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Noether symmetries and the Swinging Atwood Machine
I. Moreira, M. Almeida
To cite this version:
I. Moreira, M. Almeida. Noether symmetries and the Swinging Atwood Machine. Journal de Physique II, EDP Sciences, 1991, 1 (7), pp.711-715. �10.1051/jp2:1991105�. �jpa-00247551�
1Phys. II France 1(1991) 711 -715 JUIU£T 1991, PAGE 711
Classification
Pl~ysksAbstrvc~
03.20 05.45
Sho~communication
Noether symmetries and the Swnging Atwood Machine
1.C. Moreka and M.A~ Abneida
Instituto de llsica, UFRJ, Caixa Postal 68528, 21945, Rio de Janeiro, Brasil (Received l3Febmmy1991, mvbed13Mqy1991, accepted17Mqy1991)
Abstract. In this work we apply the Noether theorem with generahsed symmetries for discussing
the integrability of the Swnglng Atwood Machine (SAM) model. We analyse also the limitations of this procedure and compare it wth the Yoshida method.
In the last years several analytical procedures have been applied in tile study of tile integrability
of dy~lamical systems: the dkect method for the identification of invariants, the Painlev6 test, Melnikov's method, Ziglin's theorem, the analysis of the symmetries of the system, etc. In a recent
paper published in this journal, lbfillaro et aL ill wed some results obtained by Yoshida [2] for
discl~ssing the domain of integrability for a mo-dimensional Hamiltonian system: the so-called Swinging Atwood Machine (SAM).By the Liouville theorem, a two-dimensional Hamiltonian
system will be integrable if exists a second conserved quantity (invariant) in addition to the energy.
In this letter we apply a method based on the generalised Noetherian symmetries [3] for finding
the integrable case and its assochted symmetries, for the SAM problem. We compare the two
methods, Yoshida's theorem and the symmetry analysis, and discus their joint utilisation.
1. Noether theorem.
The procedure that we employ here starts from the generalised formulation of the Noether theo- rem [4] where we suppose the exhtence of symmetries via an explicit dependence on the veloc- ities. If we impose that the mo-dimensional Hamiltonian system is invariant under a symmetry transformation we get an overdetermined system of partial differential equations. This system of
equations, if solved, permits 1~s the identification of the symmetries and invariants for the dynam-
ical system.
If the action functional for the n-dimensional system
s = j Lit,
z~, z~)dt 11
712 JOURNALDEPHYSIQUEII N°7
is invariant (up to a constant) under the infinitesimal point transformation with the form t' = t + et (t, z~, z,)
z( = z, + eqi (t, xi, xi). (2)
then, by the Noether theorem, it exhts a conserved quantity for the system given by
I = fit/fiz< In< lx< + IL fIt, xi, z<) (3)
The conditions for the invariance of S are
Lfif/fik, + fiL/fi+j (fiqj lax, zj fit/fizi) = al /fik,, (4) ffiL/fit + q,fiL/fizi + L jai /fit + z,fit /fizi) +
+fiL/fi£i Ian, /fit + zjfiqi /fizj zi(fit/fit+
+ zj fit/fizj)] = fi flat + z,fi flax,. (5)
We can choose f equal to zero. Furthermore, the symmetry which is related to the invariant I is
given by
ni " ~g"fiI/fi~J 16)
where g,j is defined by the relation
gJ~fi~L/fiz,fizj = 6f (7)
For a mo-dimensional system, we take the following Lagrangian in polar coordinates:
L = Ar~/2 + Br~8~/2 V(r, 8) (8)
Conditions (4) and (5) when applied to this Lagrangian lead to the following system of equations:
in + I)Fn+iA~~fiV/fir + (Br~)~~ (fiV/fig)fifn/fig fifn-i /fir-
-A~~Br9~(n + 1)Fn+1 + 2r~~ifiFn-1lag 8fiFn/fig = O, (9)
where we have made the following expansion for the invariant 1 N
I = ~j Fn (r,8,9) r", (10)
n=0 and n = 0,1,2,
..,,
N. We have therefore N + 2 equations to be satisfied.
The equations (4) and (5) also fumish
N
qi = A~~ ~j nfnr"~~ Ill)
n=0 N
q2 = (Br~) ~j r~fifn/fig (12)
n=o N
j = £
(9fiFn /fig + In I)Fnj r" (13)
