Langrangian Mechanics

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Lagrangian

MechanicsBy :

Monica Yasya Alifa(13302241031)

Ilma Ihsan Majid (13302241032)

!jiana (13302241033)

L!"hf #ahmi $oshaana"!n (13302241043)International Class

of PhysicsEducation 2013



Lagrangian Mechanics

In "his sec"ion %e %ill :

• &se "he 'amil"ons aria"ional *rinci*le+ "o sho% "he" in "he s*ecifc ca-ody ,alling in a !ni,orm grai"a"ional feld+ i" is e.!ialen" "o /e%"onla% o, mo"ion

• &se 'amil"ons aria"ional *rinci*le "o derie Lagranges e.!a"ions o,

,or a consera"ie sys"em and demon"ra"e "heir !se in seeral eam*le• ho% ho% Lagranges e.!a"ions o, mo"ion need "o -e modifed %hen

generalied ,orces o, cons"rain" are an considera"ion

• resen" Alem-er"s *rinci*le and !se i" "o derie Lagranges e.!a"ioany *hysical sys"em "ha" inoles any generalied ,orce+ incl!dingnonconsera"ie ones



'amil"ons 5aria"ional rinci*le:

6am*le• 'amil"ons aria"ional *rinci*le

• 7his e.!a"ion "a8en along a *a"h o, "he *ossi-le mo"ion o, a *hysical sys"ee"rem!m %hen eal!a"ed along "he *a"h o, mo"ion "ha" is "he one ac"!all

• L9 75 "he lagrangian (di;erence -e"%een i"s 8ine"ic and *o"en"ial ener

• In o"her %ords+

• o!" o, "he myriad %ays in %hich a sys"em co!ld change i"s confg!ra"ion d"ime in"eral "2"1+ "he ac"!al mo"ion "ha" does occ!r is "he one "ha" ei"her

maimies or minimies "he *receding in"egral




6am*le

79 mgy

59 my2<2

• 7he aria"ion in "he in"egral o,"he Lagrangian is gien -y




6am*le




6am*le

• I" is "r!e ,or any in"egral = o, a ,!nc"ion o, "he,!nc"ion y (and i"s frs" deria"ie) "ha" has "he

*arame"ric ,orm gien -y

• 7he res!l"ing=(a) does no" de*end on a "o frs" orderand i"s *ar"ial deria"ie anishes a" a 9 0+ ma8ing"he in"egral an e"rem!m only %hen y is e.!al "o "h

sol!"ion o-"ained ,rom /e%"on>s second la% o,mo"ion




6am*le



?eneralied @oordina"es

• @oordina"es are !sed "o defne "he *osi"ion in s*ace ensem-le o, *ar"icles

• #or eam*le+ consider "he mo"ion o, "he *end!l!m




• nly one scalar coordina"e is really needed "o s*eci,y "he *osi""he *end!l!m

• 7he *end!l!m really has only one degree o, ,reedom "ha" is+ i"moe only one %ay and "ha" %ay is along an arc o, radi!s r

• ?eneralied coordina"es are any collec"ion o, inde*enden" coor.+ (no" connec"ed -y any e.!a"ions o, cons"rain") "ha" j!s" s!;is*eci,y !ni.!ely "he confg!ra"ion o, a sys"em o, *ar"icles

• 7he re.!ired n!m-er o, generalied coordina"es is e.!al "o "hesys"em>s n!m-er o, degrees o, ,reedom




• #or eam*le+ a single *ar"icle a-le "o moe ,reely indimensional s*ace ehi-i"s "hree degrees o, ,reedomre.!ires "hree coordina"es "o s*eci,y i"s confg!ra"io

• 7he *osi"ion o, *ar"icle 1 co!ld -e s*ecifed -y "he

coordina"es (1+ y1+ 1)

• 7he *osi"ion o, *ar"icle 2 co!ld -e s*ecifed -y "hecoordina"es (2+ y2+ 2)




• An e.!a"ion o, cons"rain" eis"s+ ho%eer+ "ha" conn"he coordina"es




• In s*eci,yig "he *osi"ion o, "he sys"em+ %e *ic8ed "he a-oecoordina"es+ one -y one

• Ce %o!ld no" hae com*le"e ,reedom o, choice in "he selec*rocess+ -eca!se "he choice o, "he si"h coordina"e %o!ld -,orced on !s a,"er "he frs" fe had -een made

• I" %o!ld ma8e more sense "o choose ini"ially only fe inde*coordina"es+ say (D+ Y+ E+ θ,Φ)+ !nconnec"ed -y any e.!a"ioncons"rain"+ %here (D+ Y+ E) are "he coordina"es o, "he cen"ermass and (θ,Φ) are "he eni"h and aim!"hal angles+ %hichdescri-e "he orien"a"ion o, "he d!m--ell rela"ie "o "he er"9 0o %hen *ar"icle 1is direc"ly a-oe *ar"icle 2+ and Φ 9 0o "he *rojec"ion o, "he line ,rom *ar"icle 2 "o *ar"icle 1 on"o "*lane *oin"s *arallel "o "he ais)




