Homework 2

# Homework 2 -...

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Unformatted text preview: Physics 601 Homework 2­­­Due Friday, September 17    !"#\$%&\$'()*'+,-./,01'*22234.''50%67#'8.9:.-;.0'*<' !Hint:  For some of these problems  it will be helpful to use Mathematica or some  !other symbolic manipulation program.  If you make use of such a program please  "#\$%!&'(!)*&+&'!,-#.*%/\$!!0102!!0132!0140! include the output with your homework solutions.  !"#\$%&\$'()*'+,-./,01'*22234.''50%67#'8.9:.-;.0'*<' !  ! 5'!&((6+6#'7! ! !Goldstein  1.22,  2.20, 13.42 !!0132!0140! !,-#.*%/\$!!010   "#\$%!&'(!)*&+&':%!\$;#:%(!+;&+!<#-!%=%->!6'=&-6&'9%!#<!&!?&@-&'@6&'!A'(%-!&! ! 81 5'!9*&\$\$! In addition:  9#'+6'A#A\$!,#6'+!+-&'\$<#-/&+6#'!+;%-%!6\$!&'!&\$\$#96&+%(!9#'\$%-=%(!BA&'+6+>1!! 5'!&((6+6#'7!  ! C;%!9#'=%-\$%2!;#:%=%-2!6\$!'#+!+-A%DDD%=%->!9#'\$%-=%(!BA&'+6+>!(#%\$!'#+!;&=%! 81 &'!&\$\$#96&+%(!6'=&-6&'9%!A'(%-!&!,#6'+!+-&'\$<#-/&+6#'1!!E#-!%F&/,*%2!:%! 5'!9*&\$\$!:%!\$;#:%(!+;&+!<#-!%=%->!6'=&-6&'9%!#<!&!?&@-&'@6&'!A'(%-!&! G'#:!+;&+!+;%!%'%[email protected]>!6\$!9#'\$%-=%(!H<#-!\$>\$+%/\$!:6+;!'#!%F,*696+!+6/%! 9#'+6'A#A\$!,#6'+!+-&'\$<#-/&+6#'!+;%-%!6\$!&'!&\$\$#96&+%(!9#'\$%-=%(!BA&'+6+>1!! C;%!9#'=%-\$%2!;#:%=%-2!6\$!'#+!+-A%DDD%=%->!9#'\$%-=%(!BA&'+6+>!(#%\$!'#+!;&=%! (%,%'(%'9%I!%=%'!+;#[email protected];!6+!6\$!'#+!&\$\$#96&+%(!:6+;!6'=&-6&'9%!A'(%-!&!,#6'+! &'!&\$\$#96&+%(!6'=&-6&'9%!A'(%-!&!,#6'+!+-&'\$<#-/&+6#'1!!E#-!%F&/,*%2!:%! +-&'\$<#-/&+6#'1!!J'%!/[email protected];+!9#'\$6(%-!+;%!,#\$\$6.6*6+>!+;&+!!"!#\$!<-#/!%'%[email protected]>!&**! G'#:!+;&+!+;%!%'%[email protected]>!6\$!9#'\$%-=%(!H<#-!\$>\$+%/\$!:6+;!'#!%F,*696+!+6/%! 9#'\$%-=%(!BA&'+6+6%\$!&-%!'%9%\$\$&-6*>!&\$\$#96&+%(!:6+;!6'=&-6&'9%!#<!&! (%,%'(%'9%I!%=%'!+;#[email protected];!6+!6\$!'#+!&\$\$#96&+%(!:6+;!6'=&-6&'9%!A'(%-!&!,#6'+! ?&@-&'@6&'!A'(%-!9#'+6'A#A\$!,#6'+!+-&'\$<#-/&+6#'\$1!!K#:%=%-!+;6\$!+##!6\$! +-&'\$<#-/&+6#'1!!J'%!/[email protected];+!9#'\$6(%-!+;%!,#\$\$6.6*6+>!+;&+!!"!#\$!<-#/!%'%[email protected]>!&**! A'+-A%!&\$!:6**!.%!6**A\$+-&+%(!6'!+;6\$!,-#.*%/1!!L#'\$6(%-!&[email protected]%'%-&*!+:#D 9#'\$%-=%(!BA&'+6+6%\$!