n=0
N°7 NOETHER SYMMETRIES AND SAM 713
2. The Swinging Atwood Machine.
The Lagrangian for this system, studied by lbfillaro et al. ill, is
L = 11/2)lM + m)r~ + 11/2)mr~i~ grim m COSI°)1 l14)
This Lagrangian is a particular case of (8), with A = M + m, B
= m and V = gr[M mcos(9)]
If we assume that N
= I, i-e-, if I has a linear dependence on r, the system (9) is reduced to
2r~~9fiFi/fig rift/fir = 0
,
(15) (Br~) fiV/fig (rift/fig) + 2r~~9fiF0/fig 9fiFi/fig 3Fo/fir = 0
,
(16) A~~FIfiV/fir + (Br~) ~~ fiV/fig (fife/fig) A~~Br9~Fi 9fiF0/fig = 0 (17) Solving (15) we get
Fi " Fi (9,r~9) (18)
By supposing a 1blear dependence of Fi on r29
Fi " P0(9) + Pi(9)r~9 (19)
we find, from (14) and (16),
F0 " w(9, r) + qi(9, r)9 + q2(9, r)9~ (20)
and
4r~~q2 fiq2/fir r~dpi/d9
= 0 (21)
2r~~qi fiqi/fir dp0/d9
= 0 (22)
fiqo/fir rPig Sllli°) " ° 123)
Equation (17) leads to the following additional conditions:
fiq2/fig + >~r~pi " 0
,
(24)
fiat/fig + >~rp0 " 0 (25)
fiq0/fig >~r~piglp cos(9)] 2q2r~~gsin(9)
= 0
,
(26) qjsin(9) >~~p cos(9)]rp0 " 0
,
(27)
where >
= [m/(m + M)]~/~ and p = M/m
The compatibility of equations (22), (25) and (27) imposes qi
= Po = 0 The solution for
equations (23), (21) and (24) is
pi =Ci cos(>9) + C2sin(>9)
,
q~ = >r~ci sin(>9) + lC2r~cos(19)
,
qo =(g/2)r~sin(9)[Cl cos(19) + C2sin(19)]
,
(28)
714 JOURNAL DE PHYSIQUE II N°7
where Cl C2 are constants. From (28) and (26) we find
Cl cos(9)cos(18) [l~+1/2] + C2 cos(9)sin(19) [>~ + l/2) +
+(3>/2)Ci sin(9)sm(>9) (3>/2)C2 sin(9)cos(>9)-
[1 >~] Cl cos(>9) [1 >~] C2 sin(>9) = 0 (29)
This is a transcendental equation for > it must be satisfied for any value of 9. If we make 9 = 0 (29) leads to
I) Cl =0 or ii) > =1/2.
In the first case the equation (29) reduces to
C2 cos(f)sin(>9) [>~ + l/2j (3>/2)C2 sin(9)cos(>9)-
[1- >~) C2sin(>9) = 0 (30)
Making 9 = ~~ /2 we get that the unique solution occurs for the trivial cases: >
= 0, corresponding
to m
= 0, and >
= I, corresponding to M
= 0 In the second case we find, from (29), that >
= 1/2
1s a solution for any 9 ifC2 = 0 This solution corresponds to p = M/m = 3 From (19), (20) and
(28) we get
Fi "r~9 cos(9/2)
Fo " (r~/2) 8~ sin(8/2) + gr~sin(9/2)cos~(9/2) (31)
Equations (10), (11) and (12) give us the invariant [5] and the associated symmetries for the
integrable case (p = 3)
1 = (r~/2) 9~ sin(9/2) + gr~ sin(9/2) cos~(9/2)+
+r~r9 cos(9/2)
,
(32)
qi " (1/4)r~9 cos(9/2)
,
q2 =r9 sin(9/2) r cos(9/2)
,
(33)
The symmetry analysis permits us to conclu(e that there is no second mvariant, with a linear
dependence on r, except for the case p = 3 and for the trivial cases where M or m are equal to
zero. We can now compare this result with the analysis made by lbfillaro, Casasayas and Nunes
[Ii. They applied Yoshida's theorem [2] to the SAM potential; the integrability coefficient h given by
fl = 2M/(M m) (34)
As the degree of the system is I, the systems where fl # J(J +1)/2,
with j = 0, 1, 2,..., are non-integrable systems.
Consequently, Yoshida's theorem defines the following domain of possible ate grability for the SAM potential:
p E 11,3j, m the points where p = j(j + lj /lJ j + lj 4j j35j
N°7 NOETHER SYMMETRIES AND SAM 715
We note tha~ ifp = M/m = I, Yoshida's theorem does not apply. We showed that, in this case, there is no second invariant with a 1blear dependence on +, but we do not have a definitive answer about the integrabifity (or not) of this system. There are, however, some numerical results which
suggest the bltegrability of the system in this case [6].
The symmetry analysis enables us to find invariants and their associated symmetries for n- dimensional Hamiltonian systems. The main deficiency of this procedure ties on the necessity
of a polynomial expansion on the velocities and on the resolution of the system of diflerenthl
equations coming from tlds expansion. On the other hand, Yosldda's theorem is a direct and
powerful method for the analysh of the bltegrability domain for two-dimensional homogeneous
Hamiltonian systems. However, it gives 1~s only the necessary condition for a system being an
Integrable one and does not fumish the form of the second invariant. The joint application of
the two methods, as proposed here for the SAM potential, helps us to contour some of these difficulties.
We would like to thank the referee for some useful comments.
References
ill CAS~WYAS J., NUNES A- and TUFILLARO N
,
J Phys. France 51 (1990) 1693.
[2] YOSHIDA H., Physica 29D (1987) 128.
[3] MOREIRA I.C., Sup. Cienc. Cult. 38 (1986) 320 and "Noether Theorem and Two-dimensional Inte- grable Systems", preprint 90/10, UFRJ (1990).
[4] CANTMN F and SARLEr W, SLAM Rev 23 (1981) 467.
[5j TUFILLARO N.,J Phys. France 46(1985) 1495.
[fl TUFiLLARO N. (private communication).
