Transcript Chaos in Cosmos

The complex dynamics of spinning tops
Peter H. Richter
University of Bremen
Physics Colloquium
Jacobs University Bremen
February 23, 2011
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Outline
Rigid bodies: configuration and parameter spaces
-
SO(3)→S2, T3→T2
Moments of inertia, center of gravity, Cardan frame
SO(3)-Dynamics
-
Euler-Poisson equations, Casimir and energy constants
Relative equilibria (Staude solutions) and their stability (Grammel)
Bifurcation diagrams, iso-energy surfaces
Integrable cases: Euler, Lagrange, Kovalevskaya
Liouville-Arnold foliation, critical tori, action representation
General motion: Poincaré section over Poisson-spheres→torus
T3-Dynamics
-
canonical equations
3D or 5D iso-energy surfaces
Integrable cases: symmetric Euler and Lagrange in upright Cardan frame
General motion: Poincaré section over Poisson-tori+2cylinder connection
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Rigid bodies in SO(3)
One point fixed in space, the rest free to move
3 principal axes with respect to fixed point
center of gravity anywhere relative to that point
planar
4 essential parameters after scaling of lengths, time, energy:
two moments of inertia a, b (g = 1- a- b)
two angles s,t for the center of gravity s1, s2, s3
linear
Euler
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Lagrange
General
3
Rigid bodies in T3
a little more than 2 SO(3)
→ classical spin?
Cardan angles (j, q, y)
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
(j + p, 2p - q, y + p)
Euler: symm up – Integr
asymm up – Chaos
Lagrange: up – Integr
two angles s,t for the center of gravity
angle d between the frame‘s axis and the direction of gravity
tilted – Chaos
General: horiz – Interm
horiz – Chaos
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SO(3)-Dynamics: Euler-Poisson equations
  
g = g 

   
A = A   - g  s
coordinates

g = (g 1 , g 2 , g 3 )

 = (1 , 2 , 3 )


angular momentum l = (a11,a22 ,a33 ) = A
angular velocity
Casimir constants
 
g g = 1
 
g l = l
→ four-dimensional reduced phase space with parameter l
   
1
energy constant
h = 2   A - g  s
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Relative equilibria: Staude solutions
  
g = g  = 0

   
A = A   - g  s = 0
angular velocity vector constant, aligned with gravity
high energy: rotations about principal axes
low energy: rotations with hanging or upright position of center of gravity
intermediate energy: carrousel motion
possible only for certain combinations of (h,l ): bifurcation diagram
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Typical bifurcation diagram
A = (1.0,1.5, 2.0)
l
s = (0.8, 0.4, 0.3)
2a 3h
2a 2 h
2a1h
l2
h
h
stability?
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A
Integrable cases
P
Euler: „gravity-free“

s = (0,0,0)
E
4 integrals
Lagrange: „heavy“, symmetric
A1 = A2

s = (0,0,-1)
3 integrals
L
Kovalevskaya:
A1 = A2 = 2 A3

s = (-1,0,0)
3 integrals
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Euler‘s case
-motion decouples from g-motion
Poisson sphere potential
(h,l)-bifurcation diagram

S3
S1xS2
RP3
iso-energy surfaces in reduced phase space: , S3, S1xS2, RP3
foliation by 1D invariant tori
B
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Lagrange‘s case
Poisson sphere potentials
disk: ½ < a < ¾
2S3
¾<a<1
S3
cigar: a > 1
S1xS2
S1xS2
RP3
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S3
RP3
B
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Kovalevskaya‘s case
Tori in phase space and Poincaré
surface of section
Action integral:
B
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Energy surfaces in action representation
Euler
Lagrange
Kovalevskaya
B
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S=
Poincaré section
d
cos  = 0
dt
E3h,l
S=0
U2h,l
P2h,l
V2h,l
R3()
S2(g)
Poisson sphere
accessible velocities
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Topology of Surface of Section if lz is an integral
SO(3)-Dynamics
-
1:1 projection to 2 copies of the Poisson sphere which are punctuated at their
poles and glued along the polar circles
this turns them into a torus (PP torus)
at high energies the SoS covers the entire torus
at lower energies boundary points on the two copies must be identified
P
T3-Dynamics
-
1:1 projection to 2 copies of the Poisson torus plus two connecting cylinders
the Poincaré surface is not a manifold!
but it allows for a complete picture at given energy h and angular momentum lz
S
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Examples
non-integrable
integrable
(a,b,g) = (0.4, 0.4, 0.2)
(s1,s2,s3) = (1,0,0)
(a,b,g) = (0.49, 0.27, 0.24)
(s1,s2,s3) = (1,0,0)
black: in
black: out
black: in
black: out
dark: out
dark: in
dark: out
dark: in
light: –
light: –
light: –
light: –
In both cases is the surface of section a torus:
B
part of the PP torus, outermost circles glued together
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Summary
•
•
•
•
•
•
•
Rigid bodies fixed in one point and subject to external forces need a
support, e. g. a Cardan suspension
This changes the configuration space from SO(3) to T3, and the
parameter set from 4 to 6 dimensional
Integrable cases are only a small albeit highly interesting subset
Not much is known about non-integrable cases
If one degree of freedom is cyclic, complete Poincaré surfaces of
section can be identified – always with SO(3), sometimes with T3
The general case with 3 non-reducible degrees of freedom is beyond
currently available methods of investigation
Very little is known about the quantum mechanics of such systems
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Thanks to
•
Nadia Juhnke
• