The Hamiltonian Hopf bifurcation in the Lagrange top

The Hamiltonian Hopf bifurcation in the Lagrange top

Citation for published version (APA):

Cushman, R. H., & Meer, van der, J. C. (1988). The Hamiltonian Hopf bifurcation in the Lagrange top. (RANA : reports on applied and numerical analysis; Vol. 8814). Technische Universiteit Eindhoven.

Document status and date: Published: 01/01/1988

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RANA 88-14 August 1988

THE HAMILTONIAN HOPF BIFURCATION IN THE LAGRANGE TOP by RCushman and J.C. vanderMeer

Reports on Applied and Numerical Analysis

Department of Mathematics and Computing Science Eindhoven University of Technology

P.O. BoxS13 S600 MB Eindhoven The Netherlands

IN THE LAGRANGE TOP

R.Cushman

Mathematisch Instituut, Rijksuniversiteit Utrecht P.O. Box 80.010,3508 TA UTRECHT, The Netherlands

J.C. van der Meer

Faculteit der Wiskunde en lnfonnatica, Technische Universiteit Eindhoven

P.O. Box 513. 5600 MB EINDHOVEN, The Netherlands 1980 Mathematics subject classification: 70E.58F.

Key words & phrases : Lagrange top, Hamiltonian Hopf bifurcation, energy-momentum map-ping.

Notes : This report will appear in the proceedings of the

ve

Conoque International Geometrie Symplectique et Mecanique held from May 23 to May 27. 1988, in La Grande Motte, France.

ABSTRACf

We show that the Lagrange top undergoes a Hamiltonian Hopf bifurcation when the angular momentum corresponding to rotation about the symmetty axis of the body passes through a value where the sleeping top changes stability.

O. Introduction

Consider the equilibrium where the Lagrange top is sleeping. In this paper we will show that a Hamiltonian Hopfbifu.rcation takes place when this equilibrium changes its stability.

It is well known that the Lagrange top. a heavy symmetric rigid body with one point fixed. is a completely integrable Hamiltonian system [6]. Besides the Hamiltonian itself there are two additional integrals of angular momentum: one associated to rotation about the vertical axis fixed in space and the other associated to rotation about the symmetry axis of the body. After removing the symmetry of rotation about the body axis using the reduction theorem [1]. one has a two degree of freedom system whose motions are described by the Euler-Poisson equations [4], These equation are in Hamiltonian fonn with respect to a nonstandard Poisson structure on 11 6,

In a neighborhood of an equilibrium point corresponding to sleeping motion of the top. the Euler-Poisson equations become a parameter dependent Hamiltonian system on 114. whose Pois-son structure is induced from that on JR6. In order to prove the existence of a Hamiltonian Hopf bifurcation, only the constant part of the Poisson structure on JR4 is of importance. The reason for this is that the normalization of the energy-momentum mapping in the sense of Vander Meer [8, ch.3] makes no use of the Poisson structure.

1. The Euler-Poisson formulation of the Lagrange top [7]

On JR6 let (C-( JR 6) •• ) be the commutative, associative algebra of smooth functions under pointwise multiplication.. Let z = (Zlo %2, z3, %4. z5, %6) = (xl. X2. X3. Ylt Y2, Y3) be co-ordinates on JR6. Define a Poisson bracket { } on R 6 by

where the bracket of the co-ordinate functions is given by Table 1.

{A. B} Xl X2 X3 YI Y2 Y3 BI Xl 0 0 0 0 -X3 -X2 X2 0 0 0 X3 0 Xl X3 0 0 0 X2 -Xl 0 Yl 0 -X3 -X2 0 -Y3 -Y2 Y2 X3 0 -Xl Y3 0 YI Y3 X2 -Xl 0 Y2 -Yl 0 A Table 1.

It is easily checked that (C-(R6), ( }) is a Lie algebra. Because the bracket also satisfies Leib-niz identity, namely

{f,g-h}={j,g}·h+g- {f,h}.

it follows that (C-(R6),., { }) is a Poisson algebraA.

