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by: Dr. Carissa Rowe


Dr. Carissa Rowe
GPA 3.97


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Class Notes
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This 16 page Class Notes was uploaded by Dr. Carissa Rowe on Tuesday October 20, 2015. The Class Notes belongs to PHYS608 at San Diego State University taught by M.Bromley in Fall. Since its upload, it has received 50 views. For similar materials see /class/225318/phys608-san-diego-state-university in Physics 2 at San Diego State University.


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Date Created: 10/20/15
Lecture 20 Outline end of Coupled Bits a Normal Coordinates Section 63 0 Linear Triatomic Molecule Section 64 0 LC coupled circuits incl Section 25 o Oscillations theory of Section 6162 LTM Example T and V tensors V I g 77 277 77 771772 772771 772773 7737722 T 77 77 773 0 so we have the tensor forms k k 0 m 0 0 V k 2k k and T 0 M 0 0 k k 0 0 m 1 v 39 1 I 39 I p v v 2 p v v a nd the elgenvaslues w Vla V19677 w T19677 0 k w2m k 0 IV w2T k 2k w2M k 0 0 k k w2m o is cubic eqn w2k w2mkM 2777 w2Mm 0 Eigenfrequencies So differential equations T j77j V j77j 0 Equations of motion satis ed by 77 Cam m with n 7free Vibration frequencies cal wn Since the differential equations are satisfied by 77 then the linear combination of 77 s also is and since actual motion only dependent on real part of 77 7771 Z Ca k wkt Z fka k cosw t 6k k k initial co ordinate conds set amplitude fk and phase 67 Problem is that 77 25 won t necessarily repeat unless the frequencies wk are related by fractions LTM Example Eigenmodes 0 To work out how the system is actually oscillating o Transform into normal coordinates 77 awe ie 77 AC LTM Example Eigenmode 2 LTM Example Eigenmode 3 Congruence Transformation a We are interested in set of n eigenvalues Ak w a each of which has it s associated eigenvector 55k with soln VEL k ATEL k 0 Since T and V are real symmetric matrices All the M are real and so are components of 5k 0 see Goldstein Page 242 243 for proofs of which c There are issues when considering degenerate systems a We can construct a matrix A from n eigenvectors 6 ATA I and VA TAA gt AVA ATAA 0 ie congruence transforms A is a diagonal matrix 0 Choose Normalisation such that ATA I LTM Normalisation Transformation 0 Choose Normalisation such that ATA I Normal Coordinates in a slide To work out how the system is actually oscillating Transform into normal coordinates 77 aijg ie 77 A5 using matrix transpose property 77 A2 E 3 simplifies V Vz jnmj v ZAVAE gm 2 goalie and we can do a similar trick for kinetic energy 1 1 17 7 1 T Tzquotz 39 39T ATA E 2 mm 277 77 26 C 26ka Again all just about simultaneous transformations of both T and V in which we use the transformation A to rotate our axes to simultaneously diagonalise them compare with similarity transforms ATA lL Easy Lagrange Our new Lagrangian L wig so we can determine the equations of Ck motion easily 5k CUng 0 ie normal coordinate solutions Ck Olga WM All particles in each 7normal mode Vibration with wk amplitudes of each particle s motion determined by ajk Tying it all together General soln xi 330 77 2k Ckaike Wkt normal coords j ajmj Cke z wkt with components with magnitudes C A77 1 G11 Q21 G31 771 C2 Q12 Q22 Q32 772 C3 Q13 a23 Q33 773 so for cal 0 the Cl WU WH M772 771773 ie since w2 0 centre of mass is constant k k k a1s062 071 773 amp11dC3 771 2772 773 Linear Triatomic Molecule outtro Note that 1 C2 3 were all axis Vibrations in general n atoms then 3n coords removing centre of mass 3n 6 degrees of freedom For n 3 we have 3 dof but want to remove C1 Wh nm M 772 771773 by introducing coords 21 512 5131 with 22 513 512 and eliminate 5132 by COM ma1 5133 M5132 0 so 21 22 coords PLUS one transverse coord yl ie y1y2y3 simpli es with COM my1 241 Myg 0 RL circuits Lagrangian Way Electrical analog of mechanical problems GPS sect 25 Battery of voltage V inductance L and resistance R Consider RL circuit using dynamical variable q charge 1 2 l 2 T Lq and 7 Rq 2 2 a Potential energy qV with current I q a which means that the Lagrange equation of motion d 83 83 87 o dt aq aq Qf aq gt g q o With the usual steady state soln I l exp LC circuits coupled with capacitance C gives potential energy Q22C so for LC circuit again we have T ch d 83 813 q Lquot 0 dt aq aq q C ie oscillating solution q qo coswt with wo 1 LC so if we have a number of coupled circuits mutual inductance M17 providing kinetic coupling 1 2 1 q 3 52Lij gZMjkqjqk 2 j 97 j Abstract eigenexample Consider particle mass m L Vijxixj rescale mass such that diagonals of T 1 ATA I then AVA A has soln when V AI 0 Vll A V12 0 V11 Agtltv22 A we V21 V22 A ie 0 A2 V11 V22 V12V21 tWO 801118 M V11 V22 II V11 V222 4V12V21 use in some limits to nd eigenvectors using Zj O and 121 122 1 Abstract contd


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