Goldstein 2e errata

Errata Sheet fo r Herb ert Goldstein/'sClassical Mechanics /, Second Edition /#28/1/9/8/0/#29b yF o rrest M/. Ho/#0Bman and Michael A/. UnserenOak Ridge National Lab o rato ryRevised/: July /2/1/, /1/9/9/2Belo wa re some of the mo re p rominent erro rs from Herb ert Goldstein/'s Classical Mechanics /#28Cop yright /1/9/8/0/,ISBN /0/#7B/2/0/1/#7B/0/2/9/1/8/#7B/9 /#29/, a text used in almost every graduate physics p rogram in the U/.S/. and referenced b ymany p racticing engineers and physicists w o rldwide/. This errata sheet is p robably not complete/. If additionalerro rs o rp roblems a re found in Classical Mechanics /, please send them to F o rrest Ho/#0Bman via electronic mail atforrest/@esdhof/.es d/.orn l/.gov o r to Michael Unseren at myu/@mars/.epm/.ornl/.g ov /.I f y ou do not have accessto electronic mail/, send co rrections to F o rrest Ho/#0Bman/, Oak Ridge National Lab o rato ry /, MS /6/0/3/6/, Oak RidgeTN /3/7/8/3/1/#7B/6/0/3/6/. If y ou w ould lik e to receive revised copies of this errata sheet in the future/, please send mail toF o rrest Ho/#0Bman and y ou will b e added to the mailing list/. This errata sheet ma y b e freely distributed so long asthe info rmation ab ove remains with it/./#0F P age /8/, v /0equation/, middle of page y /:v /0i /= d r /0idt/#0F P age /1/8/, equation /#28/1/#7B/4/8/#29/:Xi Fi /#01 /#0E ri /= Xi/;j Fi /#01 /@ ri/@qj /#0Eqj/= Xj Qj /#0Eqj /; /#28/1 /, /4/8/#29/#0F P age /1/9/, ab ove equation /#28/1/#7B/5/1/#29/:ddt /#12/@ ri/@qj /#13/= /@ /_ri/@qj /= Xk /@ /2ri/@qj /@qk /_ qk /+ /@ /2ri/@qj /@t /;/= /@ vi/@qj /;/#0F P age /1/9/, ab ove equation /#28/1/#7B/5/2/#29/:Xj /#28ddt /#20/@/@ /_ qj /#20Xi /1/2 mi v /2i /!/!/, /@/@qj /#20Xi /1/2 mi v /2i /!/, Qj /#29/#0Eqj /:/#0F P age /2/4/, equation /#28/1/#7B/6/9/#29/:Qj /= Xi Fif /#01 /@ ri/@qj /= /, Xrv F/#01 /@ ri/@qi/= /, Xrv F/#01 /@ /_ri/@ /_ qj /; b y/#28 /1 /, /5/1/#29 /;/= /, /@ F/@ /_ qj /: /#28/1 /, /6/9/#29y a dagger indicates an erro r in b oth the co rrected /2nd p rinting and later p rintings of the second edition/./1 /#0F P age /4/5/, equation /#28/2/#7B/1/9/#29/:I /= Z/2/1 L /#28 qi /; /_ qi /;t /#29 dt /#28/2 /, /1/9/#29/#0F P age /6/0/, equation reference b elo w equation /#28/2/#7B/5/1/#29 y /:and /#28/2/#7B/5/1/#29 can b e rewritten as /#0F P age /7/2/, fo otnote y /:F ormally/: /_r /=/_ r nr /+ r /_/#12 n/#12 /,h e n c e r /#02 /_ r /= /0 requires /_/#12 /=/0 /./#0F P age /7/5/, b elo w equation /#28/3/#7B/2/0/#29/:E/; l /; r/0 /;/#12/0 /. These constan ts are not the only ones that can b e considered/. W e/#0F P age /7/7/, b elo w equation /#28/3/#7B/1/5 /0/#29 y /:/#28F or p ositiv e k the min us sign ensures that the force is towar d the cen ter of force/./#29/#0F P age /8/7/, equation /#28/3/#7B/3/6/#29/:/#12 /= Zrr/0 drr /2 q/2 mEl /2 /, /2 mVl /2 /, /1r /2 /+ /#12/0 /#28/3 /, /3/6/#29/#0F P age /9/1/, ab ove equation /#28/3/#7B/4/3/#29/, equation /#28/3/#7B/4/3/#29/, and equation /#28/3/#7B/4/3 /0/#29 y /:d /2V /0dr /2 /#0C/#0C /#0C /#0Cr /= r/0 /= /, d fdr /#0C/#0C /#0C /#0Cr /= r/0 /+ /3 l /2mr /4/0 /#3E /0 /:d fdr /#0C/#0C /#0C /#0Cr /= r/0 /#3C /, /3 f /#28 r/0 /#29r/0 /; /#28/3 /, /4/3/#29d ln fd ln r /#0C/#0C/#0C/#0Cr /= r/0 /#3E /, /3 /: /#28/3 /, /4/3 /0/#29/#0F P age /1/3/1/, ab ove equation /#28/4/#7B/5/#29/:i /=/#28 i /#01 i /0/#29 i /0/+/#28 i /#01 j /0/#29 j /0/+/#28 i /#01 k /0/#29 k /0/#0F P age /1/4/9/, equation /#28/4/#7B/5/4/#29/:/#0E /= /, /#0B /#03 /#0C/#0D /#03 /; /#28/4 /, /5/4/#29/#0F P age /1/5/7/, section numb er in heading at the top of the page/:/4/#7B/5 THE CA YLEY/-KLEIN P ARAMETERS AND RELA TED QUANTITIES /1/5/7/#0F P age /1/5/9/, sentence ab ove equation /#28/4/#7B/8/2/#29 y /:homogeneous/, the Eqs/. /#28/4/#7B/8/1/#29 can ha v e a nonzero/, non trivial solution only when the determinan to f/#0F P age /1/7/6/, /!x /0and /!y /0equations in equation /#28/4/#7B/1/2/5/#29/:/2 /!x /0/= /_/#1E sin /#12 sin /#20 /+ /_/#12 cos /#20/!y /0/= /_/#1E sin /#12 cos /#20 /, /_/#12 sin /#20 /#28/4 /, /1/2/5/#29/#0F P age /1/8/6/, /!x equation in p roblem /1/9/:/!x /= /_/#12 cos /#1E /+ /_/#20 sin /#12 sin /#1E/;/#0F P age /1/8/9/, second pa ragraph/, seventh line/:almost all problems soluble in practice will allo w for suc h a separation/. In suc hc a s e/#0F P age /1/9/2/, equation /#28/5/#7B/1/0/#29/:T /0ij k /:/:/: /#28 x /0/#29/= ail ajm akn /:/:/: Tlmn/:/:/: /#28 x /#29 /: /#28/5 /, /1/0/#29/#0F P age /1/9/6/, third equation/:I /= /2 T/! /2/#0F P age /1/9/7/, ab ove last equation y /:The inertia tensor for the origin O /,i nt h e d y adic form of Eq/. /#28/5/#7B/1/6/#29/, can b e written/#0F P age /1/9/8/, second equation/:Ixy /= Iyx/#0F P age /2/1/1/, fo otnote equation y /:/_/! /! /!/= /#0A /#02 /! /! /!/;/#0F P age /2/1/2/, second equation y /:/#0A/= /!/3/3/0/5 /: /8/1/0/3/9 /#19 /!/3/3/0/6/#0F P age /2/1/2/, second sentence b elo w second equation y /:of precession of appro ximately /3/0/6 da ys or ab out /1/0 mon ths/. If some circumstance disturb ed the/#0F P age /2/1/3/, equation /#28/5/#7B/5/0/#29/:T /= I/1/2 /#10/_/#12 /2/+ /_/#1E /2sin /2/#12 /#11/+ I/3/2 /#10/_/#20 /+ /_/#1E cos /#12 /#11/2/; /#28/5 /, /5/0/#29/#0F P a g e/2 /2 /1 /,b e l o w equation /#28/5/#7B/7/7b/#29/:/#12/; /#1E /; /#20/; /_/#12/; /_/#1E /; and sa y /, either /_/#20 or /!/3 at the time t /= /0/. Because they are cyclic the/#0F P age /2/2/7/, middle pa ragraph y /:non v anishing correction term in Eq/. /#28/5/#7B/8/4/#29 to the p oten tial for a sphere/. No w/, the/#0F P age /2/2/8/, equation /#28/5/#7B/8/7/#29 y /:V /= /, GM mr /+ GM/2 r /3 /#5B/3 Ir /, /#28 I/1 /+ I/2 /+ I/3 /#29/#5D /: /#28/5 /, /8/7/#29/3 /#0F P age /2/2/8/, equation /#28/5/#7B/8/8/#29 y /:V /= /, GM mr /+ GM /#28 I/3 /, I/1 /#29r /3 P/2 /#28 /#0D /#29 /: /#28/5 /, /8/8/#29/#0F P age /2/2/9/, equation /#28/5/#7B/8/9/#29 y /:V/2 /= GM /#28 I/3 /, I/1 /#29r /3 P/2 /#28 /#0D /#29 /: /#28/5 /, /8/9/#29/#0F P age /2/4/0/, equation in p roblem /1/9b y /:sin /#12 /0/= /#0A/_/#1E sin /#12 /0/0/;/#0F P a g e/2 /4 /1 /,p roblem /2/5/, sixth line y /:T o pro v e this statemen t/, calculate /#12 and /_/#1E as a function of time for a hea vy symmetrical/#0F P a g e/2 /4 /1 /,p roblem /2/5/, /#0A equation y /:/#0A/= I/3 /, I/1I/1 /!