• As a fnal eam*le+ consider "he si"!a"ion o, a *ar"iclecons"rained "o moe along "he s!r,ace o, a s*here Againcoordina"es (+ y+ ) do no" cons"i"!"e an inde*enden" se"

• 7hey are connec"ed -y "he cons"rain"

$ is "he radi!s o, "he s*here

• 7he *ar"icle has only "%o degrees o, ,reedom aaila-le ,omo"ion+ and "%o inde*enden" coordina"es are needed "o scom*le"ely i"s *osi"ion on "he s*here




• In general+ i, / *ar"icles are ,ree "o moe in "hreedimens*ace -!" "heir 3/ coordina"es are connec"ed -y m condo, cons"rain"+ "hen "here eis" n 9 3/ Fm inde*enden"generalied coordina"es s!;icien" "o descri-e !ni.!ely "*osi"ion o, "he / *ar"icles and n inde*enden" degrees o,,reedom aaila-le ,or "he mo"ion+ *roided "he cons"raino, "he "y*e descri-ed in "he *receding eam*les

• !ch cons"rain"s are called holonomic 7hey m!s" -ee*ressi-le as e.!a"ions o, "he ,orm




• @ons"rain"s "ha" canno" -e e*ressed ase.!a"ions o, e.!ali"y or "ha" are nonin"egra-le,orm are called nonholonomic+ and "he e.!a"iore*resen"ing s!ch cons"rain"s canno" -e !sed "

elimina"e ,rom considera"ion any de*enden"coordina"es descri-ing "he confg!ra"ion o, "hesys"em

• "he classic eam*le o, a nonholonomic cons"ra

a -all rolling along a ro!gh



@alc!la"ing Gine"ic and o"en"ial 6nergi7erms o, ?eneralied @oordina"es : A

6am*le

• A *end!l!m o, mass m a""ached "o a s!**or" o, massmoe in a single dimension along a ,ric"ionless+ hor

• 6ach mass needs "hree @ar"esian coordina"es+ -holon




6am*le• 7here are "%o degrees o, ,reedom ,or "he mo"ion an

generalied coordina"es necessary "o descri-e "heconfg!ra"ion o, "his sys"em

• 7he *o"en"ial and 8ine"ic energies o, "his sys"em can

e*ressed in "erms o, @ar"esian coordina"es and elas




6am*le

1 7he 8ine"ic energy is e*ressi-le as a .!adra",orm in "he generalied eloci"ies+ incl!ding across "erm

2 7he *o"en"ial energy is de*enden" on a singlegeneralied coordina"e+ in "his case+ "he cosin

o, an angle




6am*le• 7he eloci"y o, mass M rela"ie "o o!r fed iner"ia

,rame o, re,erence is

• 7he eloci"y o, mass m can -e e*ressed as "he eloci"y o, mass M *l!s "he eloci"y o, mass m

rela"ie "o "ha" o, mass M+ "ha" is

• eH is a !ni" ec"or "angen" "o "he arc along %hich "h

*end!l!m s%ings




6am*le

• An een more direc" %ay "o genera"e "he corree*ression ,or "he 8ine"ic energy can -e o-"ain-y no"ing "ha"



• 7rans,orma"ion rela"ing@ar"esian o, an esem-le o,*ar"icle "o "heir generaliedcoodina"es+ eam*le in ingle*ar"icle :

7hree degrees o, ,reedom!ncons"rained mo"ion in s*ace

• 7%o degrees o, ,reedomcos"rained "o a

• ne degrees o, ,reedomcons"rained



7he eloci"y "rans,orma"ion o, "hecoordina"e

7rans,orma"ion :

%here n is "hen!m-er o, degreeso, ,reedom



7he Lagrangian is a 8no%n ,!nc"ion o,"he generalied coordina"es and

eloci"ies

(1)

• 7he are ,!nc"ion o, "ime• 7he can -e e*ressed

• !-"i"!"es e.!a"ion a-oe ine.!a"ion (1)

Lagranges 6.!a"ion o, Mo"ion

@onsera"ie ys"em• Beca!se 9 0 + %e

• 6ach generalied coois inde*ende

o"hers+ as is each ar

• Lagrangian e.!mo"ion ,or a cons



?eneralied Momen"a : Ignora-@oordina"es

• Lagrangian in a ,ree *ar"iclemoing in s"raigh" line I"senergy 8ine"ic :

