MEKANIKA LAGRANGE DAN HAMILTON.docx

8/16/2019 MEKANIKA LAGRANGE DAN HAMILTON.docx

1/13

MEKANIKA LAGRANGE DAN HAMILTON

PARADIGMA LAGRANGIAN DAN HAMILTONIAN

NAMA : SABTI WIDIYATI SUMARAH

NIM : 14302244006

PENDIDIKAN FISIKA A 2014

FAKULTAS MATEMATIKA DAN ILMU PENGETAHUAN ALAM

UNIVERSITAS NEGERI YOGYAKARTA

2016


2/13

BAB I

PENDAHULUAN

LATAR BELAKANG

Per!"!#!$!% &!%' (!%&!) (er$*(*%'!% +e%'!% ,-#! "*!.* +!.! !+!#!$ /*%'"

&!%' e#(!.)!% +!.! U%.*) +!,!. e(*!. ,er(!%+%'!% !%.!r! e)!%)! L!'r!%'!%

+!% $!#.-%!% +e%'!% (!) !)! ,er#* +#!)*)!% .e#") "e!r! e%+!"!r .er$!+!, !r!

,!%+!%' )e+*!%&! Per(!%+%'!% &!%' (!) .+!) +!,!. +!,! $!%&! +e%'!% e%&!)!%

-%.-$5-%.-$ ,e%&e#e"!!% !.!" )!"*" /"" &!%' "!! &!%' -(! +"e#e"!)!% +e%'!% !r!

!#! L!'r!%'e +!% $!#.-% !r! ,!%+!%' )e+*!%&! ,er#* +*%')!, "e(!( !r! ,!%+!%'

%#!$ &!%' e%*%.*% (!'!!%! "e(*!$ /e%-e%! /"" "e$!r*"%&! +,!%+!%' +!%!)$r%&! +e%'!% !r! (!'!!%! $!r*" +"e#e"!)!% !r! ,!%+!%' % -#e$ T$-!" S

K*$% +"e(*. "e(!'! ,!r!+'! 7K*$% 20028

Ke+*! e.-+e % (*)!% $!"# +!r .e-r5.e-r (!r* Mere)! (er!"!# +!r $*)*

)e+*! Ne9.-% +!% ere)! e!,!r)!% (!%&!) )e*+!$!% +!#! e%!%'!% !"!#!$

&!%' "!%'!. "*#. &!%' (er"/!. /") Per.!! .e)%) % e%''*%!)!% )--r+%!. **

Ar.%&! (*)!% +!r &!%' .er(!.!" ,!+! ,e%''*%!!% )--r+%!. ,er"e' ,!%!%' !.!* )*.*(

+!% "ee%"%&! )*!%..!" !,!,*% &!%' --) "e,er. )ee,!.!% -e%.* #%er

-e%.* "*+*. !.!* 7,!%!%'8 &!%' +'*%!)!% +!#! ee!$)!% !"!#!$ K--r+%!.

** .er"e(*. (!"!%&! +#!(!%')!% +e%'!% K + !%! 1 *%')% ; 2 *%')%


3/13

BAB II

PEMBAHASAN

A MEKANIKA LAGRANGED!#! )-%+" )$*"*" .er+!,!. '!&! &!%' .!) +!,!. +)e.!$* e#!#* ,e%+e)!.!%

Ne9.-% Se$%''! +,er#*)!% ,e%+e)!.!% (!r* +e%'!% e%%!* )*!%..!" /"" #!%

&!%' er*,!)!% )!r!).er".) ,!r.)e# "!# "e#"$ E%er' K%e.) +e%'!%

P-.e%"!#

Per"!!!% L!'r!%'!: L = T > V

+!%! T = E%er' K%e.) 7 ?-*#e8 T = 1

2m v

2

V = E%er' P-.e%"!# 7?-*#e8 V =mgh

Pe%+e)!.!% L!'r!%'!% +e%'!% Ne9.-%!:

*#! +!r e(!%+%')!% ,e%+e)!.!% Me)!%)! Ne9.-%! ,!+! Me)!%)!

