6th International Summer School / Conference „Let‘s Face Chaos through Nonlinear Dynamics“ CAMTP University of Maribor July 5, 2005 Peter H. Richter -

Peter H. Richter - Institut für Theoretische PhysikPeter H. Richter - Institut für Theoretische Physik

6th International Summer School / Conference 6th International Summer School / Conference

„„Let‘s Face Chaos through Nonlinear Dynamics“ Let‘s Face Chaos through Nonlinear Dynamics“

CAMTP University of Maribor July 5, 2005CAMTP University of Maribor July 5, 2005

(1.912,1.763)VII

S3,S1xS2 2T2

Rigid Body DynamicsRigid Body Dynamics

SS33

RPRP33KK33

3S3S33

dedicated to my teacher

Maribor, July 5, 2005 3

Rigid bodies: parameter spaceRigid bodies: parameter space

Rotation SO(3)Rotation SO(3) or Tor T3 3 with one point fixedwith one point fixed

principal moments of inertia: principal moments of inertia: 321 AAA 1312 /,/ AAAA

321 ,, ssscenter of gravity:center of gravity:

With Cardan suspension, additional 2 parameters:With Cardan suspension, additional 2 parameters:

1 for moments of inertia and 1 for direction of axis1 for moments of inertia and 1 for direction of axis

22

,2 angles 2 angles

4 parameters:4 parameters:


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


Phase space and conserved quantitiesPhase space and conserved quantities

3 angles + 3 momenta3 angles + 3 momenta 6D phase space6D phase space

energy conservation h=constenergy conservation h=const 5D energy surfaces 5D energy surfaces

one angular momentum l=constone angular momentum l=const 4D invariant sets4D invariant sets

3 conserved quantities3 conserved quantities 3D invariant sets3D invariant sets

4 conserved quantities4 conserved quantities 2D invariant sets2D invariant sets super-integrablesuper-integrable

integrableintegrable

mild chaosmild chaos


Reduced phase spaceReduced phase space

The 6 components of The 6 components of and and ll are restricted by are restricted by

(Poisson sphere) and (Poisson sphere) and l l ··ll (angular (angular momentum) momentum) effectively only effectively only 4D phase 4D phase spacespace

energy conservation h=constenergy conservation h=const 3D energy surfaces3D energy surfaces

integrableintegrable2 conserved quantities2 conserved quantities 2D invariant sets2D invariant sets

super integrablesuper integrable 3 conserved quantities 3 conserved quantities 1D invariant sets1D invariant sets


Euler-Poisson equationsEuler-Poisson equations

coordinatescoordinates

Casimir constantsCasimir constants

effective potentialeffective potential

energy integralenergy integral


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)






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)


Poincaré section SPoincaré section S22

=

Skip 3Skip 3


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


To do listTo do list

• explore the chaosexplore the chaos

• work out the quantum mechanicswork out the quantum mechanics

• take frames into account take frames into account


Thanks toThanks to

Holger Dullin Holger Dullin

Andreas WittekAndreas Wittek

Mikhail Kharlamov Mikhail Kharlamov

Alexey Bolsinov Alexey Bolsinov

Alexander Veselov Alexander Veselov

Igor GashenenkoIgor Gashenenko

Sven SchmidtSven Schmidt

… … and Siegfried Großmannand Siegfried Großmann

Documents

6th International Summer School / Conference „Let‘s Face Chaos through Nonlinear Dynamics“ CAMTP University of Maribor July 5, 2005 Peter H. Richter -