Chaos in Cosmos

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Chaotic motion in rigid body dynamics
Peter H. Richter University of Bremen

• 7th International Summer School/Conference
• Lets face Chaos through Nonlinear Dynamics
• CAMTP, University of Maribor
• July 1, 2008

Demo 2 - 4
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Outline

• Parameter space
• Configuration spaces SO(3) vs. T3
• Variations on Euler tops
• with and without frame
• effective potentials
• integrable and chaotic dynamics
• Lagrange tops
• Katoks family
• Strategy of investigation

Thanks to my students Nils Keller and Konstantin
Finke
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Parameter space
6 essential parameters after scaling of lengths,
time, energy
two moments of inertia a, b (g 1-a-b)
at least one independent moment of inertia r for
the Cardan frame
two angles s,t for the center of gravity
angle d between the frames axis and the
direction of gravity
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Configuration spaces SO(3) versus T3
Euler angles (j, q, y)
Cardan angles (j, q, y)
(j p, 2p - q, y p)
coordinate singularities removed, but Euler
variables lost
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surprise, surprise!
Demo 9, 10
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Eulers top no gravity, but torques by the frame
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Nonsymmetric and symmetric Euler tops with frame
integrable only if the 3-axis is symmetry axis
Demo 5 - 8
VB Euler
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Lagrange tops without frame
Three types of bifurcation diagrams 0.5 lt a lt
0.75 (discs), 0.75 lt a lt 1 (balls), a gt 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
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A nonintegrable Lagrange top with frame
Q1 Q3 2.5 Q2 4.5QR 2.1 (s1, s2, s3)
(0, -1, 0)
8 types of effective potentials, depending on pj
lz
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The 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?
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Strategy of investigation

• search for critical points of effective potential
  Veff(y,q lz) no explicit general method seems
  to exist numerical work required
• generate bifurcation diagrams in (h,lz)-plane
• construct Reeb graphs
• determine topology of energy surface for each
  connected component
• for details of the foliation of energy surfaces
  look at Poincaré SoS as section condition take
  extrema of szproject the surface of section onto
  the Poisson torus
• accumulate knowledge and develop intuition for
  how chaos and order are distributed in phase
  space and in parameter space

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Rigid Body Dynamics
dedicated to my teacher
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Rigid bodies parameter space
Rotation SO(3) or T3 with one point fixed
4 parameters
principal moments of inertia
2
center of gravity
With Cardan suspension, additional 2
parameters 1 for moments of inertia and 1 for
direction of axis
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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
• Katoks more general cases
• Effective potentials
• Bifurcation diagrams
• Enveloping surfaces
• Poincaré surfaces of section
• Gashenenkos version
• Dullin-Schmidt version
• An application

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Phase space and conserved quantities
3 angles 3 momenta 6D phase
space
energy conservation hconst 5D energy
surfaces
one angular momentum lconst 4D invariant sets
mild chaos
integrable
super-integrable
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Reduced phase space
The 6 components of g and l are restricted by
g 2 1 (Poisson sphere) and l g l
(angular momentum) ? effectively only
4D phase space
energy conservation hconst 3D energy
surfaces
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Euler-Poisson equations
coordinates
Casimir constants
energy integral
effective potential
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Invariant sets in phase space
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(h,l) bifurcation diagrams
Equivalent statements
(h,l) is critical value
relative equilibrium
g is critical point of Ul
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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
• Katoks more general cases
• Effective potentials
• Bifurcation diagrams
• Enveloping surfaces
• Poincaré surfaces of section
• Gashenenkos version
• Dullin-Schmidt version
• An application

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Integrable cases
A
Euler gravity-free
E
L
K
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Eulers case
l-motion decouples from g-motion
B
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Lagranges case
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Enveloping surfaces
B
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Kovalevskayas case
Tori projected to (p,q,r)-space
Tori in phase space and Poincaré surface of
section
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Critical tori additional bifurcations
Fomenko representation of foliations (3 examples
out of 10)
atoms of the Kovalevskaya system
elliptic center A
pitchfork bifurcation B
period doubling A
double saddle C2
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Energy surfaces in action representation
Euler
Lagrange
Kovalevskaya
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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
• Katoks more general cases
• Effective potentials
• Bifurcation diagrams
• Enveloping surfaces
• Poincaré surfaces of section
• Gashenenkos version
• Dullin-Schmidt version
• An application

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Katoks cases
7
s2 s3 0
5
6
1
4
3
2
7 colors for 7 types of bifurcation diagrams
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7colors for 7 types of energy surfaces
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4
5
6
S1xS2
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Effective potentials for case 1
(A1,A2,A3) (1.7,0.9,0.86)
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71 types of envelopes (I)
(A1,A2,A3) (1.7,0.9,0.86)
S1xS2
M32
(2,1.8)
III
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71 types of envelopes (II)
(A1,A2,A3) (1.7,0.9,0.86)
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2 variations of types II and III
A (0.8,1.1,0.9)
A (0.8,1.1,1.0)
Only in cases II and III are the envelopes free
of singularities. Case II occurs in Katoks
regions 4, 6, 7, case III only in region 7.
This completes the list of all possible types of
envelopes in the Katok case. There are more in
the more general cases where only s30
(Gashenenko) or none of the si 0 (not done
yet).
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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
• Katoks more general cases
• Effective potentials
• Bifurcation diagrams
• Enveloping surfaces
• Poincaré surfaces of section
• Gashenenkos version
• Dullin-Schmidt version
• An application

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Poincaré section S1
Skip 3
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Poincaré section S1 projections to S2(g)
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Poincaré section S1 polar circles
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Poincaré section S1 projection artifacts
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Poincaré section S2
Skip 3
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Explicit formulae for the two sections
with
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Poincaré sections S1 and S2 in comparison
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From Kovalevskaya to Lagrange
(A1,A2,A3) (2,?,1) (s1,s2,s3) (1,0,0)
? 2 Kovalevskaya
? 1.1 almost Lagrange
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Examples From Kovalevskaya to Lagrange
(A1,A2,A3) (2,?,1) (s1,s2,s3) (1,0,0)
E
B
? 2
? 2
? 1.1
? 1.1
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Example of a bifurcation scheme of periodic orbits
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To do list

• explore the chaos
• work out the quantum mechanics
• take frames into account

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Thanks to
Holger Dullin Andreas Wittek Mikhail Kharlamov
Alexey Bolsinov Alexander Veselov Igor
Gashenenko Sven Schmidt
and Siegfried Großmann
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