, T a a whee a is the th component of n and the complex amplitude C will be detemined by IC. t

22 Nomal Modes Then, we can wite, t a t (sum ove ) -Each of the th genealized coodinate is now witten as a linea combination of the nomal coodinates. - This is a linea tansfomation between two coodinate systems and this paticula linea tansfomation is specified by the set of a eigenvectos: o. - The th nomal coodinates is puely peiodic depending on the th eigenfequency only. A a a t - This linea tansfomation can be inveted to give in tems of

23 3 Nomal Modes Now, we ae going to explicitly show that the system decoupled in the nomal coodinates! t a We have and (sum ove ) Now, calculate T : t a T T k k T k k T ka a ks s s s a a ss ata s s a ' s -the ae othogonal!

24 4 Nomal Modes Now, calculate V: V Vk k V a aks V kaa ks s k s To simplify the matix poduct (blue tem), we will use the eigenvalue equation fo a paticula th eigenmode, V a T a (sum ove ) k k

25 5 Nomal Modes Multiply a diffeent eigenvecto on both sides, we have, a s V a T a k k V a a T a a k ks k ks Using the othogonality condition again, we have V a a k ks s ( a also diagonalizes V!) k Substituting this back into ou equation fo the potential enegy, we have, V V kaaks s s s

26 6 Nomal Modes Foming the Lagangian, LT V Calculating the EOM using EL equation fo each th nomal mode, d L L dt 0 0 (NO sum ove ) -EOM fo each of the nomal mode is decoupled! -Each nomal mode evolutes in time as a simple hamonic oscillato with a single eigenfequency.

27 7 Nomal Modes Note: In geneal, fo a system with n genealized coodinates, we will have n nomal modes and using the linea coodinate tansfomation t a (sum ove ) the geneal solution will be a linea combination of these nomal modes. i * t it t a a C e C e (sum ove ) Recall that the complex coefficients C ae detemined by IC. Fo typical IC, all nomal modes will pesent, i.e., C 0 so that ALL nomal modes will paticipate in the motion.

28 8 Nomal Modes - Howeve, one can imagine situations in which the IC will only lead to the excitation of a cetain nomal mode. - Let say, we have all except, - Then, the motion is vey simple with all genealize coodinates popotional to ONE non-zeo nomal mode C 0 C 0 t a (fo all ) a in popotions given by. a a - As an example in D: if A, then a (symmetic motion) a a a a if A, then a (anti-sym motion) a a

29 9 Rotation, Squeeze, and Simultaneous Diagonalization of V and T The eduction of the small oscillation poblem into a set of equations fo its nomal modes has an elegant geometic intepetation We stat with ou Lagangian: L T V L k k k k To ease in visualization late, let conside a dofs case: ηtη ηvη L T T T V V V (matix notation) both T and V ae symmetic

30 30 Rotation, Squeeze, and Simultaneous Diagonalization of V and T L T T T V V V (in going to the nomal coods) L The eduction to the nomal modes equation coesponds to a linea tansfomation A which simultaneously diagonalizes the two ealsymmetic quadatic foms: T and V. t Now, we will ty to visualize this geometically in D. a η Aζ (no cossed tems)

31 3 Rotation, Squeeze, and Simultaneous Diagonalization of V and T Fo any positive definite quadatic foms: y Note:, F x y ax bxy cy F x, y One can visualize F as the contou cuve of the function: in D as an ellipse. Geometically: x a c b 0 - If and the ellipse is a cicle - If b 0, the ellipse will be otated Matix Diagonalization semi-mao & semi-mino axes coodinate axes

32 3 Rotation, Squeeze, and Simultaneous Diagonalization of V and T Finding the nomal modes coesponds to the simultaneous diagonalization of both T and V. Let see now, how this is done geometically in 3 steps: Step 0: in oiginal genealized coodinates: η fo T and η fov T Both quadatic foms V ae NOT the same and the ellipses ae oiented diffeently

33 33 Rotation, Squeeze, and Simultaneous Diagonalization of V and T Step : Rotate T so that it is aligned with cood axes T ' ' R η ' η' aligns T in space and otates V in space but V is still tilted. V ' ' η' R η R cos sin sin cos η' R η L a b c ' ' a' ' b' ' ' c' '

