C H A P T E R 111

APPLICATIONS OF VARIATIONAL SYMMETRY APPROACH

TO CERTAIN FINITE DIMENSIONAL SYSTEMS

3.1 INTRODOCTION

In Chapter 11, we have discussed the basic concepts of

variational principles and presented a method to find the

variational symmetries and first integrals of a system with

two degrees of freedom. In the present chapter, we wish to

apply the method presented in the previous chapter to

certain physically important systems such as Toda lattice

and generalized Aenon - Beilea system and study their

integrability. We mention brlefly the historical

developments of these two systems. Among the finite

dimensional systems, the Toda lattice is a ubiquitous one,

which describes a system of particles actlng on a line with

exponential interactions of nearest nelghbours I 1 8 1 . For

Che last two decades, various investigations were conducted

on the Toda lattice from mathematical and physical points of

view [ 2 0 - 2 7 1 . To mention a few, Henon 1221 first

analytically derived the integrals. Consequently, he also

found a transformation connecting Toda and Volterra systems.

rhis investigation further leads Flaschka I Z O , 2 1 1 to

:onsider the Toda system as a finite dimensional analog of

the Korteweg - de - Vrles equatlon (KdV) and he derived a

transformation connecting KdV and Toda. Based on thle

observation many authors studled various aspects of the Toda

problem through the results of KdV 120, 21, 27, 771. Apart

from the classical Toda problem, many generalizations of the

Toda lattlce have emerged from different polnts of view. In

particular, a class of integrable and nonlntegrable Toda

systems have been classified by Dorlzzl et al., 1111.

Bountis et dl., I61 and Yoshida 187, 881 through Painleve

analysis and Zigllna theorem, respectively. A dlrect

generalization for n degrees of freedom was established by

Bogoyavlensky 1261 through root systems of simple Lie

algebras. Further generalizations have been carried out by

Reyman and Semenov - Tlan - Shansky I771 for affine Lle

dlgebras.

Our next example is Henon - Aelles system I81 whlch 1s

also physically very important and considered as a prototype

of coupled nonlinear systems descrlblng the motlon of

gravitlng stars in a cylindrical galaxy. Henon and Helles

first studied numerically the integrability of thls system.

Later, all known three ~nkqrable cases have been discovered

and their flrst Integrals have been constructed [9-171.

The organization of this chapter is as follows. In

section 3.2, we apply the method presented in Chapter I1 to

Toda lattice and study their integrability through finding

their symmetries and first ~ntegrals. We also illuetrate

our procedure on the generalized Henon - Reiles system and a system possessing a non - polynomial type potential and

obtain their infinitesimals and their integrals of motion

explicitly in sections 3.3.1 and 3.3.2. In section 3.3.3,

we dlscuss the so called super Integrable systems. In

section 3.4. we give our concluding remarks.

3.2 TODA LATTICE

In this section, we apply the method described in

Chapter I1 to the Toda potential considered recently by

Sklyanin I261 and Reyman and Semenov - Tian - Shansky [ 7 7 1

In the form

where c,, c a r cs and c* are parameters. In recent times, a

lot of attention has been paid to investigate integrability

aspects of different forms of Toda lattice 110, 27, 771.

Not much is known about the system 0.1). From equation

(2.46a) together with (3.1), we get

I n t e g r a t i n g ( 3 . 2 ) w l t h r e s p e c t t o 4 , we h a v e

w h e r e gaol 1s a f u n c t r o n o f q o n l y . From e q u a t i o n ( 2 . 4 7 b ) .

we f i n d

h = ( 1 2 0 ~ ~ - 13, ) e4-% - 2 4 ~ , ( 2 c ~ e - ~ % + c e-' ) I 4

1 + g S~2c,e24 + c 2 e 4 ) . ( 3 . 4 )

Integrating ( 3 . 4 ) w l t h r e s p e c t t o q , we o b t a i n

hl = - ( 1 2 0 ~ ~ - p I ) e '-% - 2Q, + c e -'=% 4 '9

1 + g 13, ( q e + c Z e ( 3 . 5 )

w h e r e hll i s a f u n c t i o n o f o n l y . From equations ( 2 . 4 6 c ) ,

( 2 . 4 6 d ) a n d ( 2 . 4 6 b ) , we a r r i v e a t

3 9, -q2 3 c g2* = 16 ( 8 4 . + ~ O P ~ - h a ) e + - -5( -z0 . q 2 + h 6 q , ) e 29 5 4 1

3cz + ifi- ( - ~ ~ q , ~ + h ~ q ~ ) e ' + k l ( c l e + c 4 e

where g 2,,, giti and go,* are all functions of q only. By

making use of equations (3.5) and (3.8) into (2.47a), we

obtain

P - ( - -5 - 4001 + + ha + 1oaZ - 1 h s ) e a - 9 12

+ 24a,(4~~e-~' + c4e-4)4 + i z c i ( - ~ 4 4 4

Differentiating equatlon (3.9) with respect to 9 , we get

P ( - -L - 40p t B P 4 + 5 ~ a + l k l - 1 k I ) e % - 9

12

+ z&.(~c,F-~' + cle-') + ! $ ~ ~ c - ~ q , 4

+ 24gmos4q = 0. (3.10)

Equating the various powere of q in (3.10) to zero. we

obtain

u = a = 0, 0, -60(ua + 2u2 - 2n3) = 0, 1 4

(3.11a)

g ~ o t = 2c,re2' + 2cz%e9 + n &I ' (3.11b)

where nL2, ..., n are all constants. $5

Substituting

equations (3.11a) - (3.11e) in (3.91, we find

where nSo is a constant. Substituting equations (3.3).

