Teoría de la perturbación en mecánica

8/13/2019 Teora de la perturbacin en mecnica

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Application of Perturbation theory in Classical Mechanics

Classical Mechanics

Instructor: Dr.Myles

by

Shashidhar Guttula


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Application of Perturbation theory in Classical Mechanics

Shashidhar Guttula.,Dept. of Physics, Texas Tech Uniersity

Abstract:! basic theoretical and "athe"atical oerie# of the utility of perturbation

theory in si"ple "echanical syste"s had been described in this paper.

Introduction:

Perturbation theory is a ery broad sub$ect #ith applications in "any areas of the

physical sciences. The basic principle is to find a solution to a proble" that is si"ilar to

the one of interest and then to cast the solution to the tar%et proble" in ter"s ofpara"eters related to the &no#n solution. Usually these para"eters are si"ilar to those of

the proble" #ith the &no#n solution and differ fro" the" by a s"all a"ount. The s"all

a"ount is &no#n as a perturbation and hence the na"e perturbation theory. The #ord

'perturbation( i"plies a s"all chan%e. Thus, one usually "a&es a s"all chan%e in so"e

para"eter of a &no#n proble" and allo#s it to propa%ate throu%h to the ans#er. )ne"a&es use of all the "athe"atical properties of the proble" to obtain e*uations that are

solable usually as result of the relatie s"allness of the perturbation.

It is the theory #hich is the study of the effects of s"all disturbances .If the effects ares"all, the disturbances are said to be re%ular, other#ise they are said to be sin%ular. The

basic idea in this theory is to obtain an approxi"ate solution of a "athe"atical proble"

by exploitin% the presence of a s"all di"ensionless para"eter+the s"aller the para"eter,

the "ore accurate the approxi"ate solution.

asically the perturbation theory can be diided into t#o approaches: ti"e dependent andti"e independent perturbations. There are "any point of analo%y bet#een the classicalperturbation techni*ues and their *uantu" counterparts. Generally, classical perturbation

theory is considerably "ore co"plicated than the correspondin% *uantu" "echanical

ersion.

!ll of the proble"s in classical "echanics fro" ele"entary principles, central force

proble"s, ri%id body "otion, oscillations, and theory of relatiity had al"ost exactsolutions but in chaos and adanced topics the %reat "a$ority of proble"s in classical

"echanics cannot be soled exactly and here the perturbation theory co"es into play to

sole the respectie solution in an approxi"ate fashion.

Perturbation -a"iltonian:

In a physical proble" that cannot be soled directly the -a"iltonian differs only sli%htlyfro" the -a"iltonian for a proble" that can be soled ri%orously. The "ore co"plicated

proble" is then said to be a perturbation of the soluble proble" and the difference

bet#een the t#o is called the perturbation -a"iltonian .Therefore the perturbation theoryconsist of techni*ues for obtainin% approxi"ate solutions based on the s"allness of the

perturbation -a"iltonian and on the assu"ed s"allness of the chan%es in the solutions.

In %eneral, een #hen the chan%e in the -a"iltonian is s"all, the eentual effect of the


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perturbation on the "otion can be lar%e. This su%%ests that any perturbation solution "ust

be carefully analyed to be sure that it is physically correct.

The differential e*uations that describe the dyna"ics of a syste" of particles are

definitely nonlinear and so one "ust be so"e#hat "ore cleer in applyin% the concept of

perturbation theory.

Perturbation expansion:

! re%ular perturbation case is an e*uation of the for": D/x01234 containin% a para"eter

such that the full solution xsol approaches the solution x4 of the si"plified e*uation

D/x0104234 #hich tends to 4.

The basic re%ular perturbation "ethodolo%y is si"ple:

5. 6rite the solution as a po#er series in

........777

8

77

9

7

54 ++++= xxxxxsol

9. Insert the po#er series into the e*uation and rearran%e to a ne# po#er series in:

D/xsol0(23 D/ ........777

8

77

9

7

54 ++++ xxxx 2

3P0/ 4x ,42P1/ 4x 0 5x 2P2 ( 4x 0 5x 0 9x 2;;..

