from spinning tops to rigid body motion

Groningen, June 3, 2009 2

From Spinning Tops to Rigid Body Motion

Department of Mathematics, University of Groningen, June 3, 2009

Peter H. Richter University of Bremen S3

2S3

S1xS2


OutlineOutline

• Demonstration of some basic physics• Parameter sets • Configuration spaces SO(3) and S2 vs. T3 and T2

• Phase space structure• Equations of motion• Strategies of investigation


Parameter spaceParameter space

at least one independent moment of inertia for the Cardan frame

6 essential parameters after scaling of lengths, time, energy:

angle between the frame‘s axis and the direction of gravity

two moments of inertia

two angles for the center of gravity


Configuration spaces SO(3) versus Configuration spaces SO(3) versus TT33

after separation of angle : reduced configuration spaces

Poisson ()-sphere

„polar points“ defined with respect to an arbitrary direction

Poisson ()-torus

„polar -circles“ defined with respect to the axes of the framecoordinate singularities removed, but Euler variables lost

Cardan angles ( )

Euler angles ( )



• Phase space structure • Equations of motion • Strategies of investigation


Phase space and conserved quantities

3 angles + 3 momenta 6D phase space

4 conserved quantities 2D invariant sets super-integrable

one angular momentum lz = const 4D invariant sets mild chaos

energy conservation h = const 5D energy surfaces strong chaos

3 conserved quantities 3D invariant sets integrable


Reduced phase spaces with parameter lz

2 angles + 2 momenta 4D phase space



energy conservation h = const 3D energy surfaces chaos


( l ) - phase space

3 i-components + 3 momenta li 6D phase space


2 Casimir constants · = 1 and ·l = lz 4D simplectic space


energy conservation h = const 3D energy surfaces chaos



• Phase space structure • Equations of motion• Strategies of investigation


Without frame: Euler-Poisson equations in (,l)-space

1 zll Casimir constants: Casimir constants:

),,( 321

),,( 321

AAAAllll ),,(),,( 332211321

Coordinates: Coordinates:

Energy constant: Energy constant: slA

lA

lA

h 23

3

22

2

21

1 2

1

2

1

2

1

Effective potential: Effective potential: sAAA

lU zl

)(2 233

222

211

2

motion: motion: 22

21

2211

sll


With frame: Euler – Lagrange equations

2)cos sin sin (2

1 T

2)sin cos sin (2

1

22

2

1)cos (

2

1

)cos ,sin sin ,cos sin( s

)0( cos cos sin )(sin sin V

wherewhere

Reduction to a Hamiltonian with parameter , Coriolisforce and centrifugal potential

),,,( ppHH zlp

zl 2zl

DemoDemo



• Phase space structure • Equations of motion • Strategies of investigation


• topological bifurcations of iso-energy surfaces

• their projections to configuration and momentum spaces• integrable systems: action variable representation and

foliation by invariant tori

• chaotic systems: Poincaré sections • periodic orbit skeleton: stable (order) and unstable (chaos)

Search for invariant sets in phase space, and their bifurcations

KatokKatok

EnvelopeEnvelope

ActionsActions

ToriTori

PoincaréPoincaré

PeriodsPeriods


Katok‘s cases s2 = s3 = 01

3

5

2

3

4 5 6 7

7 colors for 7 types of bifurcation diagrams

6colors for 6 types of energy surfaces

S1xS2

1 2S3

S3

RP3K3

3S3

24

6

7

13

5


7+1 types of envelopes (I) (A1,A2,A3) = (1.7,0.9,0.86)

(h,l) = (1,1)I

S3 T2

(1,0.6)I‘

S3 T2

(2.5,2.15)II

2S3 2T2

(2,1.8)III

S1xS2 M32


7+1 types of envelopes (II)

(1.9,1.759)VI

3S3 2S2, T2

(1.912,1.763)VII

S3,S1xS2 2T2

IV

RP3 T2

(1.5,0.6) (1.85,1.705)V

K3 M32

(A1,A2,A3) = (1.7,0.9,0.86)


Euler Lagrange Kovalevskaya

Energy surfaces in action representation


Examples: From Kovalevskaya to Lagrange

BB EE

(A1,A2,A3) = (2,,1)

(s1,s2,s3) = (1,0,0)

= 2 = 2

= 1.1 = 1.1


Example of a bifurcation scheme of periodic orbits


Lagrange tops without frameLagrange tops without frame

Three types of bifurcation diagrams:

0.5 < < 0.75 (discs), 0.75 < < 1 (balls), > 1 (cigars)

five types of Reeb graphs

When the 3-axis is the symmetry axis, the system remains integrable with the frame, otherwise not.