&-%!'%9%\$\$&-6*>!&\$\$#96&+%(!:6+;!6'=&-6&'9%!#<!&! ˙˙ ˙ ˙ (6/%'\$6#'&*!;&-/#'69!#\$96**&+#-7! L( x, y; x, y ) = 1 m( x 2 + y 2 ) " 1 m# 2 ( x 2 + \$y 2 ) ! 2 2 ?&@-&'@6&'!A'(%-!9#'+6'A#A\$!,#6'+!+-&'\$<#-/&+6#'\$1!!K#:%=%-!+;6\$!+##!6\$! :;%-%!! 6\$!&!,&-&/%+%-!\$,%96<>6'@!+;%!(%@-%%!#<!&'6\$#+-#,>1!!!! D A'+-A%!&\$!:6**!.%!6**A\$+-&+%(!6'!+;6\$!,-#.*%/1!!L#'\$6(%-!&[email protected]%'%-&*!+:# 2 ˙ + &1 );#:!+;&+!+;%!%'%[email protected]>2! E L 1 , y ˙ 2 ˙ ) y 22 + 1 m"˙2 " 2 #y 2 ) 2!6\$!9#'\$%-=%(1! ˙ ˙ x (6/%'\$6#'&*!;&-/#'69!#\$96**&+#-7!= (2xm(;x , y+ = 1)m( x 2 + y 2()x 1 m# 2 ( x 2 + \$y 2 ) ! 2 .1 ! 6\$!&!,&-&/%+%-!\$,%96<>6'@!+;%!(%@-%%!#<!&'6\$#+-#,>1!!!! :;%-%!M\$%!+;%!%BA&+6#'\$!#<!/#+6#'!+#!\$;#:!+;&+!+;%-%!6\$!&'#+;%-!9#'\$%-=%(! !2 ˙ ˙2 2 2 ˙ 2 x2 2 2 &1 BA&'+6+>! " # 1 m( x \$ E = + m( x%+ (y 2 ) \$ &y "12! ( x 2 + #y 2 ) 2!6\$!9#'\$%-=%(1! );#:!+;&+!+;%!%'%[email protected]>2! y ) 1 1 m 2 ˙ + 1 m ) 2 .1 M\$%!+;%!%BA&+6#'\$!#<!/#+6#'!+#!\$;#:!+;&+!+;%-%!6\$!&'#+;%-!9#'\$%-=%(! 91 );#:!+;&+!&'>!9#'\$%-=%(!9A--%'+!#.+&6'&.*%!<-#/!&!,#6'+! !! 2 ˙ y2 + 1 ˙ BA&'+6+>! " # 1 m( x 2 \$ \$L) \$Q1m% 2 ( x 2L \$Q2 ) 1! \$ \$ &y 2 2 +-&'\$<#-/&+6#'2! " # :6+;! Q1 ( x, y;"), Q2 ( x, y;") ! + 91 );#:!+;&+!&'>!9#'\$%-=%(!9A--%'+!#.+&6'&.*%!<-#/!&!,#6'+! ˙ ˙ \$x \$% % = 0 \$y \$% % = 0 ! ! \$Q2 :;%-%! Q1 ( x, y;0)" #x\$L \$Qx, y;0) \$Ly 2!6\$!'%9%\$\$&-6*>!*6'%&-!6'!+;%!! +-&'\$<#-/&+6#'2! = , Q2 ( 1 :6+;! Q1 ( x, y;"), Q2 ( x, y;") += ˙ ˙ \$˙ ˙ =%*#96+6%\$! x !&'(! y 1!! x \$% % = 0 \$y \$% % = 0 ! :;%-%! Q1 ( x, y;0) = x, Q2 ( x, y;0) = y 2!6\$!'%9%\$\$&-6*>!*6'%&-!6'!+;%! ! (1 NF,*&6'!:;>! " !9&''#+!.%!#<!+;%!<#-/!#<! " 1! !˙ ˙ =%*#96+6%\$! x !&'(! y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email protected];+<#-:&-(!+#!6'9*A(%!-%*&+6=6+>!\$6'9%!\$,&9%!&'(! O!,-6'96,*%!&(=&'+&@%!#<!A\$6'@!?&@-&'@6&'\$!6'!(%\$9-6.6'@!9#'+6'AA/!\$>\$+%/\$! +6/%!&-%!+-%&+%(!6'!&'!&'&*#@#A\$!:&>1!!C;6\$!:6**!.%!6**A\$+-&+%(!6'!+;%!<#**#:6'@! H<6%*(!+;%#-6%\$I!6\$!+;&+!6\$!\$+-&[email protected];+<#-:&-(!+#!6'9*A(%!-%*&+6=6+>!\$6'9%!\$,&9%!&'(! +6/%!&-%!+-%&+%(!6'!&'!&'&*#@#A\$!:&>1!!C;6\$!:6**!.%!6**A\$+-&+%(!6'!+;%!<#**#:6'@! \$6/,*%!,-#.