8. Set each coefficient in the po#er series e*ual to 4 and sole the resultin% syste"s :

P0/x00423D/x004234

P1/x00x1234

P2/x00x10x2234

This deter"ines x00x10x2.The idea applies in "any different contexts :

a.approxi"ate solutions to al%ebraic and transcendental e*uations

b.approxi"ate expressions to definite inte%rals

c. ordinary and partial differential e*uations

The perturbation analysis is often co"ple"entary to nu"erical techni*uies.The

perturbation analysis so"eti"es %ies asy"ptotic relationships that are "ore useful thana s"all nu"ber of nu"erical experi"ents. -o#eer, in other cases there are really no

s"all para"eters and #e hae to rely on nu"erical solutions.

Applications in Classical Mechanics:


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a. Projectile Motion:

The pro$ectile "otion in 9+D #ithout considerin% air resistance #ith initial elocity of the

pro$ectile as 0 and the angle of elevation as ,then the force F=mg, the

force component becomes :

x-direction..

4 xm=

y-direction..

ymmg =

eglecting the height , ass!me x=y=0 at t=0,then

4..

=x , cos,cos 44.

tvxvx == "

sin9

,sin,4

9...

tvgt

yvgtygyo

+

=+==

The speed and total displace"ent as functions of ti"e are found to be

9


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cossin9

2/9

4

g

vRTtx ===

?ext ,if #e add the effect of air resistance to the "otion of the pro$ectile then there #illbe decreases in ran%e under the assu"ption that the force caused by air resistance is

directly proportional to the pro$ectile@s "otion

The initial conditions are the sa"e as aboe initial case

Vvty

Uvtx

tytx

===

===

====

sin24/

cos24/

424/24/

4

.

4

.

Then the e*uations of "otion, beco"e

mgykmym

xkmmx

=

=

...

...

The solution is 25/ kte

k

Ux = and 25/

9

ktek

gkV

k

gty

++=

The ran%e >@ #hich is the ran%e includin% the air resistance ,can be found as preiously

by calculatin% the ti"e T re*uired for the entire tra$ectory and then substitutin% this alueinto aboe e*uation for x. The ti"e T is found as preiously by findin% t3T #hen

y34.therefore fro" aboe e*uation #e find

25/ ktegk

gkVT

+=

This is a transcendental e*uation and #e cannot obtain an analytic expression for 'T(.

Therefore perturbation "ethod is used to find an approxi"ate solution .To use this"ethod,#e find an expansion para"eter #hich is nor"ally s"all and in the present case

this para"eter is the retardin% force constant & assu"in% it to be s"all. Axpand theexponential ter" of the transcendental e*uation in a po#er series #ith the intention of

&eepin% only the lo#est ter"s of &n ,#here & is the expansion para"eter.

............B

5

9

5C/ 8899 +

+= TkTkkT

gk

gkVT 2

If only the ter"s in the expansion throu%h ,this e*uation can be rearran%ed to yield


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9

8

5


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28

=5/7

g

kVRR = , the expansion #ill not coner%e unless k

g

kV ,5

sin4v

g

V

g=

The ran%e alues calculated approxi"ately: perturbation "ethod is plotted as a functionof the retardin% force constant &:

Note:!s the retardin% force constant is increased the ran%e >@ %ets decreased linearly

The si"ulation is carried out in Matlab E Si"ulin& Tool.

>ef: Fi%9+0 p%: B0 Marion H Thornton, Classical Dyna"ics of particles and syste"s,

=th Adition

b. Damped Harmonic Oscillator:

The oscillation in a si"ple har"onic oscillator is a free oscillation, once it is set into

oscillation, the "otion #ould neer cease. To analye the "otion in this case a ter"


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representin% the da"pin% force is incorporated into the differential e*uation. It is

assu"ed that the da"pin% force is a linear function of the elocity. Thus if a particle of

"ass " "oes under the co"bined influence of a linear restorin% force E&x and a

restorin% forcedt

dxb ,the differential e*uation describin% the "otion is :

49

9

=++ kxdt

dxb

dt

xdm

#hich can be #ritten as

49 9

49

9

=++ xdt

dx

dt

xd

-ere 3b


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The three+body proble" is one of the "ost celebrated proble"s in celestial "echanics..

