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Proceedings of XLIII International Summer SchoolConference APM 2015

Multidimensional pendulum in a nonconservative force eld
wxim F hmolin shmolindrmlerFruD shmolindimeFmsuFru

Abstract

sn this tivityD we systemtize the results on the study of the equtions of motion of dynmilly symmetri multidimensionl rigid odies in nononE servtive fore (eldsF he form of these equtions is tken from the dynmis of rel lowerEdimensionl rigid odies interting with resisting medium y lws of jet )ows where ody is in)uened y nononservtive tring foreY under tion of this foreD the veloity of some hrteristi point of the ody remins onstntD whih mens tht the system possesses nonintegrle servo onstrintF

1

Introduction

sn the erlier tivitiesD the uthor hs lredy proved the omplete integrility of the equtions of plneEprllel motion of ody in resisting medium under the jet )ow onditions when the system of dynmil equtions possesses (rst integrlD whih is trnsendentl @in the sense of the theory of funtions of omplex vrileA funtion of qusiEveloities hving essentil singulritiesF st ws ssumed tht the intertion of the medium with the ody is onentrted on prt of the surfe of the ody tht hs the form of @oneEdimensionlA plteF sn the sequelD the plnr prolem ws generlized to the sptil @threeEdimensionlA seD where the system of dynmil equtions possesses omplete set of trnsendentl (rst integrlsF sn this seD it ws ssumed tht the intertion of the medium with the ody is onentrted on the prt of the surfe of the ody tht hs the form of plnr @twoEdimensionlA diskF woreoverD we study the dynmi prt of equtions of motion of di'erent fourEdimensionl dynmilly symmetri rigid ody where nononservtive fore (eld is onentrted on prt of the surfe of the odyD whih hs the form of twoEdimensionl @threeE dimensionlA diskD nd the tion of the fore is onentrted in the twoEdimensionl plne @oneEdimensionl lineA perpendiulr to this diskF sn this workD we disuss resultsD oth new nd otined erlierD onerning the se where the intertion of the medium with the ody is onentrted on the prt of the surfe of the ody tht hs the form of (n - 1)Edimensionl disk nd the fore ts in the diretion perpendiulr to the diskF e systemtize these results nd formulte them in the invrint formF

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Multidimensional pendulum in a nonconservative force eld

2

Certain General Discourse

pirst of ll for nEdimensionl rigid odyD we will e interested the se @I"(n - 1)AD iF eFD when in some oordinte system Dx1 . . . xn tthed to the odyD the opertor of inerti hs the form dig{I1 , I2 , . . . , I2 }, @IA

iF eFD the ody is dynmilly symmetri in the hyperplne Dx2 . . . xn @Dx1 is the xe of dynmil symmetryAF he on(gurtion spe of freeD nEdimensionl rigid ody is the diret produt

Rn в y(n)

@PA

of the spe Rn D whih de(nes the oordintes of the enter of mss of the odyD nd the rottion group y(n)D whih de(ned rottions of the ody out its enter of mss nd hs dimension n(n + 1)/2. hereforeD the dynmil prt of equtions of motion hs the sme dimensionD wheres the dimension of the phse spe is equl to n(n + 1). sn prtiulrD if is the tensor of ngulr veloity of nEdimensionl rigid ody @it is seondErnk tensorD see ID PD QAD so(n)D then the prt of dynmil equtions of motion orresponding to the vie lger so(n) hs the following form @see PD QD RAX

+ + [, + ] = M , = dig{1 , . . . , n }, 1 =

@QA @RA

-I1 + I2 + . . . + In I1 - I2 + I3 + . . . + In , 2 = ,..., 2 2 I1 + . . . + In-2 - In-1 + In I1 + . . . + In-1 - In n-1 = , n = , 2 2 M = MF is the nturl pro jetion of the moment of externl fores F ting to the ody in Rn on the nturl oordintes of the vie lger so(n) nd [., .] is the ommuttor in so(n)F yviouslyD the following reltions holdX i - j = Ij - Ii for ny i, j = 1, . . . , nF por the lultion of the moment of n externl fore ting to the odyD we need to onstrut the mpping Rn в Rn - so(n), tht mps pir of vetors (DN, F) Rn в Rn from Rn в Rn to n element of the vie lger so(n)D where DN = {1 , 2 , . . . , n }, F = {F1 , F2 , . . . , Fn }, nd F is n externl fore ting to the odyF rere DN is the vetor direting from the point D of the oordinte system Dx1 . . . xn to the point N of fore tingAF por this endD we onstrut the following uxiliry mtrix 1 2 . . . n F1 F2 . . . Fn .
@SA