L!'r!%'! &!%' ,e%+e)!.!% #e($ r% H*)* II Ne9.-% : H*)* II

Ne9.-%(er(*%&: @Per*(!$!% -e%.* &!%' .er!+ .!, ,er*(!$!% 9!).*

Se!r! M!.e!." !+!#!$ : F =

d ́p

dt 718

?!+ H*)* II Ne9.-% % (er#!)* ,!+! "*!.* (e%+! +e%'!% !""! .er.e%.*

e%'!#! ,er*(!$!% -e%.* .!, "er%' ,er*(!$!% 9!).* N!*%

,er"!!!% H*)* II Ne9.-% &!%' #e($ .er)e%!# !+!#!$ : Σ F =m á … (2 )

"e$%''! ,er"!!!% 718 $!%&! e%!+ -&!%' (!' ,er"!!!% 2 )!re%!

,er"!!!% 728 ++!,!.)!% +!r ,e%*r*%!% ,er"!!!% "e(!'! (er)*.: F =d ́p

dt

F =d (m v́ )

dt

¿m d v́

dt .

Σ F =m á

F = !


4/13

D!%! +,er.(!%')!% "e(*!$ ,!r.)e# (er,%+!$ )e +!#! R P-""%&!

"!#)!% .er'!%.*%' ,!+! 9!).* . ∈ R !+ /*%'" ++e/%")!%:

: R R %

+!r +e/%" % +!,!. +.e%.*)!% )ee,!.!%

; = q́ : R R %

,er"!!!% Euler Lagrangian

d

dt ( ∂ L∂ ´ x )= ∂ L∂ x S-#*" ,er"!!!% 'er!) e%''*%!)!% e.-+e L!'r!%'e +!,!. +!r +e%'!%

e#$!. ,er"!!!% E*#er L!'r!%'e +!% ,er"!!!% 'er!) ,e'!" + !.!" &!.* :∂L

∂ ´ x=m ´ x ;

∂ L

∂ x=−kx Ke*+!% +!r "-#*" !"%'5!"%' e%!+ :

∂ L

∂ ´ x=m ´ x

∂ L=m ´ x ∂ ´ x

∫∂ L=m∫ ´ x d ́x

L=m

(

1

2´ x2

)

T =1

2m ´ x

2

∂ L

∂ x=−kx

∂ L=−kx∂x

∫∂ L=−k ∫ x dx

L=−k (1

2 x

2

) V =

−1

2k x

2

?!+ "-#*" ,er"!!!% 'er!) ,e'!"

L=1

2m ´ x

2−1

2k x

2…

De%'!% e.-+e L!'r!%'e % ).! +!,!. e%!r "-#*" ,er"!!!% 'er!) +!% *'!

).! +!,!. e%!r ,er"!!!% 'er!) +!r "-#*" ,er"!!!% 'er!)%&! +!% ,er"!!!%

'er!)%&! +(er)!% -#e$ ,er"!!!% E*#er L!'r!%'e D,er-#e$ :


5/13

d

dt ( ∂∂ ´ x (12 m ´ x2−12 k x2))= ∂∂ x ( 12 m ´ x2−12 k x2)d

dt ( 12 m2 ´ x)=1

2k 2 x

d

dt m ´ x=−kx

m d ́x

dt =−kx

m ´ x=−kx

B GAYA PADA SISTEM KOORDINAT UMUM

?)! "e(*!$ ,!r.)e# e%'!#! ,er'e"er!% "e!*$ r +(!9!$ ,e%'!r*$

"e(*!$ '!&! !)" F '!&! &!%' (e)er! ,!+!%&! +%&!.!)!% +e%'!%

δW = F .δr= F x δx+ F y δy+ F z δz

D!#! (e%.*) &!' #e($ "e+er$!%! +%&!.!)!% +e%'!%

δW =∑i

F i δ x i

T!,!) (!$9! ,er"!!!% + !.!" .+!) $!%&! (er#!)* *%.*) ,!r.)e# .*%''!#

.e.!, *'! *%.*) "".e (!%&!) ,!r.)e# U%.*) "!.* ,!r.)e# $!r'! i !+!#!$ +!r 1

"!,! 3 U%.*) N ,!r.)e# $!r'! i !+!#!$ +!r 1 "!,! 3 N

?)! ,er.!(!$!% δ x i +%&!.!)!% +!#! )--r+%!. ** !)! +,er-#e$

δW =∑i ( F i∑

k

∂ xi

∂ qk δ qk )

