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This 16 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 19 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 15 3 Dead Rotating Bodies 0 end of Scattering Cross Sections Section 310 Quick look at the Lab point of View Section 311 Brief look at three bodies Section 312 Rotation Coordinates Section 41 Orthogonal Transformations Section 42 o more basic Linear Algebra Section 43 Euler Angles Section 44 Example II homework problem a tough but fun7 problem V7 E k Vr gt a 0 7quot a 7 Flashback to Lab Frame De ne 90 as the angle measured in the lab ie can convert between 9 and go In collision the scattering particle slows down since m2 gains kinetic energy T elastic scattering is Where total kinetic energy is constant eg the best fast neutron moderators use light elements Inelastic scattering is Where energy is transferred eg excitation of scattering particle and or target ThreeBody Problem o Has no general soln for both classical quantum mech a To examine this de ne relative to COM and also de ne s7 Fk ThreeBody Equation of motion a Given f Fk 0 Thus the equation of motion can be changed d2F1 Gm1m2 F1 F2 Gm1m3 F1 F3 ml Z 2 2 dt lTl T Ql lTl T Ql lTl T 3l lTl T 3l d2F1 GWQltF1 F2 GW3ltF1 F3 m2 a P a p d2 d2 d2 G quot i 81 Ewm dt2 dt2 dt2 5 2 o with m m1 mg m3 and the vector a GGlt1n2 3 13 823 33 0 ie 3 coupled equations some restricted solns possible Euler s Collinear 80111 a When we have mass ratio s m1 lt m2 lt 7713 1 2 3 0 Such that F1 F2 F3 51 53 s gt G all lie on the same line 0 Period 739 is the same with Confocal points ie same 0 Energy is negative so system is bound Lagrange s Equilateral 80111 0 Given vector 3 6 eqns decouple 0 Equivalent to two body problem Where each mass moves in elliptical orbit in a plane same focal point and period 0 this is soln also for m1 lt m2 lt m3 1 2 3 Restricted 3body problem a When Two masses are large and bound consider the a potential that a small mass feels as the system rotates httpWWWwikipedia orgWikiLagrangianpoint o Saddle points L2 L1 L3 unstable against perturbation a but L4 L5 are stable Lagrangian A solns Rotating Lagrangian 0 Consider the problem in cylindrical o ords p 6 z a given rotating frame 6 6 wt 1 dr 2 d6 2 c T V 2 V 62 2D 2mltdt 7 6 7 7 7 I l 2 2 d9 2 Q 2 I 31 2m dt p dt w dt V p 6 z 1 12 d1 e2 2m cit p dt dz d6 1 mwp2a Emp2w2 V p 6 0 ie we pick up the Coriolis and Centrifugal terms 0 At Lagrange Points we have p39 Z 0 Rotation Direction Cosines o Specify orientation With 9 direction cosines 0 eg 5133 axis speci ed by 3 direction cosines a11a12a13 o with eg all 008611 i z39cos 11 i iEii Zquot COS 611 i COS 612jCOS 613 k a11i 117 a13k 0 eg if Q13 008613 COS7T2 0 rotation 511 5132 plane Transformation Matrix 0 linear algebra of orthogonal transformations With o the direction cosines as matrix elements aw cos 6 5131 Q11 Q12 Q13 5131 I 7 A7 5132 Q21 Q22 Q23 5132 5133 Q31 Q32 Q13 5133 0 Warning Goldstein uses Einstein summation notation p v v p v v 972 E awa gt awxj j 0 ie sum is implied when index appears more than once u 2 u a note problem With xi gt xix 1e not Euler Angles o the 9 aw are not independent since length preserved 2 6 r 513513 513513 ajajakak ajakajak jkazjazk 0 ie we have siX equations awe 6 V k 1 2 3 o introduce the three Euler angles gb 6 gb a there are other common conventions headingpitchroll Euler Angle Transformations a We have vectors 5 D517 7 C5 and a B57 0 so the overall transform is 517 A51 BCDa cosgb singb 0 l 0 0 1 30er sinw 0 A singb cosgb OJ 0 cos6 sin6J sinw 0er OJ 0 0 l 0 sin6 cos6 0 0 l Full matrix expansion in Goldstein 446 eg all coswcosgb cos6singbsinrb Q33 cos6 0 De ne inverse A4L that takes 7 back to F ie ie need akiagj 6 so the matrix product AA lL I o Defn Orthogonal matrix has A4L A transpose of A Passive vs Active 0 Can think of transform either on axes or on vector o 18 passive vs active transforms Finite Rotations Consider active rotation of vector clockwise by angle 1 ie the co ordinate system is xed gt gt so we have NV NPcosltIgt F Fcoslt13 gt which with VQ F x 7 sin I gives Rotation formula F Fcoslt13 aF1 COSCIFX 81an Rotations don t commute 0 Problem is that rotations dont commute AB y BA 0 eg consider two rotations one 900 about 2 o and also one 900 about new y
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