In particular it focuses on the se"inal contribution of the French "athe"atician -enri

PoincarN #hose atte"pt to find a solution led hi" to the discoery of "athe"aticalchaos. The %eneral "athe"atics of the proble" is discussed in "any classic texts on both

analytical dyna"ics and celestial "echanics. The three+body proble" can be si"ply

stated: three particles "oe in space under their "utual %raitational attraction0 %ientheir initial conditions, deter"ine their subse*uent "otion. It can therefore be described

by a set of nine second+order differential e*uations. The proble" naturally extends to any

nu"ber of particles, and in the case of n particles it is &no#n as the n+body proble".

)er the years atte"pts to find a solution to the three+body proble" has spa#ned a#ealth of research. et#een 5O4 and the be%innin% of the t#entieth century "ore than

Q44 papers relatin% to the proble" #ere published ino&in% a roll call of distin%uished

"athe"aticians and astrono"ers, and hence, as is often the case #ith such proble"s, itsi"portance is no# perceied as "uch in the "athe"atical adances %enerated by

atte"pts at its solution, as in the actual proble" itself. These adances hae co"e in

"any different fields, includin%, in recent ti"es, the theory of dyna"ical syste"s.

! special case of the three+body proble" #hich has featured pro"inently in research as aresult of its si"plified for" and its practical applications is #hat PoincarN called the

7restricted7 three+body proble". In this for"ulation t#o of the bodies, &no#n as the

pri"aries, reole around their centre of "ass in circular orbits under the influence oftheir "utual %raitational attraction and hence for" a t#o body syste" in #hich their

"otion is &no#n. ! third body, %enerally &no#n as the planetoid, assu"ed "assless #ith

respect to the other t#o, "oes in the plane defined by the t#o reolin% bodies and,

#hile bein% %raitationally influenced by the", exerts no influence of its o#n. Theproble" is then to ascertain the "otion of the third body.

This particular case of the three+body proble" is the si"plest one of i"portance and in

the context of PoincarN@s #or& is especially si%nificant since "ost of his results pertain tothis for"ulation. !part fro" its si"plifyin% characteristics, it also proides a %ood

approxi"ation for real physical situations, as, for exa"ple, in the proble" of deter"inin%

the "otion of the "oon around the earth, %ien the presence of the sun. In this instance,

the proble" is al"ost circular /the eccentricity of the earth@s orbit is approxi"ately4.45O2, al"ost planar /both the earth@s orbit and the "oon@s orbit are nearly in the plane

of the ecliptic2, and the alues of the "ass ratios and the "ean distances bet#een the

bodies satisfy the conditions. The for"ulation also proides a reasonable approxi"ationto the syste" consistin% of the sun, Rupiter and a s"all planet.

!part fro" its intrinsic appeal as a si"ple to state proble", the three+body proble" has a

further attribute #hich has been responsible for the abundance of potential solers: its

inti"ate lin& #ith the funda"ental *uestion of the stability of the solar syste". That is,the *uestion of #hether the planetary syste" #ill al#ays &eep the sa"e for" as it has

no#, or #hether eentually one of the planets #ill escape fro" the syste" or, perhaps

#orse, experience a collision. It is a *uestion #hich has concerned astrono"ers forcenturies, eer since it #as first obsered that the "otions of the earth and of the other

planets #ere not precisely re%ular and periodic.