In A the Hamiltonian vector field XH of a Hamiltonian function BE Coo(R6) is the deriva-tion

XH(f) = adH

f

= {B. f} •

Using Table 1 we see that Hamilton's equations for X_H are x={B.x}=xx

a::

. aB aB

y={B,y}=xx ax +yx

ay ,

(1)

where x is the usual vector-product on IR 3 and daB = [ adB • adB , ddB] . x Xl x2 X3

The Lagrange top is described by the Hamiltonian

- 1 2 2 1 2

B= 211 (Yl +Y2)+

213 Y3 + Ax3 , A>O, (2)

where I = diag(/t,/2 , 13) is the moment of inertia tensor of the top_ For 1 to be the moment of

inertia tensor of a physically realizable body

0<13<211

(see [6], p.l00). From (1) we see that Hamilton's equations for the Lagrange top are x=xxly

y=xxAe₃+yxly.

(3)

These are exactly the Euler-Poisson equations of the Lagrange top [4]. To remove as may param-eters from B as possible, we change the times scale by setting tnew = 11 t and the length scale by All = 1. The resulting rescaled Hamiltonian is

(4) and Hamilton's equations are

x=xxJy

Y

=x x e3 +y xTy ,

where]

=

diag(1, I, y). From (3) it follows that

11

y = ->1. .

h

1

A straightforward calculation shows that the manifold Ta S2 t;; JR6 defined by

xi

+~

+x1

=1

XIYl +X2Y2 +X3Y3 = a

(6)

is invariant under the flow of XH' In [3] it is shown that Ta S2 is the reduced phase space obtained from the original phase space TSO (3) by removing the S 1 symmetry associated to rota-tion about the symmetty axis of the top at the corresponding angular momentum value a.

The third integral of the Lagrange top is the angular momentum

L : Ta S2 ~ JR6 -? R: (x, y) -? Y3

associated to the S 1 symmetty of rotation about a vertical axis fixed in space. From (1) it follows that Hamilton's equations for XL are

X=xxe3

y=y

xe3'

2. Reduction, relative equilibria and the swallowtail

(8)

In this section we review the salient facts about the relative equilibria and critical values of the energy-momentum mapping of the Lagrange top. We follow [3]. We will show that near the two points where the thread attaches. the set of critical values looks like part of a swallowtail sur-face. This fact has been observed before by Prof. H. Knorrer of ETH and Prof. D. Olillingworth of the University of Southhampton. We begin by studying the relative equilibria of the S 1 action

'P_{t :} R3

x

R3 -+ JR3 X R3 : (X, y) -+ (R,x. Rty) ,

where Rt =

[-~

:

~l

and

c

=

cost.

S = sint. 'Pt is the flow of XL- It leaves Ta S2 invariant and

o

0 IJ

We want to use this S 1 symmetry to reduce H to a one degree of freedom Hamiltonian system on

a second reduced phase space P a, b = L -1 (b) (") Ta S2/ S 1. However. there is a difficulty: the

action of 'PI on Ta S2 has fixed points. namely (O,O,l,O,O.a). whenb=a.

and

(0,0,1. O. 0, -a). when b =-a .

Thus the usual reduction theorem [1] does not apply for all values of

a

and b. To get around this problem we use invariant theory.

The algebra of polynomials on L -1 (b) (") Ta S2. which are invariant under the S 1 action generated by the How'P" is generated by

'Itt =

xi

+

~

•

'lt2 =

yt

+

y~ • 'lt3 = XIY 1

+

X2Y2 ,

subject to the relations

Xl

+

X~ = 1 • X3

+

7tS1t6 =

a •

116 = b .

(9)

(10) (11) Note that (9) and (10) define the algebra of 'P,-invariant polynomials on 1R6. The extra relations (11) define the algebra of polynomials on L -1 (b) (") Ta S2.

Eliminating the variables 'ltl • 'lt3 and 116 from (10) and (11) gives

(12) which defines the second reduced phase space Pat b. On P a, b the Hamiltonian induced by H is

(13) The relative equilibria of H on L -1 (b) (") Ta S2 are S 1 orbits of XL which conespond to critical

points of Eon Pa, b. E has critical points only on Pat b (") {X4

=

OJ. Solving (13) for 'lt2. setting X4 = 0 in (12), and then eliminating Xz from (12) gives

(14) where a = E - ~ "( bZ• The critical points of E on P a, b (") {7t4 = O} correspond to multiple roots 7tS of the polynomial

f

which satisfy I 'Its I::; 1. The critical values of the energy-momentum map of the Lagrange top conespond to a piece of the discriminant locus ~ = {(a. b, E) e R3 I discr(f)(a, b. E) = O} which is pictured below.

