Homework 2 -...

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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 
 
 ( € € ) ...
View Full Document

{[ snackBarMessage ]}

Ask a homework question - tutors are online