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This 24 page Class Notes was uploaded by Jamaal McGlynn 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 24 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 24 Generating functions a Canonical Transforms Section 91 o Generating Functions Section 93 o Symplectic Approach Section 94 Poisson Brackets Section 95 Eqn s of Motion and Solution Section 96 Generating functions a The 71 contribs to variation of action only at end points a if F is a function of q p Q 132 it vanishes at end points 0 ie F is used as a canonical variables bridge between half of the old set and half of the new set 0 Simplest general example F F1q7 Q t then dFl aFl aFl 3171 PM PM IC 2M H Q IC dt Q ataqiqaQiQ pg 0 collecting the like terms since independent variables an an 8F 8 DP 862 a 0 ie F generates the eqns of canonical transforms Pr SHO Gen Funcs SHO Canonical Transform SHO Solutions SHO Phase Space Four basic canonicals trannies 0 see Goldstein Table 91 page 373 in my edition Four basic canonicals trannies 0 see Goldstein Table 91 page 373 in my edition Problem 922 point transforms a Two degrees of freedom given point transforms Q1 9 a1T1dQ2ZC11q2 i Find most general canonical transform eqn s for P1 P2 ii Show that with a choice for P1 P2 that 2 p p H 1 2 p291922 291 a can be transformed such that Q1 and Q2 are ignorable iii then solve the problem nding q1q2p1p2 as functions of time Point transforms Example Point transforms Example Point transforms Example Symplectic Approach lf time indep transforms Qi Qiq p B P q p Where Hamiltonian stays constant C H 39 1 H Q zampv 8Q 8Qq g acgyz j j a but inverse functions qj qjQ P and pi pjQ P a so we can consider Hqp t gt HQ7 132 ME a but we still want canonical transform eg Qi 3 so 3 3132 819339 8P O 3 an 8193 an o ie direct conditions quick time indep equations check of canonical transforms Poisson Brackets Defn Poisson brackets of two functions u v with respect to canonical variables q p is Wm 2 am 81 81 m k 8 519k 561k 5 Syniplectic structure q coupled with p and p with q eg if u and v are just q s or p s 8 86139 36139 8 Webb 7 7 k 8 519k 561k 52 v 8qz39 H 8q239 quot since aqj 613 but apj 0 for all z j Similarly 0 pupil and Qiapjl 517 ll9j7 Qz l IMPORTANTLY All poisson brackets are invariant under canonical transformations du dt Not another equation of motion rewrite Hamiltonian mechanics with Poisson brackets Following on from Hamilton s Eqns of motion 8U 8u 8u 8U 8H 8U 8H8u u H8u aq l an at 6 3m 3m aqi at at Simple cases are q i q H pi p H and 2 1 29 1 with transform Q 1C 3 IC and 39 Poisson s theorem Poisson bracket of any two constants of motion is also a constant of motion H u 21 07 can proove this tediously using Jacobi s identity u 21 2 10 10 u 21 0 and yet another equation of motion 0 assuming that in nitesimal change in u is indep of time a From Poisson bracket form of eqn s of motion 2 luHl r Z U HLH a so Taylor series expanding about t 0 du t2d2u t2 t t tH HH u0 dt 02dt2 0 u0 u 02 0 0 eg for constant acceleration F ma 2 1 Hf m max gtxH gt 97H17HEHla so soln 51313 930 p502 a 930 not 252 o a big Sledgehammer for such a little problem Lecture 19 Outline Coupled Oscillators o Oscillations theory of Section 6162 0 Normal Coordinates Section 63 0 Linear Triatornic Molecule Section 64 Oscillations formalism kinetic o In the distant past 171 Cg Zj 3quotqu 39 J 0 see Goldstein equations 171 and 172 on page 25 0 our generalised coords don t explicitly depend on time 2 1 1 87 1 T Z 5W3 Z in Z 8 Z 5 Z Mijqiqj 139 i j M o The coeffs Mm are functions of qk 3771 MijQ17Q277Qn mijQO17QO277QO7L 3 13 77k Qk 0 0 only include constants since T has quadratic terms in gl 0 so we have Mm E 7711701017 C102 7 Clan Tij and 1 1 T 5mmin 537mm LTlVI Example Linear Triatomic Molecule 0 Three atoms two of m and one of M 0 At equilibrium distance I apart both springs 16 0 potential energy in terms of 771 91 901 is k ltltx2 x1 22 5 mg x2 22 V 2 2 E 2 k 772 7712 E 773 7722 k 2 5 77 2773 7722 2771772 2772773 0 Where b 902 901 903 9302 Also T gag 933 923 so 77 7 73gt Oscillations formalism Lagrangian 0 So we have Lagrangian L T j77mj Vijnmj 0 so if the 77 s are the generalised coords d8 83 1 1 E677 a n EWWW 0 0 ie n equations of motion note Tm and lij Note that most of the time Tm is diagonal 5 017712 Vijnmj 0 so equations of motion Tm lljnj O with sum over j Eigenvalue equations So differential equations Tijijj lijnj O Suggests oscillating solution m Where Caz is some complex amplitude with C C is just some scale factor V coords But note that actual motion is given by real part so n linear homogeneous equations for ai s Vijaj w2TZjaj O This is just an eigenproblem V0 ATE requiring V11 ATM V21 AT21 l12 T12 leg Ang 0 LTM Example T and V tensors V 5 77 277 7722 2771772 27727732 T 77 773 0 so we have the tensor forms is lc O m 0 O V k 2k g and T O M O O lc k 0 O m 0 so we need to nd the eigenvalues w Via k w2m c 0 IV w2T lc 2k w2M k 0 O c k w2m 0 ie cubic eqn w2c w2mkM 2m w2Mm O The REAL new deal 0 We are interested in set of n eigenvalues Ak to each of which has it s associated eigenvector 07k with soln Vd k ATOTk Since T and V are real symmetic matrices All the Ak are real and so are components of 07k see Goldstein for proofs of which There are issues when considering degenerate systems at least two Ak s same 0 We can construct a matrix A from n eigenvectors 07k ATA I and VA TAA gt AVA ATAA A 0 ie similarity transforms A is a diagonal matrix
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