Transcript Chaos in Cosmos

Chaotic motion in rigid body dynamics
Peter H. Richter
University of Bremen
7th International Summer School/Conference
Let‘s face Chaos through Nonlinear Dynamics
CAMTP, University of Maribor
Demo 2 - 4
July 1, 2008
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Outline
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Parameter space
Configuration spaces SO(3) vs. T3
Variations on Euler tops
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with and without frame
effective potentials
integrable and chaotic dynamics
Lagrange tops
Katok‘s 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 frame‘s 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)
after separation of angle j: reduced configuration spaces
Poisson (q, y)-sphere
Poisson (q, y)-torus
„polar points“ q = 0, p
defined with respect to an
arbitrary direction
„polar y-circles“ q = 0, p
defined with respect to the
axes of the frame
coordinate singularities removed,
but Euler variables lost
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surprise, surprise!
Demo 9, 10
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Euler‘s top: no gravity, but torques by the frame
lz
centrifugal
potential
2
lz 2
Vc =
r + (a sin 2 y + b cos2 y ) sin 2  + g cos2 
Euler-Poisson (y,q)-torus
h
p
Q = (1,1.5, 2)
Q = (2,1,1.5)
Q = (2,1.5,1)
q
Euler-Poisson
(y,q)-sphere
Reeb graph
-p
-p
E
y
p
S1 x S2
2 S3
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Nonsymmetric and symmetric Euler tops with frame
3
3
integrable only if
the 3-axis is
symmetry axis
3
Demo 5 - 8
VB Euler
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Lagrange tops without frame
Three types of bifurcation diagrams:
0.5 < a < 0.75 (discs), 0.75 < a < 1 (balls), a > 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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Q1 = Q3 = 2.5
A nonintegrable Lagrange top with frame
Q2 = 4.5
QR = 2.1
(s1, s2, s3) = (0, -1, 0)
8 types of effective potentials, depending on pj = lz
pj = 0
pj = 7
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pj = 3
pj = 7.1
pj = 4.5
pj = 6
pj = 8
pj = 50
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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
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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 sz
project 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
S3
K3
3S3
S3,S1xS2
VII
RP3
6th International Summer School / Conference
„Let‘s Face Chaos through Nonlinear Dynamics“
CAMTP University of Maribor July 5, 2005
Peter H. Richter - Institut für Theoretische Physik
2T2
(1.912,1.763)
Rigid bodies: parameter space
Rotation SO(3) or T3 with one point fixed
4 parameters:
2 principal moments of inertia:
center of gravity:
2 angles
aA1=A2A/2 A1 , Ab3 = A3 / A1
s1 ,rs,2s, s3
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)
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Phase spaces and basic equations
• Full and reduced phase spaces
• Euler-Poisson equations
• Invariant sets and their bifurcations
Integrable cases
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•
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Katok‘s more general cases
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Effective potentials
Bifurcation diagrams
Enveloping surfaces
Poincaré surfaces of section
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•
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Euler
Lagrange
Kovalevskaya
Gashenenko‘s version
Dullin-Schmidt version
An application
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Phase space and conserved quantities
3 angles + 3 momenta
6D phase space
energy conservation h=const
5D energy surfaces
one angular momentum l=const 4D invariant sets
mild chaos
3 conserved quantities
3D invariant sets
integrable
4 conserved quantities
2D invariant sets
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 h=const
3D energy surfaces
2 conserved quantities
2D invariant sets
3 conserved quantities
1D invariant sets
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integrable
super integrable
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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
Ul
Momentum map
F : (, g )  (h, l )
R 3 ( ) S 2 (g )
Equivalent statements:
 (, g ) : dF = 0
(h,l) is critical value
 = 0, g = 0
relative equilibrium
g : dU l = 0
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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
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•
•
-
Katok‘s more general cases
•
•
•
-
Effective potentials
Bifurcation diagrams
Enveloping surfaces
Poincaré surfaces of section
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•
•
Maribor, July 1, 2008
Euler
Lagrange
Kovalevskaya
Gashenenko‘s version
Dullin-Schmidt version
An application
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Integrable cases
A
Euler: „gravity-free“