A striking feature of 11 is that it has a one dimensional piece

a

isolated. The thread attaches to the two dimensional part of A at the points P i2 = (± 2. ±2, 1 + 2y) where 1 is a uiple root of

f.

When I

a

I :! 2 T lies in the two dimensional part of A.

Next we give a local description of A near P2. A similar argument works about P -2. Intro-duce new parameters (AI.

Az.

A3)

as follows:

u

= 1 +;''It

a

=

Az

+ ;.., + 2. b = ;"3 + 2 .

~E

Figure I. The critical values of the energy-momentum mapping of the Lagrange top. Then P 2 corresponds to (AI.

Az.

;"3) = (0. O. 0) and

g(%)=-/(I-x)=x3 +2p x2 _4rx+q2,

where

x

= 1 -Ks and

2p

=

Al

+

l.z

+

~

Ai, -4r

= -

2Al

+

2Az

+

Al

Az ,

q

=

*'

Az •

We have

Lemma 1. [8, p.77J. The polynomials ft(x)

=

x3 +2p x2 -4r x+q2

h(x)

=

x 3 -2p x 2 -4r x_q2 =-ft(-x)

h(x)

=

X4

+

p x2

+

q X

+ (r+t

p2)

have the same discriminant

Proof: We compute the invariants g 2 and g3 of

11

and

13.

because in both cases we have 4-4(discr)=g~ -27g}

(see [2. p.182 & 185]). In the cubic case

ft(x-; p)=x3 -4(t p2+ r)x+ 16(2;p3+ !pr- 116 q2)

we obtain

while in the quartic case

13(x)=x4 +6( !P)X2 +4(tq)x+(r+t p2) we find that and 1 2 1 1 3 1 1 3 g3=(-p)(r+.!.p )-(.!.q)l_(_p)3=_p +-pr--q . 6 4 4 6 27 6 16 []

Since the discriminant locus of a quartic polynomial is a swallowtail ([2, p.189]). near P2 the discriminant locus d is a piece 1: of a swallowtail. The piece 1: is detennined by the require-ment that all the roots of (15) are nonnegative. This condition arises from the requirement that, 7ts

'S

1. 1: is pictured below. Near P -2 we have the same picture.

r p q I

,

I :/

"

./

--.'

,

,;"--

...

,

\ ~~

,

Figure 2. The swallowtail. 1: is the part corresponding to nonnegative multiple roots of g.

The thread T= {I =r=O,p

>

OJ.

In the remainder of this paper we will show that 1: comes from a Hamiltonian Hopf bifurca-tion in the Euler-Poisson equabifurca-tions of the Lagrange top as the parameter I a I increases through the value 2.

3. The Hamiltonian Hopf bifurcation

In this section we describe the theoretical background for the Hamiltonian Hopf bifurcation in the Lagrange top. For a romprehensive treatment see [8].

We begin by showing that the Euler-Poisson equations (7) undergo a Hamiltonian Hopf bifurcation. For this it is sufficient to study the linearized equations. In order to obtain the swal-lowtail, we have to show that

we

are in the generic situation. This means that we have to check if the higher order terms satisfy certain conditions (see (23) below).

Observe that Pa = (e3, ae3) e Ta S2 is the equilibrium point of XH which corresponds to a sleeping top. (Note that the value of the energy momentum mapping at Pa lies on the thread). We have to study what happens when I a I increases through 2. Linearizing XH at Pa gives

-ay 0 1 0

=

_o

1 0 a(y-l)'

-1 0 -a(y-l) 0

where Tp. (TaS2) has co-ordinates (%10.%2. 0,YI.Y2. 0) e R6. We identify Tp. (TaS2) with

R4 • XHo is a linear Hamiltonian vector field on (R 4 • m) where m is the symplectic form

m(z.w)=wt

o

-a 0 1 a 0 -1 0

o

1 0 0 z. -1 0 0 0 w. z e R4. (16)

In fact m is equal to (W(Pa) I T_p.(TaS2

»+,

where W is the structure matrix of the Poisson bracket { } on C""'(R6) and

+

denotes the operation of inverse 1ran.SJX>se. A short calculation

shows that the characteristics polynomial of XHo is

[).2

+1-

(a2(2y-l)2 _(4_a2»]2 +1. a2(2y-l)2(4-a2)

4 4

whose roots are

{

±t

rJ4-a2

±

ia(2y-I», if I a IS; 2

±t

i(a(2y-I)±Va2-4 ), if I a I

~

2.