34 34 Rotation, Squeeze, and Simultaneous Diagonalization of V and T Step : Squeeze (expand and contact) T so that it becomes a cicle with unit adius T '' G expands and contacts V '' T so that it becomes a '' cicle. G applies the '' same scaling to V., η'' G η' 0 G, '', 0 L a b c '' ' '' '' '' ' '' '' '' η G η'

35 35 Rotation, Squeeze, and Simultaneous Diagonalization of V and T Step 3: Rotate again but this time we otate so that V is in, T V R ζ aligns V in space. Since T is a cicle, the otation does NOT affect T. ζ R η '' R cos sin sin cos ζ R η'' L

36 36 Rotation, Squeeze, and Simultaneous Diagonalization of V and T T, ζ Aη ζ Aη V,, A R GR, L - So the sequence of linea tansfomations needed to diagonalize both T and V consists of a otation followed by a squeeze and by anothe otation. - The key step is the squeeze so that T becomes a cicle befoe the last otation to align V along its axes.

37 37 Rotation, Squeeze, and Simultaneous Diagonalization of V and T Unlike the standad eigenvalue poblem, the linea tansfomation A R GR needed to simultaneously diagonalize both eal-symmety quadatic foms (T and V) is NOT othogonal in the taditional sense, i.e., AA (the squeeze matix is not othogonal) Howeve, it is othogonal in a moe genealized sense A is othogonal with espect to the metic tenso T ATA ecall ata s s

38 38 Rotation, Squeeze, and Simultaneous Diagonalization of V and T A metic tenso is ust a genealization of ou Euclidean distance measue. With egula Euclidean space, the distance between two points is, ds ds ds ds dx dy dz kdqd qk Fo othe non-euclidean space, distance in geneal can be measued by a metic tenso G give by, ds g dq dq k k whee g k is the matix element of G.

39 39 Rotation, Squeeze, and Simultaneous Diagonalization of V and T Positive definite metics Othe familia examples of metix tensos: Cylindical cood: Spheical cood: ds d d dz ds d d sin d G 0 0 G sin Mikowski space (Loentzian metic): (non-positive definite) ds dx dy dz c dt

40 40 Rotation, Squeeze, and Simultaneous Diagonalization of V and T So, in ou discussion fo small oscillations, the condition fo the kinetic enegy matix given by ATA simply means that the set of eigenvectos ae othonomal with espect to a diffeent metic given by the metic tenso T o. AND, the set of eigenvectos fom T diagonalizes simultaneously the potential enegy matix V, a T k AVA λ

41 4 Degeneacy When one o moe of the oots fom the eigenvalue chaacteistic equation, s is epeated i.e.,,, then ou agument in showing that s ata 0 s fom ata 0 s s does not follow diectly. Howeve, one can still constuct a FULL set of othonomal eigenvectos fom the degeneate set of allowed vectos. To see how to do that, let conside a simple 3D case when we have distinct eigenvalue 3 and a double oot fo the othe two eigenvalues 3

42 4 Degeneacy a 3 Let be a nomalized eigenvecto associate with Thee is an infinite set of eigenvectos associated with in the degeneate plane to ata a 3 a 3 a a ata 3 0 ata 3 0 a 3 a is othogonal to both and a Va Ta,, 3 But, a might NOT be othogonal to a

43 43 Degeneacy Fotunately, one can always constuct an othogonal pai fom any andomly chosen vectos in the plane, e.g., a and a a 3 a' - Any linea combination of will also be an eigenvecto fo a and a a a ' Let be a new eignenvecto given by, a Check: a' ca c a Va ' cva c Va c Ta c Ta c c ' T a a Ta

44 44 Degeneacy Now, we enfoce the othogonality between a and a' a is nomalized ata ' cata c ata 0 c c ata 0 a a' 3 a c c ata a This new vecto must also be popely nomalized, a' Ta' ca c a T ca c a c ata cc ata c ata c c cc c c T is symmetic ata ata c c

45 45 Degeneacy Thus by solving the two equations c and fo, we have successful constuct a set of othonomal eigenvectos fo the degeneate eigenvalue c c ata c and c c a a' 3 a a The standad Gam-Schmidt othogonalization pocedue can be used when the degeneacy space is moe than dimensions.