VB Lagrange


The Katok family – and othersThe Katok family – and others

arbitrary moments of inertia, (s1, s2, s3) = (1, 0, 0)

Topology of 3D energy surfaces and 2D Poincaré surfaces of section has been analyzed completely (I. N. Gashenenko, P. H. R. 2004)

How is this modified by the Cardan frame?


Invariant sets in phase spaceInvariant sets in phase space


(h,l) bifurcation diagrams(h,l) bifurcation diagrams

)(3 R

0,0

0:),( dF

)(2 S

lU

),(),(: lhF

MomentumMomentum map map

EquivalentEquivalent statements: statements:

(h,l) is critical value(h,l) is critical value

relative equilibriumrelative equilibrium

is critical point of Uis critical point of U ll0: ldU


Rigid body dynamics in SO(3)Rigid body dynamics in SO(3)

- Phase spaces and basic equations• Full and reduced phase spaces• Euler-Poisson equations• Invariant sets and their bifurcations

- Integrable cases• Euler• Lagrange• Kovalevskaya

- Katok‘s more general cases• Effective potentials• Bifurcation diagrams• Enveloping surfaces

- Poincaré surfaces of section• Gashenenko‘s version• Dullin-Schmidt version• An application


Integrable casesIntegrable cases

Lagrange: Lagrange: „„heavy“, symmetricheavy“, symmetric

21 AA )1,0,0( s

Kovalevskaya: Kovalevskaya:

321 2AAA )0,0,1(s

Euler:Euler: „gravity-free“„gravity-free“

)0,0,0(s EE

LL

KK

AA


Euler‘s caseEuler‘s case

ll--motionmotion decouples from decouples from --motionmotion

Poisson sphere potentialPoisson sphere potential

admissible values in (p,q,r)-space for given l and h < Uadmissible values in (p,q,r)-space for given l and h < U ll (h,l)-bifurcation diagram(h,l)-bifurcation diagram

BB


Lagrange‘s caseLagrange‘s case

effective potentialeffective potential (p,q,r)-equations(p,q,r)-equations

integralsintegrals

I: ½ < I: ½ < < < ¾¾

II: ¾ < II: ¾ < < 1 < 1

RPRP33

bifurcation diagramsbifurcation diagrams

SS33

2S2S33

SS11xSxS22

III: III: > 1 > 1

SS11xSxS22

SS33 RPRP33


Enveloping surfacesEnveloping surfaces

BB


Kovalevskaya‘s caseKovalevskaya‘s case

(p,q,r)-equations(p,q,r)-equations

integralsintegrals

Tori projected Tori projected to (p,q,r)-spaceto (p,q,r)-space

Tori in phase space and Tori in phase space and Poincaré surface of sectionPoincaré surface of section


Fomenko representation of foliations (3 examples out of 10)Fomenko representation of foliations (3 examples out of 10)

„„atoms“ of the atoms“ of the Kovalevskaya systemKovalevskaya system

elliptic center A elliptic center A

pitchfork bifurcation Bpitchfork bifurcation B

period doubling A* period doubling A*

double saddle Cdouble saddle C2 2

Critical tori: additional bifurcationsCritical tori: additional bifurcations


EulerEuler LagrangeLagrange KovalevskayaKovalevskaya

Energy surfaces in action Energy surfaces in action representationrepresentation


Katok‘s casesKatok‘s cases ss22 = s = s33 = 0 = 01

23

45 6

7

2

3

4 5 6 7

7 colors for 7 types of 7 colors for 7 types of bifurcation diagramsbifurcation diagrams