(3.6) and (3.11~) in (2.46h3 and solving the resulting

equation, we obtain

2 ( ~ - 4 ) gso = t ( h 2 + no)e + 4ca2cs(2e

-4 - 4 + cBe - 44

+ 2c.e - 3 4 ) + 2 1 2 4 ~ ~ ~ + ygcs )e-2' + PC,(%

-2'4-5) t 2a2 - as )e '+% + hocl (cse

+ c4e -Ti'%) + c Z ( 4 + 'a, - 2as)e 9,

where giO, is a function of only. Making use of

equations (3.6), (3.7). (3.11b). (3.11~) and (3.13) in

12.461), we get

-'+2' + c c e -2'+' + c c e -=I '4 ) + c c e L 1 2 I 2 4

+ % , i r ( 3 . 1 4 )

where g 18 a f u n c t l o n o f 4 o n l y . Using equations ( 3 . 5 ) , 011

( 3 . 8 ) . ( 3 . 1 1 e ) and ( 3 . 1 2 ) i n ( 2 . 4 7 f ) , we have

h = - -i(lh + 30a2 - 60aa - P l ) e n 30 a 2 '4-4 + 2cn(na

+ h2 - a 5 ) e -'-% + c.(aa + 2a2 + - 2 a 3 ) e -%

- 3% - 2 4 c I + 3c.e lg - 2(c42a5

+ 2cBn12 ) s e -24 - f(hfi2 - a ) e q1 -% *a

I P - 2a1,c.ee-' + -kc ( 2e4"+ c e 4 + 2c2e34 ) "S 30 1

where ha, is a function of q, only. Maklng use of equations

(3.7). (3.8). (3.11d). (3.11e). (3.14) and (3.15) in

(2.47e). we arrive at 2rs,-4)

:(la- 40a2 + 2 0 ~ 1 ~ - 4 h B ) e + 8c1 ( 2a5cS4

+ + cs (+,4 + aid) le-" + +(12ai5 + a La - g, s, -9 - h19 ) e + 2c4 (al5% + aid )e-' + fk(ll- 1 2 h 2

- 6 h 6 + 120as)e '" + 32ci2(4q,2 + h a q ) e 44

- 3cLc2(12~q, - 9q,2)e3'% t i f r 1 (-2. 12 s,2 + hLdq,

2 - %,,I + 4c2 (hnq, - a,%2) 1e2' + >(-aiz4 2 + 6014qi

- lh,, )e' + 2c4 ( -a6 + as + 2az )e-' + C= ( - b 6 - Q2

-9 + 4us)eT + c4(6a2 + %)q,e + hns4 - g ,.<% 4

+ 2gOfi9 = 0. (3.16)

Differentiating (3.16) with respect to 4 , and equating the

varxous powers of q, in the resulting equation to zero, w e

where a*, and P are constants. Solving equations (3.11.3) i.

and (3.17b), we find

a = p l = O , o = 2 a . (3.18)

Substituting equations (3.17) and (3.18) in (3.16), we

obtain -49%

hnl = k , c a (cue + 2c.e -l') + 2 ( k p 4 2 + c,a14)e -24

+ 2a14 cd e-' + 2nSc2eql + aso, (3.19)

where ado is a constant. Substituting equations (3.11a) -

(3.11e), (3.12), (3.17) - (3.19) In (3.2), (3.6) - ( 3 . 8 ) ,

(3.51, (3.13) - (3.15) respectively , we obtain

together with

a = a = a = aS0= a = 0, 7 . 0 11

/3,r 0, go2= gzO= g,*= goo= h = h = 0 . 2 1 (3.21)

From equations (2.48a) and (2.48b3, we also obtain

From the compatibility condltron for d, we obtain a = a 1. LD.

naklng use of equations (3.20) and 0.21) in equations

(2.42) and (2.44), we get the infiniteslmals

n1 = a1,4;+ P , , F , + a S 4 ~ , + as^,, (3.23a)

where F a , P a , ..., Fa are given by

+ 2cz e' + 2c,e-" + 2c4 e - 4 ) - 3 4 % ' e4-%

+ 4 q 3 ( c , e -2% + c4e-') + ~ q ~ % ~ + 444. s u b s t ~ t u t i n g n,. nz a n d 6 i n t o e q u a t i o n ( 2 . 3 6 ) . u e f i n d t h e

f i r s t i n t e g r a l I :

I = a 1.