• Ass!me "ha" lagrangian is "he

*ar"ic!larly sim*le ,orm L 9 7+so

• Linear momen"!m * o,

*a

•

?eneralied coordina"es"he .!an

decalled ?enemomen"a c"o "he genecoordina"e



• A consera"ie sys"em can "hen-e %ri""en as

• 7he lagrangian does no"e*lici"y con"ain "he coordina"e

.8 + "hen



leng"h o, "he *end!l!m

r =l 9cons"an"

com*onen"s o, "he eloc

7he heigh" o, "he -o-meas!red ,rom "he y *

The Spherical Pendulum

Figure 10.6.1 The spherical pendulum



The Spherical Pendulum:Lagrangian function in xy plan



The Spherical Pendulum: The angular momentum about the vertical z



The Spherical Pendulum:!onical motion of the pendulum

• 7he al!e o, remains close"o "he al!e o, 0 6*ression,or s.!are is



The Spherical Pendulum:Po"er erie e#uation in $



The Spherical Pendulum:Period of ocillate harmonical



The Spherical Pendulum: The e%ective potential

Ac"!ally+ "he in"egra"ed e.!a"ion o, mo"ionis j!s" "he energy e.!a"ion in %hich "he"o"al energy



&orce of !ontraint:Lagrange 'ultiplier



&orce of !ontraint:Lagrange 'ultiplier Term

7hese "erms Ji are called generalied,orces o, cons"rain" 7hey are iden"ical "o,orces i, "heir corres*onding generalied

coordina"e .i is a s*a"ial coordina"e As %eshall see+ ho%eer+ "hey are "or.!es i, "heircorres*onding coordina"e .i is an ang!larcoordina"e



()*lembert) Principle: The principle of virtual "or+

By !sing /e%"on>s second la% o, mo"ion and-y incl!ding "he iner"ial "erm as a real,orce+ >Alem-er">s *rinci*le e*ressed in

e.!a"ion :



()*lembert) Principle:()*lembert) principle in (

• >Alem-er">s *rinci*le in 3 ,ora single *ar"icle e*ressed in"hreedimensional @ar"esiancoordina"es :

• 7he inde i re*resen"s one o,"hree @ar"esian coordina"es+and "he #+ are "he s!m o, all

• "he ,orce com*onen"s ac"ing on"he -ody along "he i"h direc"ion

• >Alem-er">s *rinci*le -e

• I, "he e"ernal ,orc

consera"ie+ "hey cderied ,rom a *o"en"ia,!nc"ion+



()*lembert) Principle:Lagrangian e#uation conervative ytem der

from variational principle

• I, "he *o"en"ial ,!nc"ion 5 isinde*enden" o, any generalied

eloci"y %e can incl!de "ha"



7he 'amil"onian #!nc"ion:'amil"ons 6.!a"ion

• @onsider "he ,ollo%ing ,!nc"ion o, "he generalied coordina"es

• #or sim*le dynamic sys"ems "he 8ine"ic energy 7 is a homogen.!adra"ic ,!nc"ion o, "he . >s+ and "he *o"en"ial energy 5 is a ,!o, "he .>s alone+ so "ha"

• #rom 6!ler>s "heorem ,or homogeneo!s ,!nc"ions+ %e hae

•7here,ore

• !**ose %e regard "he n e.!a"ions

• As soled ,or "he >s in "erms o, "he *>s and "he .>s:

7he ,!nc"ion ' is e.!al "o "he"o"al energy ,or "he "y*e o,sys"em %e areconsidering



• Ce can "hen e*ress ' as a ,!nc"ion o, "he *>s and "he .>s:

• 7hen calc!la"e "he aria"ion o, "he ,!nc"ion ' corres*onding " aria"ion Ce hae

•7he frs" and "hird "erms in "he -rac8e"s cancel+ -eca!se 9 aL<defni"ion Also+ -eca!se Lagrange>s e.!a"ions can -e %ri""en can %ri"e

• /o% "he aria"ion o, ' m!s" -e gien -y "he e.!a"ion

7he 'amil"onian #!nc"ion:'amil"ons 6.!a"ion



6DAML6

Solution:

• Ce hae

• 'ence

• 7he e.!a"ions o, mo"ion

• 7hen read

• 7he frs" e.!a"ioamo!n"s "o a res"a"ememomen"!mKeloci"y rel

in

• &sing "he frs" e.!

second can -

• n rearrangi

%hich is "he ,amiliar e.!"he harmonic oscilla"or

-"ain 'amil"on>s e.!a"mo"ion ,or a onedimensharmonic oscilla"or



6DAML6

Solution:

• Ce hae

• In *olar coordina"es+hence

• @onse.!en"ly

• 7he 'amil"on 6.

• 7he

• 7he las" "%o e.!a"ions yicons"ancy o, ang!lar mom

• 9cons"a

#ind "he 'amil"oniane.!a"ions o, mo"ion ,o*ar"icle in a cen"ral fe

#rom "%o gi

Langrangian Mechanics

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