7colors for 7colors for 7 types of 7 types of energy energy surfacessurfaces

SS11xSxS22

1 2S2S33

SS33

RPRP33KK33

3S3S33


Effective potentials for case 1Effective potentials for case 1 (A(A11,A,A22,A,A33) = (1.7,0.9,0.86)) = (1.7,0.9,0.86)

l = 1.763 l = 1.773 l = 1.86 l = 2.0

l = 0 l = 1.68 l = 1.71 l = 1.74

SS33

RPRP33KK33

3S3S33


7+1 types of envelopes7+1 types of envelopes (I)(I) (A(A11,A,A22,A,A33) = (1.7,0.9,0.86)) = (1.7,0.9,0.86)

(h,l) = (1,1)I

S3 T2

(1,0.6)I‘

S3 T2

(2.5,2.15)II

2S3 2T2

(2,1.8)III

S1xS2 M32


7+1 types of envelopes (II)7+1 types of envelopes (II)

(1.9,1.759)VI

3S3 2S2, T2

(1.912,1.763)VII

S3,S1xS2 2T2

IV

RP3 T2

(1.5,0.6) (1.85,1.705)V

K3 M32

(A(A11,A,A22,A,A33) = (1.7,0.9,0.86)) = (1.7,0.9,0.86)


2 variations of types II and III2 variations of types II and III

2S3 2S2

II‘ (3.6,2.8)

S1xS2 T2

(3.6,2.75)III‘

Only in cases II‘ and III‘ are the envelopes free of singularities.

Case II‘ occurs in Katok‘s regions 4, 6, 7, case III‘ only in region 7.

A = (0.8,1.1,0.9)A = (0.8,1.1,0.9) A = (0.8,1.1,1.0)A = (0.8,1.1,1.0)

This completes the list of all possible This completes the list of all possible types of envelopes in the Katok case. types of envelopes in the Katok case. There are more in the more general There are more in the more general cases where only scases where only s33=0 (Gashenenko) =0 (Gashenenko)

or none of the sor none of the sii = 0 (not done yet). = 0 (not done yet).


Poincaré section SPoincaré section S11

Skip 3Skip 3


Poincaré section SPoincaré section S1 1 – projections to S– projections to S22(())

SS--

(())

SS++

(())

00 0000


Poincaré section SPoincaré section S1 1 – polar circles– polar circles

)1,5.1,2(A

)0,0,1(s

Place the polar circles at Place the polar circles at upper and lower rims of the upper and lower rims of the projection planes. projection planes.


Poincaré section SPoincaré section S1 1 – projection artifacts– projection artifacts

)1,1.1,2(A

)61623.0,0,94868.0(s

s =( 0.94868,0,0.61623)

A =( 2, 1.1, 1)


Explicit formulae for the two sectionsExplicit formulae for the two sections

S1:with

S2:

where


Poincaré sections SPoincaré sections S1 1 and Sand S22 in comparison in comparison

s =( 0.94868,0,0.61623)

A =( 2, 1.1, 1)


From Kovalevskaya to LagrangeFrom Kovalevskaya to Lagrange(A(A11,A,A22,A,A33) = (2,) = (2,,1),1)

(s(s11,s,s22,s,s33) = (1,0,0)) = (1,0,0)

= 2 Kovalevskaya= 2 Kovalevskaya = 1.1 almost Lagrange= 1.1 almost Lagrange


Examples: From Kovalevskaya to LagrangeExamples: From Kovalevskaya to Lagrange

BB EE

(A(A11,A,A22,A,A33) = (2,) = (2,,1),1)

(s(s11,s,s22,s,s33) = (1,0,0)) = (1,0,0)

= 2= 2 = 2= 2

= 1.1= 1.1 = 1.1= 1.1


Example of a bifurcation scheme of periodic orbitsExample of a bifurcation scheme of periodic orbits

from spinning tops to rigid body motion

Documents

d phase space g

phase spaceh

investigationparameter

basic physicsparameter

invariant sets

d energy surfaces

phase space3 gi components

types of envelopes