hynmil systems studied in this tivityD re dynmil systems with vrile dissiption with zero men @see RD SAF e need to exmine y diret methods prt of the min system of dynmil equtionsD nmelyD the xewton equtionD whih plys the role of the eqution of motion of the enter of mssD iFeFD the prt of the dynmil equtions orresponding to the spe Rn X

mwC = F,

@TA

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Proceedings of XLIII International Summer SchoolConference APM 2015
where wC is the elertion of the enter of mss C of the ody nd m is its mssF woreoverD due to the higherEdimensionl ivls formul @it n e otined y the opertor methodA we hve the following reltionsX

wC = wD + 2 DC + E DC, wD = vD + vD , E = ,

@UA

where wD is the elertion of the point DD F is the externl fore ting on the ody @in our seD F = SAD nd E is the tensor of ngulr elertion @seondErnk tensorAF vet the position of the ody in iuliden spe En is de(ned y the funtions whih re the yli in the following senseX the generlized fore F nd its moment (DN, F) depend on generlized veloities only @qusiEveloitiesAD nd do not depend on the position of the ody in the speF henD the system of equtions @QA nd @TA on the mnifold Rn в so(n) is losed system of dynmil equtions of the motion of free nEdimensionl rigid ody under the tion of n externl fore FF his system hve een seprted from the kinemti prt of the equtions of motion on the mnifold @PA nd n e exmined independentlyF

3

General Problem on the Motion Under a Tracing Force

gonsider motion of homogeneousD dynmilly symmetri @se @IAAD rigid ody with front end fe @ (n - 1)Edimensionl disk interting with medium tht (lls the nE dimensionl speA in the (eld of resistne fore S under the qusiEsttionrity onditions @see TD UAF vet (v , , 1 , . . . , n-2 ) e the @generlizedA spheril oordintes of the veloity vetor of the enter of the (n - 1)Edimensionl disk lying on the xis of symmetry of the odyD e the tensor of ngulr veloity of the odyD Dx1 . . . xn e the oordinte system tthed to the ody suh tht the xis of symmetry C D oinides with the xis Dx1 @rell tht C is the enter of mssAD nd the xes Dx2 , Dx3 , . . . , Dxn lie in the hyperplne of the diskD nd I1 , I2 , I3 = I2 , . . . , In = I2 , m re hrteristis of inerti nd mssF e dopt the following expnsions in the pro jetions to the xes of the oordinte system Dx1 . . . xn X DC = {-, 0, . . . , 0}, vD = v iv (, 1 , . . . , n-2 ) , where cos sin cos 1 sin sin 1 cos 2 iv (, 1 , . . . , n-2 ) = @VA ... sin sin 1 . . . sin n-3 cos n-2

sin sin 1 . . . sin n

-2

is the single vetor on the xe of vetor vF sn the se @IA we dditionlly hve the expnsion for the funtion of the in)uene of the medium on the nEdimensionl odyX S = {-S, 0, . . . , 0}, iFeFD in this se F = SF purtherD the uxiliry mtrix @SA for the lultion of the moment of the resistne fore hs the form

0 x2N -S 0

. . . xnN ... 0

,

@WA

then the prt of the dynmil equtions of motion tht desries the motion of the ody out the enter of mss nd orresponds to the vie lger so(n)D n e otinedF e

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Multidimensional pendulum in a nonconservative force eld
note tht system @QAD due to the existing dynmil symmetry

I2 = . . . = In ,
possesses yli (rst integrls

@IHA



k1

0 k1 = onst, . . . ,

k

s



0 ks

= onst, s =

(n - 1)(n - 2) . 2

@IIA
1

rere k1 = 1, . . . , ks re the ertin s nonreurrent numers from the set W {1, 2, . . . , n(n - 1)/2}F sn the sequelD we onsider the (rst integrls @IIA of the system on its zero levelsX

=



0 k1

= ... =

0 ks

= 0.