As a function of the parameter a the eigenvalues of XHo behave as depicted in figure 3.

This behaviour characterizes the Hamiltonian Hopf bifurcation. To make this rigorous. we have to show that the curve

r:

R -+ sp (m. R) : a -+ XHo(a) is generic, that is.

r

has a transversal intersection with the orbit 0 = {P XHo (2p) p-1 e sp(m. R) I P e SP(m. R)} of. the

linear Hamiltonian vector field XH.(2p).

p2 =1. under the group of linear symplectic mappings

Sp(m, R). The matrix XH_o(2p) bas purely imaginary eigenvalues and a semisimple-nilpotent decomposition

.~ ~

.

(2)

.

~ ~.,

(2)

I

a\

< 2

\al=

2

Figure 3. Eigenvalues ofXH.(a) in the complex plane. where

Xs=

o

p 00

-p

0 00 00 0 p 00 -p 0

o

p 0 -1

-p

0 1 0

o

1 0

-p .

-lOp 0

I al

> 2 (17)

One can show 1hat a complement to T T(2p) 0 in sp(m, IR). the space of linear Hamiltonian vector

fields on (JR4. 0». is spanned by Xs and

o

000

o

000 X",= 0 -1 00 . 1 000 Consequently

X=aXs+'Xn-+PX",

is a versal deformation of XH • (2p). Equating the characteristic polynomials of

X

and XH. (2 p)

Since

!!!

I

~ 0 _{for every} 'Y~ ~ _,

do a=2p

rintersec~

o

transversally atXH_o

(2p)-It remains to investigate the higher oIder tenns of H at Pa-Recall that the Hamiltonian H of the Lagrange top is invariant under the Sl action 'Pt. Near Pa we can describe Ta S2 by the chart: (Dl x JR2, .a) where._{a :} DI x R2 t;;; R4 ~ Ta S2 t;;; R6 is given by

(xhXl.YhY2) ~ (X"X2. (l-xt _x~)1I2. YloY2. (a- XtYI-X2Y2)(l-xt _~rlJ2)

with .a(O,O)=Pa and Dt={(XltX2)e JR2 1 (xt+~)<l}- Expanding (1_xt-x~:fll2 in a power series in xt

+~.

we find that

iI

= .: H is a power series in

~,~,~,~ (l~

(see (9». which are invariant under the Sl action ;" =

action 'Pt -More precisely

where A H=H₁ +Hl +H2

+ - _. ,

H o(x. Y)

=

1. a2y+ 1 2 c s 00 -s cOO on JR4 induced by the S 1 00 c s 00

-s

c

Hl(X.Y)= t(a2'Y-l)(xt+~)-a')(XlYl +X2Y2)+t(Yt+Y~) (19) H2(x, Y)= j-(a2

y-t

)(xt +x~)2 -a')(xt +xh(xtYt +X2Y2) +j- ')(xIYl +X2Y2)2 -Furthennore. in this chart: the integral L

=

Y3 becomes

L =L(x.

y)=(a-xIYI-X2Y2)(1 +t (xt

+~)+

!

(xt

+~)2+

" ' ) .

Thus, near Pat the energy·momentum mapping (H, L) of the Lagrange top on Ta S2 is

(H. L)

near 0 in R4.

Because the functions

xt +x~ +xJ andXIYI +X2Y2 +X3Y3

the inclusion mapping i : Ta S2 -+ ~6 is a Poisson mapping. In other words, restricting { } to

Ta S2 defines a Poisson bracket (

lTa

S'2 on COO(Ta S2). In the chart (D 1 x R 2, C\)a) we have an

induced Poisson bracket [ ] on R 4 given by

[f, g] = C\): ({(C\);;I)*

I,

(C\);;I)* g

lTaS

2 ) .

I.

g E COO(DI X R2).

The structure matrix W of [ ] is given in Table 2.

[ ] Xl X2 Y1 Xl 0 0 0 X2 0 0 -(1_x?_~)112 Y1 0 (I-x! -X~ )112 0 Y2 -(I-x! _~)112 0 -(a-x1YI-X2Y2)(I-xt-x~rll2 Table 2 Expanding W in a power series in x?

+

~ we get

where W=WO+W₁ + ... 0 0 0 1

o

0 -1 0 Wo= 0 lOa -1 0 -a 0 Y2 (1_x?_X~)ll2 0 (a-XIYI-X2Y2XI-x?-~rll2 0

Note that Wo induces the symplectic form co given by (16) and is the structure matrix of the Pois-son bracket [ ]0.

