(eBook) Classical Mechanics - SOLUTIONS (Cambridge, 2006) - Gregory.pdf

文档介绍：CLASSI CAL MECHANI CSSOLUTIONS MANUALR. Douglas GregoryNovember 2006Please report any errors in these solutionsby emailingcm:******@2Contents1 The algebra and calculus of vectors32 Velocity, acceleration and scalar angular velocity 273 Newton’s laws of motion and the law of gravitation 624 Problems in particle dynamics765 Linear oscillations and normal modes 1396 Energy conservation1797 Orbits in a central ?eld2218 Non-linear oscillations and phase space 2769 The energy principle30610 The linear momentumprinciple33511 The angular momentumprinciple 38112 Lagrange’s equations and conservation principles 42913 The calculus of variations and Hamilton’s principle 47314 Hamilton’s equations and phase space 50515 The general theory of small oscillations 53316 Vector angular velocity57717 Rotating reference frames59018 Tensor algebra and the inertia tensor 61519 Problems in rigid body dynamics 646Chapter OneThe algebra and calculusof vectorsc?Cambridge University Press, 2006Chapter 1 The algebra and calculus of vectors4Problem1. 1In terms of the standard basis setfi;j;kg,aD2i?j?2k,bD3i?4kandcDi?5jC3k.(i) Find3aC2b?4candja?bj2.(ii) Findjaj,jbjanda?b. Deduce the angle betweenaandb.(iii) Find ponent ofcin the direction ofaand in the direction ofb.(iv) Finda?b,b??b/?.b?c/.(v) Finda?.b?c/?b/?cand verify that they are equal. Is the setfa;b;cgright- or left-handed?(vi) By evaluating each side, verify the identitya?.b?c/?c/b?.a?b/(i)3aC2b??j?2k/?4k/??5jC3k/D8iC17j?26k:ja??b/?.a?b/D.?i?jC2k/?.?i?jC2k/D.?1/2C.?1/2C22D6:(ii)jaj2Da??j?2k/?.2i?j?2k/D22C.?1/2C.?2/2D9:Hencejaj ??4k/?.3i?4k/D32C.?4/2D25:Hencejbj ??j?2k/?.3i?4k/D?2?3?C?.?1/?0?C?.?2/?.?4/?D14:c?Cambridge University Press, 2006Chapter 1 The algebra and calculus of vectors5The angle?betweenaandbis then given bycos?Da?bjaj jbjD143?5D1415:Thus?Dtan?11415.(iii) ponent ofcin the direction ofaisc?baDc??ajaj??5jC3k/??2i?j?2kj2i?j?2kj?D?1?2?C?.?