@IPA

he hoie of nonzero omponents r1 , (n - 1)(n - 2)/2 = n - 1 ones @here r1 , not equl to k1 , . . . , ks AF sf one onsiders more generl prolem tht lies on the stright line C D = Dx1

. . . , rp of tensor onsists of p = n(n - 1)/2 - . . . , rp re the rest p of numers from the set W1 D
on the motion of ody under tring fore T nd provides the ful(llment of the reltion @IQA

v onst,

throughout the motionD then insted of F1 system @QAD @TA ontins T - s()v 2 , = DC. ghoosing the vlue T of the tring fore ppropritelyD one n hieve the equlity @IQA throughout the motionF sndeedD expressing T due to system @QAD @TAD we otin for cos = 0, n > 2 the reltion

T = Tv (, 1 , . . . , +s()v
2

n-2

2 2 , ) = m (r1 + . . . + rp )+

1-
v

m sin v , 1 , . . . , (n - 2)I2 cos , 1 , . . . ,
n n-2

n-2

,

v

,
n-2

@IRA

,

v

= |rN | = (rN , iN (1 , . . . , , v i (1 , . . . ,

)) =
@ISA

= 0 · cos + 2

x
s=2

sN

, 1 , . . . ,

n-2

sN

n-2

).

rere isN (1 , . . . , n-2 ), s = 1, . gle vetor on the xe of vetor Sn-2 {1 , . . . , n-2 }D de(nied y ing (n - 1)Edimensionl sphere eFD

. . , n, @i1N (1 , . . . , n-2 ) 0A re the omponents of sinE rN = {0, x2N , . . . , xnN } on (n - 2)Edimensionl sphere the equlity = /2 s equtoril setion of orrespondE Sn-1 {, 1 , . . . , n-2 } @de(ned y the equlity @IQAAD iF
iN (1 , . . . , n-2 ) =

=

0 cos 1 sin 1 cos 2 ... sin 1 . . . sin n-3 cos sin 1 . . . sin n-2

= iv , 1 , . . . , 2
@ITA

n-2

n-2

@see iqF @VAAF his proedure n e interpreted in two wysF pirstD we hve trnsformed the system using the tring fore @ontrolA tht provides the onsidertion of the lss @IQA

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of motions interesting for usF eondD we n tret this s n orderEredution proedureF sndeedD system @QAD @TA genertes the following independent system of following order @due to iqsF @IQAD @IIAD @IPAAX n(n + 1)/2 - (n - 1)(n - 2)/2 - 1 = 2(n - 1). vet introdue the new qusiEveloities in system @QAD @TAF por thisD we trnsform the vlues r1 , . . . , rn-1 y the omposition of following (n - 2) rottionsX



z1 z2 . . . = Tn- zn-1

2,n-1

(-1 ) Tn-

3,n-2

(-2 ) . . . T1,2 (-n

-2

r1 r2 ) ... , rn-1

@IUA

where the mtrix Tk,k+1 ( ), k = 1, of seond order minor Mk,k+1 X 10 0 FF 0 F 0 0 0 Mk,k+1 Tk,k+1 = 0 0 0 00 0

. . . , n - 2, is otined from the unit one y the presene 0 0 mk,k mk,k+1 mk+1,k mk+1,k+1
@IVA

0 0 0 0 , M FF F 0 01 = cos , m

k,k+1

=

,

mk,k = mk

+1,k+1

k+1,k

= -mk

,k+1

= sin .

es we seeD we nnot solve the system with respet to , 1 , . . . , n-2 on the mnifold

O1 = {(, 1 , . . . , =

n-2

, r1 , . . . ,

r

n-1

) R2(

n-1)

:

k , 1 = l1 , . . . , n-3 = ln-3 , k , l1 , . . . , ln-3 Z}. @IWA 2 hereforeD on the mnifold @IWA the uniqueness theorem formlly is violtedF woreoverD for even k nd ny l1 , . . . , ln-3 D n indeterminte form ppers due to the degenertion of the spheril oordintes (v , , 1 , . . . , n-2 )F por odd k D the uniqueness theorem is oviously violted sine one of the eqution degenertesF his implies tht system @QAD @TA outside @nd only outsideA the mnifold @IWA n e redued to the following form @n > 2AX = -z
n-1