¿∑i (∑k F i

∂ x i∂ qk

δ qk ) ¿∑

i (∑

k

F i∂ x i

∂ qk )δ qk Per"!!!% + !.!" +!,!. +.*#"

δW =∑k

Q k δ qk

+!%!


6/13

Qk =∑ ( F i ∂ xi∂ qk ) Be"!r!% Qk &!%' ++e/%")!% e%*r*. ,er"!!!% + !.!" +"e(*. +e%'!%

gaya umum O#e$ )!re%! ,er)!#!% Qk δ qk e#) +e%" *"!$! !)! +e%"

Qk !+!#!$ '!&! )! qk e%&!.!)!% !r!) +!% +e%" Qk !+!#!$ .-r)! )!

qk e%&!.!)!% "*+*.

GAYA UMUM UNTUK SISTEM KONSERVATIF

?)! "e(*!$ '!&! (e)er! ,!+! "e(*!$ ,!r.)e# +!#! "e(*!$ e+!% '!&!

)-%"er;!./ (e"!r%&! '!&! .er"e(*. +%&!.!)!% -#e$ ,er"!!!%

F i=−∂ V

∂ x i

+!%! V e%&!.!)!% "e(*!$ /*%'" e%er' ,-.e%"!# O#e$ )!re%! .* ,er**"!% '!&!

** +!,!. +%&!.!)!%

Qk =−(∂ V ∂ xi∂ xi

∂ qk ) er*,!)!% .*r*%!% ,!r"!# V .er$!+!, qk !)!

Qk =−( ∂ V ∂ qk ) M"!#)!% ).! e%''*%!)!% )--r+%!. ,-#!r q1=r q2=θ !)! '!&!

** +!,!. +%&!.!)!% +e%'!% Qr=∂ V

∂ r Qθ=

∂ V

∂ θ ?)!V er*,!)!% /*%'"

r "!! 7+!#! )!"*" '!&! "e%.r!#8 !)!Q

θ=0

Per"!!!% +/ere%"!# 'er!) *%.*) "*!.* "".e )-%"er;!./ +!,!. +!r )!

).! )e.!$* /*%'" L!'r!%'!% +!#! (e%.*) )--r+%!. .er.e%.* D "" #!% )! '!&!

r!,!.!% .+!) )-%"er;!./ "!#)!% %#!%&! !+!#!$

'

) C !)! ).! +!,!. e%*#")!%

)

) ) :

VCC

∂

∂−=

'

Se#!%*.%&! ).! +!,!. e%+e/%")!% "e(*!$ /*%'" L!'r!%'!% L=T −V +!%

e%*#")!% ,er"!!!% +/ere%"!# 'er!) +!#! (e%.*)


7/13


8/13

I% .!) #!% !+!#!$ ,er"!!!% 'er!) -"#!.-r $!r-%) "!.* +e%" +e%'!% '!&!

,ere+!

2 P!r)e# &!%' (er!+! +!#! Me+!% Se%.r!#

R**")!% ,er"!!!% L!'r!%'e 'er!) "e(*!$ ,!r.)e# +!#! "e(*!$ (+!%' + (!9!$

,e%'!r*$ '!&! "e%.r!# K.! ,#$ )--r+%!. ,-#!r 1 = r 2 = θ M!)!

( 222212

21 r r ;T θ+==

)(r VV =

( ( )r Vr r L 22221 −θ+=

Se#!%*.%&! +e%'!% e%''*%!)!% ,er"!!!% L!'r!%'e +,er-#e$ :

r r

L

=

∂

∂8r 7/ r

r

L 2−θ=

∂

∂

0L

=θ∂

∂θ=

θ∂

∂

2r L

O#e$ )!re%! "".e%&! .+!) )-%"er;!./ !)! ,er"!!!% 'er!)%&! !+!#!$ :

r

L

r

L

+.

