Physics'601'H omework'2333Due ' Friday,'September'17'!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!include!the!output!with!your!homework!solutions.!!Goldstein!!1.22,!!2.20,!13.4!!In!addition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x, y;˙ x ,˙ y ) =12m(˙ x 2+˙ y 2) "12m#2(x2+$y2)!:;%-%!! 6$!&!,&-&/%+%-!$,%96<>6'@!+;%!(%@-%%!#<!&'6$#+-#,>1!!!!&1 );#:!+;&+!+;%!%'%-@>2!! E =12m(˙ x 2+˙ y 2) +12m"2(x2+#y2)2!6$!9#'$%-=%(1!.1 M$%!+;%!%BA&+6#'$!#<!/#+6#'!+#!$;#:!+;&+!+;%-%!6$!&'#+;%-!9#'$%-=%(!BA&'+6+>!! " #12m(˙ x 2$˙ y 2) +12m%2(x2$&y2)1!91 );#:!+;&+!&'>!9#'$%-=%(!9A--%'+!#.+&6'&.*%!<-#/!&!,#6'+!+-&'$<#-/&+6#'2!! " #$L$˙ x $Q1$%%= 0+$L$˙ y $Q2$%%= 0:6+;!! Q1(x, y;"),Q2(x, y;")!:;%-%!! Q1(x, y;0) = x, Q2(x, y;0) = y2!6$!'%9%$$&-6*>!*6'%&-!6'!+;%!=%*#96+6%$!! ˙ x !&'(!! ˙ y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x'="x #$"(ct) ct'="(ct) #$"x"=11#$2!:;%-%!!! "=vc!6$!&!,&-&/%+%-!$,%96<>6'@!+;%!.##$+1!!!C;%!<6%*(!A'(%-!9#'$6(%-&+6#'!6$!&!$#D9&**%(!?#-%'+P!$9&*&-!<6%*(!! "(x,t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x, y;˙ x ,˙ y ) =12m(˙ x 2+˙ y 2) "12m#2(x2+$y2)!:;%-%!! 6$!&!,&-&/%+%-!$,%96<>6'@!+;%!(%@-%%!#<!&'6$#+-#,>1!!!!&1 );#:!+;&+!+;%!%'%-@>2!! E =12m(˙ x 2+˙ y 2) +12m"2(x2+#y2)2!6$!9#'$%-=%(1!.1 M$%!+;%!%BA&+6#'$!#<!/#+6#'!+#!$;#:!+;&+!+;%-%!6$!&'#+;%-!9#'$%-=%(!BA&'+6+>!! " #12m(˙ x 2$˙ y 2) +12m%2(x2$&y2)1!91 );#:!+;&+!&'>!9#'$%-=%(!9A--%'+!#.+&6'&.*%!<-#/!&!,#6'+!+-&'$<#-/&+6#'2!! " #$L$˙ x $Q1$%%= 0+$L$˙ y $Q2$%%= 0:6+;!! Q1(x, y;"),Q2(x, y;")!:;%-%!! Q1(x, y;0) = x, Q2(x, y;0) = y2!6$!'%9%$$&-6*>!*6'%&-!6'!+;%!=%*#96+6%$!! ˙ x !&'(!! ˙ y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x'="x #$"(ct) ct'="(ct) #$"x"=11#$2!:;%-%!!! "=vc!6$!&!,&-&/%+%-!$,%96<>6'@!+;%!.##$+1!!!C;%!<6%*(!A'(%-!9#'$6(%-&+6#'!6$!&!$#D9&**%(!?#-%'+P!$9&*&-!<6%*(!! "(x,t)!:6+;!+;%!!3. In!class!we!showed!that!for!particles!in!a!magnetic!field!time‐independent!gauge!transformations!changed!the!action!for!motion!between!fixed!points!but!did!so!in!a!manner!independent!of!the!path.!!In!this!problem!you!should!show!it!is!also!the!case!for!particles!in!an!electo‐magnetic!field!with!time‐dependent!gauge!transformations.!!The!Lagrangian!is! € L =12 ˙ x 2− qφ( x ,t) + A ( x ,t) ⋅ ˙ x ( )!and!the!gauge!transformation!is!given!by! €  A ( x ,t) → A '( x ,t) = A ( x ,t) + ∇ Λ( x ,t)φ( x , y) →φ'( x ,t) −∂Λ( x ,t)∂t!.!!!!!!"#!$"%&'%()%'*+,$"'-#"$+%.''%")+/0#"1)%2#+/'! "(x,t) #"(x',t')3'4($'-)5")+52)+',$+/2%&''0#"'%($'6)7$'$8*)%2#+'2/'527$+'9&''2/'527$+'9&'' ! L =12"t#( )2$12c2"x#( )2')3 ':/$'%($';*<$"=-)5")+5$'$8*)%2#+'0#"'%($')>%2#+' ! S = dx dt"L'%#'/(#6'%()%'%($'$8*)%2#+'#0'1#%2#+''0#"'%(2/'/&/%$1'2/'%($')"$<)%272/%2>'6)7$'$8*)%2#+'! "t2# c2"x2( )$(x,t) = 0'93 ?$"20&'%()%'%($'-)5")+52)+',$+/2%&@'L@'2/'-#"$+%.'2+7)"2)+%'!"#"$%()%''%($'0#"1'#0'%($'%")+/0#"1$,'L 2/'2,$+%2>)<'%#'%($'*+%")+/0#"1$,'#+$3'>3 ?$"20&'%()%'%($')>%2#+@' ! S = dx dt"L@'2/'-#"$+%.'2+7)"2)+%3''4#',#'%(2/'#+$'+$$,/'%#'02+,'%($'A)>#92)+'#0'%($'-#"$+%.'%")+/0#"1)%2#+3''',3