+

v s() (n - 2)I2 cos

v

, 1 , . . . ,

n-2

,

v

,
2 n-2

@PHA

z

n-1

=

v2 s()v , 1 , . . . , (n - 2)I2
n-2

n-2

,

v

2 - (z1 + . . . + z

)

cos + sin
@PIA

v s() + (n - 2)I2 sin z
n-2

(-1)s z
s=1 n-2 zn-1

n-1-s



v ,s

, 1 , . . . ,
2 n-3

n-2

,

v

,

=z

cos 2 + ( z1 + . . . + z sin
n-1

)

cos cos 1 + sin sin 1
n-2

+

v s() {z (n - 2)I2 sin (-1)s
+1

v

,1

, 1 , . . . , ,

,

v

+

n-2

+
s=2

z

n-1-s



v ,s

, 1 , . . . ,

n-2

v

cos 1 }- sin 1

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Multidimensional pendulum in a nonconservative force eld
v2 s()v (n - 2)I2 z v
@PPA

-

,1

, 1 , . . . , =z
n-3 zn-1

n-2

,

,
n-3 zn-2

n-3

cos -z sin
2 n-4

cos cos 1 - sin sin 1

2 -(z1 + . . . + z

)

cos 1 cos 2 + sin sin 1 sin 2
n-2

+

v s() { (n - 2)I2 sin
n-2

v ,2

, 1 , . . . ,

,

v
n-2

-z v

n-1

+z

n-2

cos 1 + sin 1

+
s=3

(-1)s z

n-1-s

v ,s

, 1 , . . . , ,

,

1 cos 2 }+ sin 1 sin 2
@PQA

+

v2 s()v,2 , 1 , . . . , n-2 (n - 2)I2 ................... z1 = n +(-1)n
-2

, v .......................
-2

(-r1 sin n

+ r2 cos n , 1 , . . . ,

-2

)+ , v =

v2 s()v (n - 2)I2
n-2

,n-2

n-2

cos = z1 sin +

(-1)s
s=1

+1

z

n-s

cos s-1 sin 1 . . . sin

+
s-1

s() v (-1)n+1 (n - 2)I2 sin
n-1

v ,n-2

, 1 , . . . ,

n-2

,

v

в

в
s=2

(-1)s z

n+1-s

cos s-1 sin 1 . . . sin
n-2

+
s-1

+(-1)n 1 = z

v s() (n - 2)I2

2

v ,n-2

, 1 , . . . ,
v ,1

,

v

,
n-2

@PRA

n-2

v s() cos + sin (n - 2)I2 sin

, 1 , . . . ,

,

v

, ,

@PSA @PTA

2 = -z

n-3

cos v s() + v,2 , 1 , 2 , 3 , sin sin 1 (n - 2)I2 sin sin 1 v .......................................... cos n-2 = (-1)n+1 z1 + sin sin 1 . . . sin n-3
-2 v ,n-2

+

v s() (n - 2)I2 sin sin 1 . . . sin n v
v ,1

, 1 , . . . ,

n-2

,

v

,
n-2

@PUA

, 1 , . . . , , 1 , . . . ,

n-2

,

v

= (rN , iN 1 + = (rN , iN

, 2 , . . . , 2

), ),
@PVA

,2

n-2

,

v

, 2 + , 3 , . . . , 2 2

n-2

..........................................
v ,n-3

, 1 , . . . ,

n-2

,

v

= (rN , iN

, . . . , , n-3 + , n- 2 2 2

2

),

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Proceedings of XLIII International Summer SchoolConference APM 2015
v , . . . , , n-2 + ), 2 2 2



v ,n-2

, 1 , . . . ,

n-2

,

= (rN , iN

nd funtion v (, 1 , . . . , n-2 , /v ) n e represented in the form @ISAF sn rightEhnd side of the system @PHA!@PUA fter ommon multiplier

s() v , (n - 2)I2 cos
the vlues v,s (, 1 , . . . , n-2 , /v ) , s = 1, . . . , n - 2, re represented in liner form @nd lwys (n - 2) oe0ients preiselyAF por instneD in iqF @PIA @with leftEhnd side zn-1 AD the funtions @PVA re represented with ll indies s from I to n - 2 @every index per one timeAD iF eFD

1 2 3 4 . . . n - 2.