+

∂

∂=

∂

∂

θ∂

∂=

θ∂

∂ LL

+.

+

)(r / r r 2 +θ= ( ) 0r

+.

+ 2=θ

E MEKANIKA HAMILTON

Per"!!!% H!#.-% *%.*) 'er!) ,!+! "e(*!$ /*%'" +!r )--r+%!. **

∑ −=)

) ) L ,:H

U%.*) "e(*!$ "".e +%!) "e+er$!%! e%er' )%e.) "".e !+!#!$ /*%'"

)*!+r!. +!r:

+!% e%er' ,-.e%"!#%&! er*,!)!% /*%'" "!! :

8:7V8::7TL) ) )

−=

Ber+!"!r)!% .e-re! E*#er *%.*) /*%'" $--'e% +,er-#e$

∑∑∑ =∂∂

=∂

∂=−

) )

)

) )

)

)

) ) T2:

T:

:

L:L ,:

O#e$ )!re%! .* :

∑ +=−−=−=)

) ) VT8VT7T2L ,:H

Per"!!!% % .!) #!% !+!#!$ e%er' .-.!# +!r "".e &!%' ).! .%!* Se#!%*.%&!

,!%+!%' % (*!$ ,er"!!!% &!%' +.*#" "e(!'! :


9/13

)

) :

L ,

∂

∂=

7) = 12 %8

+!% %&!.!)!% +!#!:

+!#! , +!%

8: ,7:: ) ) ) ) =

De%'!% ,er"!!!% + !.!" ).! +!,!. %&!.!)!% /*%'" H &!%' (er"e"*!!% +e%'!% ;!r!"

) ) : , δδ "e(!'! (er)*. :

∑

δ

∂

∂−δ

∂

∂−δ+δ=δ

)

)

)

)

)

) ) ) ) ::

L:

:

L ,:: ,H

S*)* ,er.!! +!% "*)* )e+*! &!%' !+! +!#! .!%+! )*r*%' "!#%' e%!+!)!% -#e$

)!re%! e%*r*. +e/e%") ) :FL , ∂∂= -#e$ )!re%! .*:

[ ]∑ δ−δ=δ)

) ) ) : , ,:H

V!r!" /*%'" H "e#!%*.%&! +!,!. +%&!.!)!% +!#! ,er"!!!% (er)*. :

∑

δ

∂

∂+δ

∂

∂=δ

)

)

)

)

)

::

H ,

,

HH

A)$r%&! +,er-#e$ :

D*! ,er"!!!% .er!)$r % +)e%!# +e%'!% ,er"!!!% )!%-%) H!#.-% *%.*) 'er!)

Per"!!!%5,er"!!!% % .er+r +!r 2% ,er"!!!% +e/er%"!# -r+e51 7(!%+%')!% +e%'!%

,er"!!!% L!'r!%'e &!%' e%'!%+*%' % ,er"!!!% +/ere%"!# -r+e52 Per"!!!%

H!#.-% (!%&!) +,!)! +!#! e)!%)! )*!%.* 7.e-r +!"!r 'e!#! !.-)8

F ONTOH PEMAKAIAN MEKANIKA HAMILTON

1 O"#!.-r H!r-%)

E%er' )%e.) +!% e%er' ,-.e%"!# "".e +!,!. +%&!.!)!% "e(!'! :

2


10/13


11/13

0 , =− θ

D*! ,er"!!!% &!%' .er!)$r e%*%*))!% (!$9! -e%.* "*+*. .e.!,

2 , )-%" .!% r $θ

= = θ =&

Se+!%')!% +*! ,er"!!!% "e(e#*%&! e(er)!%

r

8r 7V

r

$ ,r

3

2

r ∂

∂−==

*%.*) ,er"!!!% 'er!) +!#! !r!$ r!+!#


12/13

BAB III

PENUTUP

KESIMPULAN

1 Per"!!!% 'er!) ,!r.)e# &!%' +%&!.!)!% -#e$ ,er"!!!% L!'r!%'e +!,!. +,er-#e$

+e%'!% e%%!* e%er' )%e.) +!% e%er' ,-.e%"!# ,!r.)e# .!%,! ,er#* e%%!* '!&!