Next introduce the functions

S

=

x? +x~ - P(XIYl +X2Y2),

N = t (x? + x~) +

t

(Yt +Y~) - p(x IY 1 + X2Y2),

M= t(xt+~).

T = X2Yl -XIY2 •

(20)

where p is 1 if a is close to 2 and p is -1 if a is close to - 2. Here S • M • N are Hamiltonian func-tions corresponding to the Hamiltonian vector fields Xs • XM. XN under the bracket [ ]0. We have

and

[T. M]o = 2M • [T. N]o = -2N • [N. M]o = T •

Using (20) we can write

where

and

H=(la ly-l)S+N+aoM+alM2+a2MS+t yS2+ ... ,

ao=ao(a)= a2Y-21 a Iy

al =al(a)= 2a2y-41 a

ly+2y-1-2

a2

=

a2(a) = 2'}( I ai-I)

A

L=S+ .•.

The coefficient a 1 (a) is uniquely detetmined and al(±2)=2y-t >0 • 1 SInCey> '2' (21) (22) (23)

Using the theory of [8, chpt 3], it follows that there is an S 1 equivariant diffeomorphism ~ of R,4, which leaves the origin fixed and a diffeomOIphism • of R,2 such that

. . A

~o (H,L) 0 +=(G,S).

Here

(H. L) :

R,4 ~ R2: (~, 11) ~ (iI(~, 11), L(~, 11» and

(G, S): R,4 ~ JR 2 : (x,y) ~ (G(x, y).

sex,

y» where

G(x, y)=N +aoM +al M2 (24)

A A

Consequently, the set of critical values of (H, L) is diffeomorphic to the set of critical values of

(G, S). In this result from singularity theory the Poisson structure of R4 plays no role.

In fact, the mapping (G, S) is an energy momentum mapping for a Hamiltonian system G

on (JR4, 00). We now recount the analysis of this standard system given in [8]. S is an integral for

G. The flow", of Xs generates an S 1 action on R,4. Since the algebra of "t-invariant polynomi-als is generated by M, N, S and T subject to the relation (21) (compare with (18», we can remove this SI symmetry using invariant theory. The reduced phase space is Ps =S-I(9)/S1

which is defined by

S2 +T2 =4MN, M~ 0, N~ O.

On P

s

the Hamiltonian induced by G is

g=J=J(M,N)=N+aoM+al M2 .

(25)

(26) The relative equilibria of G are the Xs omits on S-1 (9) which correspond to critical points of Jon

(25), having set T

=

0, gives

4a1 M3 + 4a

o

M2 +4g M +82 =0, M'f!. O. (27)

The critical points of Jon Ps f"\ {T = O} correspond to the multiple nonnegative roots of (27). Since

a

1

>

0 for I

a

I near 2, using Lemma 1 we find that the discriminant locus of (27) is the

same piece of the swallowtail surface as 1: in Figure 2. Thus 1: comes from a Hamiltonian Hopf bifurcation, which is what we wanted to show.

Acknowledgement

We would like to thank Prof. Victor Guillemin of M.lT. for introducing us to this problem and for giving us his unpublished notes [5], which treated a related problem.

References

[1] Abraham, R. and Marsden. J.E., Foundations ojmechanic8, 2nd ed. Benjamin/Cummings, Reading, Mass., 1978.

[2] Brieskom, E. and Kn6rrer, H., Plane algebraic curves, BirldUiuser, Boston, 1986.

[3] Cushman, R. and Knorrer, H., The momentum mapping oj the Lagrange top, in: Differential geometric methods in physics, ed. H. Doebner et. al., LNM 1139, (1985). 12 - 24, Springer-Verlag, New York..

[4] Golubev V., Lectures on integration ojthe equation ojmotion oj a rigid body about aflxed point, Israel program for scientific translations, 1960.

[5] Guillemin, V., Unpublished notes, M.I.T., 1987.

[6] Landau, L. and Lifschitz, E., Mechanics, Addison-Wesley, Reading, Mass., 1960.

[7] Ratiu, T. and Van Moerbeke, P .• The Lagrange rigid body motion, Ann. Inst. Fourier, Grenoble 32 (1982), 211 -234.

[8] Van der Meer, J.C., The Hamiltonian H opt bifurcation, LNM 1160, (1985), Springer-Verlag, NewYol'k..