@PWA

fut furthermoreD in iqsF @PPA!@PRAD the set of funtions @PVA is formed in nother wyF por instneD in eqution with leftEhnd side zn-2 D the set of funtions @PVA is formed with orresponding indies @PWAF fut in eqution with leftEhnd side zn-3 D it is lredy formed with the following indiesX

2 2 3 4 ... n - 2
iF eFD the funtion v,2 (, 1 , . . . , n-2 , /v ) is lredy repeted twieF ht is the generl distriution of indiesc st n e represented y tle IQF

@QHA

Table 13: General Distribution of Indices in Set of Functions (28) Left-hand Side of (20)(27) Distribution of Indices s in Set of Functions (28) 1 2 3 4 ... 2 2 3 4 ... 3 3 3 4 ... 4 4 4 4 ... ... ... ... ... ... ...

zn zn zn zn

-2 -3 -4 -5

n-2 n-2 n-2 n-2
...

...

z1

n-2 n-2 n-2 n-2

n-2

4

Case Where the Moment of a Nonconservative Force Is Independent of the Angular Velocity

imilrly to the hoie of ghplygin nlyti funtionsD we tke the dynmil funtions sD x2N , . . . , xnN in the following form @using @ITAAX

s() = B cos , r

N

= R()iN , R() = A sin , A, B > 0.

@QIA

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Multidimensional pendulum in a nonconservative force eld
rerewithD the funtions v (, 1 , . . . , n-2 , /v ) , 2, in system @PHA!@PUAD tke the following formX

v ,s

( , 1 , . . . ,

n-2

, /v ) , s = 1, . . . , n- v



v

, 1 , . . . ,

n-2

,

v

= R() = A sin ,

v ,s

, 1 , . . . ,

n-2

,

0.

@QPA

henD due to the nonintegrle onstrint @IQAD outside nd only outside the mnifold @IWAD system @PHA!@PUAA hs the nlyti form

= -z z z
n-1

n-1

+ b sin , cos , sin cos cos 1 2 2 , + (z1 + . . . + zn-3 ) sin sin 1 cos cos cos 1 - zn-3 zn-1 - zn-3 zn-2 sin sin sin 1 1 cos 2 , sin 1 sin 2 .....................................
2 n-2

@QQA

2 = sin cos - (z1 + . . . + z

)

@QRA @QSA

n-2

=z

n-2 zn-1

cos sin =

z
2 -(z1 + . . . + z

n-3

2 n-4

cos sin ..... )

@QTA

cos z1 = z1 sin 1 = z
n-2

n-2

(-1)s
s=1

+1

z

n-s

cos s-1 sin 1 . . . sin

,
s-1

@QUA @QVA @QWA @RHA @RIA

cos , sin cos 2 = -zn-3 , sin sin 1 .......................................... cos , n-3 = (-1)n z2 sin sin 1 . . . sin n-4 cos n-2 = (-1)n+1 z1 , sin sin 1 . . . sin n-3 AB (n > 2), (n - 2)I2

introduing the dimensionless vrilesD prmetersD nd the di'erentition s followsX

zk n0 v zk , k = 1, . . . , n - 1, n2 = 0 b = n0 , < · >= n0 v < > .

@RPA

e see tht the 2(n - 1)thEorder system @QQA!@RIA @whih n e onsidered s system on the tngent undle T Sn-1 {zn-1 , . . . , z1 ; , 1 , . . . , n-2 } of the (n - 1)Edimensionl sphere Sn-1 {, 1 , . . . , n-2 }D see elowA ontins the independent (2n - 3)thEorder system @QQA! @RHA on its own (2n - 3)Edimensionl mnifoldF Theorem 1. he system @QAD @TA under onditions @IQAD @IIAD @IPAD is redued to dynmi system @PHA!@PUA on the tngent undle T Sn-1 {zn-1 , . . . , z1 ; , 1 , . . . , n-2 } of (n - 1)E dimensionl sphere Sn-1 {, 1 , . . . , n-2 }F por the omplete integrtion of 2(n - 1)thEorder system @QQA!@RIAD in generlD we need 2n - 3 independent (rst integrlsF roweverD fter the hnge of vriles

w w

n-1 n-4

=z =

n-1

,w
3

n-2

=

2 z1 + . . . + z

2 n-2

,w

n-3

=

z
2 1

z +z

2 2

, . . . , w2 =

z

z2 , z1

n-3 2 n-4

2 z1 + . . . + z

, w1 =

z

n-2 2 n-3

,

@RQA

2 z1 + . . . + z

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Proceedings of XLIII International Summer SchoolConference APM 2015
system @QQA!@RIA splits s followsX