*%/!6'!8!\$,&9%!(6/%'\$6#'1!?#-%'+P!!+-&'\$<#-/&+6#'\$!&-%[email protected]=%'!.>!! \$6/,*%!,-#.*%/!6'!8!\$,&9%!(6/%'\$6#'1!?#-%'+P!!+-&'\$<#-/&+6#'\$!&-%[email protected]=%'!.>!! 1 ! x ' = "x # \$" (ct ) ct ' = " (ct ) # \$"x " = 1 x ' = "x # \$" (ct ) ct ' = " (ct ) # \$"x " = 1 # !\$ 2 1# \$ 2 v :;%-%!! " = v !6\$!&!,&-&/%+%-!\$,%96<>6'@!+;%!.##\$+1!!!C;%!<6%*(!A'(%-! :;%-%!! " = c!6\$!&!,&-&/%+%-!\$,%96<>6'@!+;%!.##\$+1!!!C;%!<6%*(!A'(%-! c 9#'\$6(%-&+6#'!6\$!&!\$#D9&**%(!?#-%'+P!\$9&*&-!<6%*(! " ( x, t ) !:6+;!+;%! 9#'\$6(%-&+6#'!6\$!&!\$#D9&**%(!?#-%'+P!\$9&*&-!<6%*(! " ( x, t ) !:6+;!+;%! !   ! ! !   !!   !"#!\$"%&'%()%'*+,\$"'-#"\$+%.''%")+/0#"1)%2#+/' " ( x, t ) # " ( x ', t ' ) 3'4(\$' -)5")+52)+',\$+/2%&''0#"'%(\$'6)7\$'\$8*)%2#+'2/'527\$+'9&''2/'527\$+'9&'' 2 2 L = 1 (" t # ) \$ 1 c 2 (" x # ) ' 2 2 )3 ':/\$'%(\$';*<\$"=-)5")+5\$'\$8*)%2#+'0#"'%(\$')>%2#+' S = " dx dtL '%#'/(#6' ! ! %()%'%(\$'\$8*)%2#+'#0'1#%2#+''0#"'%(2/'/&/%\$1'2/'%(\$')"\$<)%272/%2>'6)7\$' 2 \$8*)%2#+' (" t2 # c 2" x )\$ ( x, t ) = 0 ' 93 ?\$"20&'%()%'%(\$'-)5")+52)+',\$+/2%&@'[email protected]'2/'-#"\$+%.'2+7)"2)+%'!"#"\$%()%''%(\$' ! 0#"1'#0'%(\$'%")+/0#"1\$,'L 2/'2,\$+%2>)<'%#'%(\$'*+%")+/0#"1\$,'#+\$3' >3 !\$"20&'%()%'%(\$')>%2#[email protected]' S = " dx dtL @'2/'-#"\$+%.'2+7)"2)+%3''4#',#'%(2/' ? #+\$'+\$\$,/'%#'02+,'%(\$'A)>#92)+'#0'%(\$'-#"\$+%.'%")+/0#"1)%2#+3''' ,3 B"#1'!)"%'>3'#+\$'6#*<,'>#+><*,\$'%()%'%(\$'0#"1'#0'%(\$';*<\$"'-)5")+52)+' \$8*)%2#+'2/'%(\$'/)1\$'2+')<<'0")1\$/3''?\$"20&'%()%'%(2/'2/'%"*\$3' !   3. In class we showed that for particles in a magnetic field time‐independent  gauge transformations changed the action for motion between fixed  points but did so in a manner independent of the path.  In this problem  you should show it is also the case for particles in an electo‐magnetic field  with time‐dependent gauge transformations.  The Lagrangian is  ˙ ˙ L = 1 x 2 − q φ ( x, t ) + A( x, t ) ⋅ x  and the gauge transformation is given by  2 ∂Λ ( x , t )  .    A( x, t ) → A' ( x, t ) = A( x, t ) + ∇Λ( x, t ) φ ( x, y ) → φ ' ( x, t ) − ∂t     ( € € ) ...
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