A +al, lac all I=+ a, I , , ( 3 . 2 4 )

whe re H is t h e t o t a l e n e r g y a n d 1 , , Iz , I, a r e g i v e n by

2 ' 4 - 4 ) 4+92 + 4 c s e

- '4+4' I, = 2 ( e + 4c e + 2cS2e -4%

- 2s1 + 4 c * ~ . e - ~ 4 + 2-2. 2.e + c Z e 4 + 2c1 'e 44

+ 4c*cze34+ 2cz2e2'+ 2c4e-') + 4 ( e 4 - % + c , e - 2S,

+ c4e -' ) 4 2 + 4 ( e % % + c , e 2%+ C z e 4 ) Q 2

We have verified that Ii. IZ and Is are the first

Integrals of the system (3.1). Also, we have noticed that

the coefficient functions Fz, Fa, F, and F6 in the

infinitesimals n, and nl are related to Fi and P1 in the

following way:

(Fz, Fsl = 2H(e1, 4 ) - IF1, F4),

Further, the integrals IZ and Is are related to I* in the

following way:

21Z = 4H2 - Ii, Ip = IzH - 8cZc4.

Hence, Ii is the only independent integral, besides H =

total energy. We have verified that the lntegrals Is and H

are in involution. We have obtained the above results

without any restrictions on the parameters c,, cZ, cs and

c in the potential. Thus, the system (3.1) is integrable

in the Liouville's sense for arbitrary parameters c,, c2, c,

and c4

3.3.1 Generalized Henon - Heiles system In this section, we study the invariance of generalized

Henon - Heiles type of potential and find its first

integrals 1911 by the same procedure as in the previous

section. We consider the potential of the form 18, 87, 881

Proceeding as in the Toda system. for (3.25), we obtain the

lntegrable case

A = B, C = -D, 3E = F.

For this case. we get the inflnitewimalw

and

b. = - 2a 20 V - 2a2,(A~q - Tqs3 + FqS2% - Cq,q2

+ $q3), (3.26~)

where V is the given potential of the system and az7, a2.

are constants. Substituting equations (3.26al - (3.26~) In

(2.36). we find the first integral

I = a 27 I* + IZ,

where

1, = 4% + 44 - fT3 + ~ q , ~ ~ - c 4 q 2 + 5 q 3 , (3.27)

and

I2 = 2H.

~f we put E = F = 0 in (3.25), we obtain the usual Henon - Heiles system and recover the known three lntegrable cases:

a) C = -6D, A 6 B arbitrary,

b) C = -16D. 16A = B , (3.28)

C) C = -D, A = 0 .

For the case (a), we find the infinitesimals and first

integral as

+ % 2 + qDq2). (3.29~)

For the case (b), we have

(3.30a)

4 4 2 1 = - 5 - + :(A + 2Dq)T242 - FT3&4 +

- $(A + D q ) % h - gT6. (3.30~)

For the case (c), we get the same infinitesimals and I as in

(3.26a), (3.26b) and (3.27) with E = F = 0.

3.3.2 e n polynomial w e potential

We also apply the above procedure to a nonpolynomial

type potential. AE an example, we consider the following

potential I85, 861:

"(%, q , = (q,' + 9 ) + + 842 + c % - ~ + D4-2.

( 3 . 3 1 1

we obtain the infinitesimals and the first integral as

n, - - q 2 ~ + 444. n, = 49% + ( ( B - A ) - ~ ~ t 4 , ( 3 . 3 2 )

3 . 3 . 3 Super Integrable system

For the problems discussed in sections 3 . 3 . 1 and 3 . 3 . 2 ,

we have found that there exists two time independent first

integrals. However. there are systems for which there exist

more than two time independent first integrals. In this

section, we discuss such a system called a super lntegrable

system 1 5 5 , 5 7 - 5 9 1 . By the super ~ntegrability of a system

wlth two degrees of freedom, it is meant that for the system

the maximum number of first integrals is three. all of which

are not in involution.

We consider the following type of potential :

V ( q , 4 ) = + 4 2 ) + Bq, - 2 + c q - 2 . ( 3 . 3 3 )

Again applying invariance algorithm discussed earlier in

section 2 . 3 , w e finally obtain

where tiz6. az, and n2, are constants. Then from ( 2 . 3 6 ) . we

flnd

I = n I t o I t o 26 r 27 z zmlsr

where

1, = q2 + 2 A q 2 + 2Bq-2,

2 ~ 4 - ~ + Z A ~ ' ,

3 . 4 CONCLUSION

We have illustrated this procedure by considering the

Toda lattice in two degrees of freedom and proved the

integrability by showing the existence of generalized

variational symmetries and derived the sufficient first

integrals using Noether's theorem. Apart from the Toda

problem, we have also applied this method to certain other

well- known syetems such as generalized Renon - Heiles

system .~nd a system wlth non polynomial type potential and

consequently obtained their symmetries and recovered the

required number of first integrals. Hence, it seems that

this procedure works very well as a method to identify the

integrable systems in a systematic way.