= -w w w
n-1 n-2

n-1

+ b sin ,
2 n-2

@RRA

= sin cos - w =w
n-2

cos , sin

@RSA @RTA
n-2 2 1 + ws cos s , ws sin s ), s = 1, . . . , n - 3,

w

n-1

cos , sin )

ws = ds (w s = ds (w n
-2

n-1

, . . . , w1 ; , 1 , . . . ,

@RUA @RVA

n-1

, . . . , w1 ; , 1 , . . . ,
n-1

n-2

, . . . , w1 ; , 1 , . . . , n-2 ), cos d1 = Zn-2 (wn-1 , . . . , w1 ) , sin cos , d2 = -Zn-3 (wn-1 , . . . , w1 ) sin sin 1 ............................................................ cos dn-2 = (-1)n+1 Z1 (wn-1 , . . . , w1 ) , sin sin 1 . . . sin n-3

= dn-2 (w

@RWA

herewithD zk = Zk (wn-1 , . . . , w1 ), k = 1, . . . , n - 2, re the funtions due to the hnge @RQAF e see tht for the omplete integrtion of system @RRA!@RVA it su0es to speify two independent (rst integrls of system @RRA!@RTAD on one (rst integrl of systems @RUAD nd n dditionl (rst integrl tht tthes iqF @RVA @iF eFD n in l l AF e hve the following trnsendentl (rst integrlX

1 (w

n-1

, wn-2 ; ) =

w

2 n-1

+w

2 n-2

- bwn-1 sin + sin2 = C1 = onst. wn-2 sin

@SHA

hen the dditionl (rst integrl otined hs the following struturl formX

2 (w

n-1

, wn-2 ; ) = G sin ,

wn-1 wn-2 , sin sin

= C2 = onst.

@SIA

por the omplete integrtionD s ws mentioned oveD it su0es to (nd on one (rst integrl for @potentilly seprtedA systems @RUAD nd n dditionl (rst integrl tht tthes iqF @RVAF sndeedD we hve the desired (rst integrls s followsX

s

+2

(ws ; s ) =
n-3

2 1 + ws =C sin s

s+2

= onst, s = 1, . . . , n - 3,

@SPA @SQA

n (w

, . . . , w1 ; , 1 , . . . ,

n-2

) = Cn = onst,

herewithD we must sustitute the leftEhnd sides of the (rst integrls @SPA for s = n - 4, n - 3D in the expressions of (rst integrl @SQA insted Cn-2 , Cn-1 F Theorem 2. he 2(n - 1)thEorder system @RRA!@RVA p ossesses the su0ient quntity @nA of independent (rst integrls @SHAD @SIAD @SPAD nd @SQAF husD in the se onsideredD the system of dynmil equtions @QAD @TA under ondition @QIA hs (n2 - n + 4)/2, n > 2, invrint reltionsX the nonintegrle nlyti onstrint of the form @IQAD the yli (rst integrls of the form @IIAD @IPAD the (rst integrl of the form @SHAD the (rst integrl @SIAD whih is trnsendentl funtion of the phse vriles

330


Multidimensional pendulum in a nonconservative force eld
@in the sense of omplex nlysisA expressed through (nite omintion of elementry funtionsD ndD (nllyD the trnsendentl (rst integrls of the form @SPAD @SQAF Theorem 3. ystem @QAD @TA under onditions @IQAD @QIAD @IIAD @IPA p ossesses (n2 - n + 4)/2, n > 2, invrint reltions @omplete setAD n of whih trnsendentl funtions from the point of view of omplex nlysisF rerewithD ll reltions re expressed through (nite omintions of elementry funtionsF gonsider the following (2n - 3)thEorder systemX

Ё + b cos + sin cos - - 1 2 + 2 2 sin2 1 + 3 2 sin2 1 sin2 2 + . . . + n-2 sin2 1 . . . sin2 n 2
2 1 + cos - 1 + b 1 cos + 1 Ё cos sin 2 2 2 - 2 + 3 sin 2 + 4 2 sin2 2 sin2 3 + . . . + -3

sin = 0, cos

2 n-2

sin2 2 . . . sin2 n

-3

в

в sin 1 cos 1 = 0,
2 1 + cos + 2 + b 2 cos + 2 Ё cos sin cos 1 - + 21 2 sin 1 - 3 2 + 4 2 sin2 3 + 5 2 sin2 3 sin2 4 + . . . +