&!%' (er!)" ,!+! ,!r.)e# E%er' )%e.) ,!r.)e# +!#! )--r+%!. )!r.e"!% !+!#!$

/*%'" +!r )ee,!.!% e%er' ,-.e%"!# ,!r.)e# &!%' (er'er!) +!#! e+!% '!&!

)-%"er;!./ !+!#!$ /*%'" +!r ,-""

2 Per"!!!% L!'r!%'e er*,!)!% ,er"!!!% 'er!) ,!r.)e# "e(!'! /*%'" +!r )--r+%!.

** )ee,!.!% ** +!% *%')% 9!).* L = T > V

3 Per"!!!% H!#.-% .er+r +!r 2% ,er"!!!% +e/er%"!# -r+e51 7(!%+%')!% +e%'!%

,er"!!!% L!'r!%'e &!%' e%'!%+*%' % ,er"!!!% +/ere%"!# -r+e52 Per"!!!%

H!#.-% (!%&!) +,!)! +!#! e)!%)! )*!%.* 7.e-r +!"!r 'e!#! !.-)8

4 Per"!!!% H!#.-% *%.*) 'er!) ,!+! "e(*!$ /*%'" +!r )--r+%!. ** &!.*

∑ −=)

) ) L ,:H


13/13

DAFTAR PUSTAKA

B-!" M!r& 55 Mathematical Methods in the Physical Sciences 555

G-#+".e% He(er. 2000 !lassical Mechanics Third Edition Ne9 Y-r): A++"-% We"#e&

Gre'-r& D-*'#!" 2006 !lassical Mechanics Ne9 Y-r): !(r+'e U%;er".& Pre""

M-r% D!;+ 2004 "ntroduction to !lassical Mechanics #ith Pro$lems and Solutions

Ne9 Y-r): !(r+'e U%;er".& Pre""

Me)!%)!#!'r!%'!%,+/ $..,":999!!+e!e+*14203Me)!%)!JL!'r!%'!% 20

Me 2016

Me)!%)!#!'!r!%'e,+/ $..,:".!//*%&!+".e"+e/!*#./#e",e%++)!%S*,!r+

20MSMEKANIKA20LAGRANGE,+/ 23 e 2016

+%!)!#!'r!%'r+!%$!#.-%,+/ /#e::U"er"MYD-9%#-!+"13145.*'!"5

.r!%"#!.e5$!,.er125L!'r!%'!%5!%+5H!#.-%!%,+/ 23 e 2016
https://www.academia.edu/8142053/Mekanika_Lagrangianhttp://staff.uny.ac.id/sites/default/files/pendidikan/Supardi,%20M.Si/MEKANIKA%20LAGRANGE.pdfhttp://staff.uny.ac.id/sites/default/files/pendidikan/Supardi,%20M.Si/MEKANIKA%20LAGRANGE.pdfhttp://c/Users/MY/Downloads/173777149-tugas-translate-chapter12-Lagrangian-and-Hamiltonian.pdfhttp://c/Users/MY/Downloads/173777149-tugas-translate-chapter12-Lagrangian-and-Hamiltonian.pdfhttp://staff.uny.ac.id/sites/default/files/pendidikan/Supardi,%20M.Si/MEKANIKA%20LAGRANGE.pdfhttp://staff.uny.ac.id/sites/default/files/pendidikan/Supardi,%20M.Si/MEKANIKA%20LAGRANGE.pdfhttp://c/Users/MY/Downloads/173777149-tugas-translate-chapter12-Lagrangian-and-Hamiltonian.pdfhttp://c/Users/MY/Downloads/173777149-tugas-translate-chapter12-Lagrangian-and-Hamiltonian.pdfhttps://www.academia.edu/8142053/Mekanika_Lagrangian

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MEKANIKA LAGRANGE DAN HAMILTON.docx