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Since bodies in the solar syste" are approxi"ately spherical and their di"ensions

extre"ely s"all #hen co"pared #ith the distances bet#een the", they can be considered

as point "asses. Under ?e#ton7s la# and to a first approxi"ation, the planets "oe inelliptical orbits around the sun, the sun bein% at one of the foci of the ellipse. This

description is a first approxi"ation because it only allo#s for the interaction bet#een the

sun and the particular planet #hose "otion is bein% described and does not ta&e intoaccount the forces bet#een the indiidual planets. These other forces cause perturbations

to the ori%inal elliptical orbit so that it ery slo#ly chan%es and it is conceiable that

these ery slo# chan%es could, after a ery lon% period of ti"e, alter the present orbits insuch a #ay that a planet could be thro#n out of the syste" or a collision could occur.

!lthou%h such a scenario does not a%ree #ith obserations "ade oer the last 5,444

years, it is *uite a different thin% to proe "athe"atically that it could not happen, and it

is the search for such a "athe"atical proof that proides the connection #ith the three+body proble". I%norin% all other forces such as solar #inds or relatiistic effects and

ta&in% only %raitational forces into account, the solar syste" can be "odelled as a ten+

body proble" hain% one lar%e "ass /the sun2 and nine s"all ones, and inesti%ated

accordin%ly.

ec!lar ert!rbation theory applied to #-ody problem:

It addresses lon%+period oscillations in planetary orbits, #ith a history of "ore than 944

years. Many of the funda"ental *uestions in celestial "echanics hae been ans#ered,so"e interestin% ones re"ain $ust beyond the scope of basic secular theory. The recent

burst of extrasolar planetary syste" detections has tri%%ered rene#ed interest in this

sub$ect, as si"ple extensions of the theory "ay hae the potential to explain "any of the

orbital properties of these syste"s.

The solution to a t#o+body syste", consistin% of a planet and a star, can be described inter"s of fie fixed orbital ele"ents that define an elliptical eplerian orbit, and a ti"e+depended one that %ies the position of the planet alone the orbit. In syste"s #ith "ore

than t#o planets, eplerian orbits are no lon%er exact solutions due to %raitational

interactions bet#een the planets. -o#eer, since the %raitational forces are stilldo"inated by the central body, each of the planets follo#s a nearly eplerian orbit. !

si"ilar set of orbital ele"ents, &no#n as osculatin% ele"ents can be defined at instant in

ti"e0 these ele"ents ary slo#ly due to perturbations fro" other planets. The lon%+ter"oscillations of the osculatin% orbital ele"ents in ti"e is the sub$ect of secular perturbation

theory.

The classical secular theory, deeloped by aplace and a%ran%e , be%ins #itha%ran%e7s planetary e*uations #hich are a set of ordinary differential e*uations %uidin%

the ti"e eolution of the osculatin% ele"ents in ter"s of the perturbin% potential The

secular parts of the perturbin% potential are obtained by aera%in% oer all releant orbitalperiods so that all short+period ter"s related to planetary positions alon% their orbits

anish. The result of this aera%in% procedure is that the se"i"a$or axes of the planets

re"ain constant, the pericenters and nodes precess, and the eccentricities and inclinationsary *uasi+periodically. ! linear approach has been applied to the Solar Syste" for "ore

than one and a half centuries details of the theory, #hich is second order in


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s"all orbital inclinations and eccentricities, and first order in "asses !lthou%h the

application of this lo# order theory is li"ited, it %ies useful conclusions for the Solar

Syste".

Secular Modes in T#o+ and Three+Planet Syste"s

The discoery of "ulti+planet extra solar syste"s of the si"ple t#o+planet syste" has

dra#n rene#ed attention. The ali%ned and anti+ali%ned states of orbits are actually the

t#o ei%en"odes and confir"ed that #ith eccentricity da"pin% due to any external forces,one of the t#o ei%en"odes decays *uic&ly leadin% to either apsidal ali%n"ent or anti+

ali%n"ent, dependin% on the "asses and se"i"a$or axes of the t#o planets. In addition to

apsidal loc&, the eccentricities ratio of the t#o planets re"ains constant, if eccentricity

da"pin% is the only external effect. This result can be used to deter"ine the eccentricityof one planet in a t#o+planet syste", if apsidal loc& is obsered and the eccentricity of

the other planet can be "easured.