2 n-2

sin2 3 . . . sin2 n

-3

в

в sin 2 cos 2 = 0,
2 1 + cos + 3 + b 3 cos + 3 Ё cos sin cos 1 cos 2 + 21 3 + 22 3 - sin 1 sin 2 - 4 2 + 5 2 sin2 4 + 6 2 sin2 4 sin2 5 + . . . +

2 n-2

sin2 4 . . . sin2 n

-3

в

@SRA

в sin 3 cos 3 = 0, ..................................................................... n Ё 1 + cos2 + cos sin cos 1 cos n-5 + 21 n-4 + . . . + 2n-5 n-4 - sin 1 sin n-5 - n-3 + n-2 sin2 n-3 sin n-4 cos n-4 = 0, 2 2
-4

+ b n

-4

cos +

n-4

n Ё

1 + cos2 + cos sin cos 1 cos n-4 + 21 n-3 + . . . + 2n-4 n-3 - sin 1 sin n-4 - n-2 sin n-3 cos n-3 = 0, 2
-3

+ b n

-3

cos +

n-3

n Ё

-2

+ b

n-2

+ 21

n-2

1 + cos2 cos + n-2 cos sin cos 1 + . . . + 2n-3 n-2 sin 1

+ cos n-3 = 0, b > 0, sin n-3

whih desries (xed the moment of fores is nononservtive (eld 2(n - 1)D ut the phse

nEdimensionl pendulum in )ow of running medium for whih independent of the ngulr veloityD iFeFD mehnil system in @see ID PAF sn generlD the order of suh system is equl to vrile n-2 is yli vrileD whih leds to the strti(tion

331


REFERENCES
of the phse spe nd redues the order of the systemF he phse spe of this system is the tngent undle

TS

n-1

{ , 1 , . . . , n

-2

, , 1 , . . . ,

n-2

}

@SSA

of the (n - 1)Edimensionl sphere Sn-1 { , 1 , . . . , n-2 }F he equtions tht trnsform system @SRA into the system on the tngent undle of the twoEdimensionl sphere 2 3 . . . n-2 0, nd the equtions of gret irles 1 0, 2 0, . . . , n-2 0 de(ne fmilies of integrl mnifoldsF st is esy to verify tht system @SRA is equivlent to the dynmil system with vrile dissiption with zero men on the tngent undle @SSA of the (n - 1)Edimensionl sphereF woreoverD the following theorem holdsF Theorem 4. ystem @QAD @TA under onditions @IQAD @QIAD @IIAD nd @IPAD is equivlent to the dynmil system @SRAF sndeedD it su0es to set = , 1 = 1 , . . . , n-2 = n-2 , b = -b F

Acknowledgements
his work ws supported y the ussin poundtion for fsi eserhD projet noF IPE HIEHHHPHEF

References
I wF F hmolinD xew gse of sntegrility in the hynmis of wultidimensionl olid in xononservtive pieldD hokldy hysisD SVXII @PHIQAD RWT!RWWF P wF F hmolinD xew se of integrility of dynmi equtions on the tngent undle of QEsphereD ussin wthF urveysD TVXS @PHIQAD WTQ!WTSF Q wF F hmolinD riety of sntegrle gses in hynmis of vowE nd wultiE himensionl igid fodies in xononservtive pore pieldsD tournl of wthemtil ienesD PHRXR @PHISAD QUW!SQHF R wF F hmolinD e xew gse of sntegrility in the hynmis of wultidimensionl olid in xononservtive pield under the essumption of viner hmpingD hokldy hysisD SWXV @PHIRAD QUS!QUVF S wF F hmolinD hynmil endulumEvike xononservtive ystemsD epplied xonE viner hynmil ystemsD pringer roeedings in wthemtis nd ttistisD olF WQ @PHIRAD SHQ!SPSF T wF F hmolinD e wultidimensionl endulum in xononservtive pore pieldD hokldy hysisD THXI @PHISAD QR!QVF U wF F hmolinD glssi(tion of sntegrle gses in the hynmis of pourE himensionl igid fody in xononservtive pield in the resene of rking poreD tournl of wthemtil ienesD PHRXT @PHISAD VHV!VUHF

wxim F hmolinD vomonosov wosow tte niversityD ussin pedertion

332