/3/#0F P age /2/4/8/, fourth line y /:and subtract the result from the similar pro duct of Eq/. /#28/6/#7B/1/6/#29 from the left with a yl /./#0F P age /2/5/6/, equation/, middle of the page/:/#11 /#11 /#11/= /~B B By y y/; /~/#11 /#11 /#11/= /~y y yB B B/;/#0F P age /2/5/9/, equation /#28/6/#7B/5/3/#29 y /:j V V V/, /! /2T T Tj /= /#0C/#0C /#0C /#0C /#0C /#0C k /, /! /2m /, k /0/, k /2 k /, /! /2M /, k/0 /, k k /, /! /2m /#0C/#0C /#0C /#0C /#0C /#0C /=/0 /: /#28/6 /, /5/3/#29/#0F P a g e/2 /7 /6 /,l a s tp a ragraph y /:On the other hand/, the transformation represen ted b y Eqs/. /#28/7/#7B/2/#29 and /#28/7/#7B/4/#29/,/#0F P age /2/7/7/, second pa ragraph/, ninth line y /:systems mo ving uniformly relativ et o e a c h other/. Measuremen ts made en tirely/#0F P age /2/7/8/, second pa ragraph/, third and fourth lines y /:coincide at zero time/, as seen b y observ ers in b oth systems/. Let one system/, call itthe primed system/, mo v e uniformly with v elo cit y v relativ e to the other/, unprimed/#0F P age /2/8/5/, equation /#28/7/#7B/3/0/#29 y /:L/4/4 /= /,/1 /, /#0C /2 /#01/, /1/2/#11 /#0D/: /#28/7 /, /3/0/#29/#0F P age /3/0/0/, /#0Cfth line from the b ottom y /:measured in the rest system is alw a ys shorter than the corresp onding time in terv al/#0F P age /3/1/1/, equation /#28/7/#7B/1/0/3/#29 y /:/4 P/#16 P/#16 /= /, /,m /2/1 /+ m /2/2 /#01c /2/+/2 p/1 /#16 p/2 /#16 /: /#28/7 /, /1/0/3/#29/#0F P age /3/4/4/, ab ove equation /#28/8/#7B/1/7/#29 y /:q and t /. The conjugate momen ta/, considered as a column matrix p p p/,i s t h e n b y Eq/./#0F P age /3/5/0/, ab ove equation /#28/8/#7B/4/1/#29 y /:/_ x /0/, with the single comp onen to f a a ab eing mv/0 /. The new Hamiltonian is no w/#0F P a g e/3 /6 /3 /,b e l o w equation /#28/8/#7B/7/0/#29 y /:Equations /#28/8/#7B/6/9/#29 and /#28/8/#7B/7/0/#29 are exactly Hamilton/'s equations of motion/, Eqs/./#0F P age /3/8/3/, ab ove equation /#28/9/#7B/1/5/#29 y /:rather than /_Qi /. This can b e accomplished b y writing F in Eq/. /#28/9/#7B/1/1/#29 as/#0F P age /3/8/5/, q/2 equation y /:q/2 /= /, /@F /0/@p/2 /;/#0F P age /4/3/1/, P equation in p roblem /5/:P /= /#0Bq /2/2 /#12/1/+ p /2/#0B /2q /2 /#13/#0F P age /6/0/7/, elements /#28/3/,/1/#29 and /#28/3/,/2/#29 in equation /#28B/#7B/3y/#29 y /:A A A/= /0/@ /, sin /#20 sin /#1E /+c o s /#12 cos /#1E cos /#20 sin /#20 cos /#1E /+c o s /#12 sin /#1E cos /#20 /, cos /#20 sin /#12/, cos /#20 sin /#1E /, cos /#12 cos /#1E sin /#20 cos /#20 cos /#1E /, cos /#12 sin /#1E sin /#20 sin /#20 sin /#12sin /#12 cos /#1E sin /#12 sin /#1E cos /#12 /1A/#28B /, /3y/#29/5