6ith the rapid pace of obserational discoeries, "any additional extra solarsyste"s #ith t#o or "ore planets #ill e"er%e. It is i"portant to understand the secular

"odes and their interaction #ith other perturbations in these syste"s. To #or& in a threeplanet syste", the first proble" face is the identification of the ei%en"odes. In a t#o+

planet syste", there are t#o natural apsidal co+precession states /ali%ned and anti+ali%ned

and they co"pose the t#o ei%en"odes. -o#eer, in a three+planet syste", there are fournatural states #hile the syste" can hae only three ei%en"odes. Dependin%

on "ass ratios and orbital spacin%s, different sets of three ei%en"odes are selected fro"

the four natural states.

The secular potential application has the potential to explain the obsered si%nificant

non+ero eccentricities of the planets in syste"s. ! further step #ill be the study

of syste"s #ith four planets #ith applications to the "a$or planets in our Solar Syste".The possibility of da"pin% of inclinations and eccentricities of the %iant planets by

interactions can also be considered .

The solutions and applications for planetary syste"s #ith three or

four planets hae interestin% results fro" this theory . This approach allo#s to learn #hat

deter"ines about orbital spacin%s, eccentricities and inclinations in planetary syste"s.

imulation !esults:


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prin" #Mass system $ith no dampin" :

The sprin% "ass syste" #ith no da"pin% had been "odel and si"ulated in Si"ulin& #ithan i"pulse si%nal an as input to the syste".

Si"ulin& Model for sprin%+"ass syste" #ithout da"pin%

The e*uation of "otion for sprin%+"ass syste" hain% no da"pin% is %ien as,

49

9

=+ kxdt

xdm

)DA= /Dor"and+Prince2 al%orith" #ith relatie tolerance of 54e/+B2 is used for

si"ulations. I"pulse force is applied at t3s and total ti"e for si"ulation is 84s.


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I"pulse Input Si%nal

)utput >esponse: )scillations

prin"%mass system $ith dampin" factor:


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The e*uation of "otion for sprin%+"ass syste" #ith da"pin% is %ien as,

( ) 49

9

=+++ xkbdt

dxb

dt

xdm o

! ter" additional to & #hich constraints the position of body is also included.

)utput >esponse for Da"ped Syste"


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Syste" is run #ith the sa"e para"eters as preious. !s eident fro" the output response,

because of da"pin%, the oscillations die and body stabilies after 94s. The settlin% ti"e,

natural fre*uency and "axi"u" oershoot para"eters are function of sprin% constantand da"pin% factor of the syste" are %ien solution of second order e*uation /in laplace

do"ain2 as,

49 99

=++ nnss

Conclusions:

! si"ple approach of the use of Perturbation theory in "echanical syste"s has been

discussed and analyed. The "athe"atics inoled in this theory %ets co"plicated for

co"plex syste"s. This theory is applicable to "any areas of science li&e Vuantu"Mechanics, Se"iconductor Physics, -i%h+Aner%y particle Physics, etc;

For"al perturbation theory proides a nice ad$unct to the for"al theory of celestial

"echanics as it sho#s the potential po#er of arious techni*ues of classical "echanics indealin% #ith proble"s of orbital "otion. Due to the nonlinearity of the ?e#tonian

e*uations of "otion, the solution to een the si"plest proble" can beco"e ery

inoled.?eertheless, the "a$ority of dyna"ical proble"s inolin% a fe# ob$ects can

be soled one #ay or another. Therefore because of this nonlinearity that so "anydifferent areas of "athe"atics and physics "ust be brou%ht to%ether in order to sole

these proble"s.

!eferences:

5. Classical Dyna"ics of particles and syste"s, Marion HThornton =th Adition

9. Classical Mechanics, Golstein, Poole H Saf&o, Third Adition

8. ! First loo& at Perturbation theory, Ra"es G.Si""onds H Ra"es A.Mann,Rr

=. Perturbation theory in Classical Mechanics, F M Fernande, Aur.R.Phys.5Q /5O2

. Introduction to Perturbation Techni*ues ,?ayfeh. !.-

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