s = (0,0,0)
E
Lagrange: „heavy“, symmetric

A1 = A2 s = (0,0,-1)
Kovalevskaya:
A1 = A2 = 2A3
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L

s = (-1,0,0)
K
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Euler‘s case
Poisson sphere potential
(h,l)-bifurcation diagram
l-motion decouples from g-motion
admissible values in (p,q,r)-space for given l and h < Ul
B
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Lagrange‘s case
effective potential
(p,q,r)-equations
I: ½ < a < ¾
integrals
bifurcation diagrams
2S3
S3
II: ¾ < a < 1
III: a > 1
S1xS2
S1xS2
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RP3
S3
RP3
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Enveloping surfaces
B
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Kovalevskaya‘s case
Tori projected
to (p,q,r)-space
Tori in phase space and
Poincaré surface of section
(p,q,r)-equations
integrals
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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
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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
•
•
•
-
Katok‘s more general cases
•
•
•
-
Effective potentials
Bifurcation diagrams
Enveloping surfaces
Poincaré surfaces of section
•
•
•
Maribor, July 1, 2008
Euler
Lagrange
Kovalevskaya
Gashenenko‘s version
Dullin-Schmidt version
An application
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7
Katok‘s cases
s2 = s3 = 0
5
1
2
2S3
1
2
6
4
3
7 colors for 7 types of
bifurcation diagrams
S3
3
3S3
K3
4
7colors for
7 types of
energy
surfaces
RP3
5
6
7
S1xS2
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S3
Effective potentials for case 1
K3
3S3
RP3
(A1,A2,A3) = (1.7,0.9,0.86)
l=0
l = 1.763
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l = 1.68
l = 1.71
l = 1.74
l = 1.773
l = 1.86
l = 2.0
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7+1 types of envelopes (I)
S3
I
T2
(h,l) = (1,1)
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S3
I‘
T2
(1,0.6)
2S3
II
(A1,A2,A3) = (1.7,0.9,0.86)
2T2
(2.5,2.15)
S1xS2
III
M32
(2,1.8)
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7+1 types of envelopes (II)
RP3
IV
T2
(1.5,0.6)
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K3
V
M32
(1.85,1.705)
3S3
VI
(A1,A2,A3) = (1.7,0.9,0.86)
2S2, T2
(1.9,1.759)
S3,S1xS2
VII
2T2
(1.912,1.763)
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2 variations of types II and III
A = (0.8,1.1,0.9)
2S3
2S2
A = (0.8,1.1,1.0)
S1xS2
T2
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.
II‘
(3.6,2.8)
III‘
(3.6,2.75)
This completes the list of all possible
types of envelopes in the Katok case.
There are more in the more general
cases where only s3=0 (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
•
•
•
-
Katok‘s more general cases
•
•
•
-
Effective potentials
Bifurcation diagrams
Enveloping surfaces
Poincaré surfaces of section
•
•
•
Maribor, July 1, 2008
Euler
Lagrange
Kovalevskaya
Gashenenko‘s 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)
S (g)
-
p
S+(g)
q
0
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y
2p
0
0
2p
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Poincaré section S1 – polar circles
A = (2,1.5,1)

s = (1,0,0)
Place the polar circles at
upper and lower rims of the
projection planes.
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Poincaré section S1 – projection artifacts
A = (2,1.1,1)

1.1, 1)
sA==((02,
.94868
,0,0.61623)
s =( 0.94868,0,0.61623)
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Poincaré section S2
=
Skip 3
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Explicit formulae for the two sections
with
S1:
S2:
where
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Poincaré sections S1 and S2 in comparison
A =( 2, 1.1, 1)
s =( 0.94868,0,0.61623)
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From Kovalevskaya to Lagrange
 = 2 Kovalevskaya
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(A1,A2,A3) = (2,,1)
(s1,s2,s3) = (1,0,0)
 = 1.1 almost Lagrange
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Examples: From Kovalevskaya to Lagrange
(A1,A2,A3) = (2,,1)
B
=2
(s1,s2,s3) = (1,0,0)
E
=2
 = 1.1
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 = 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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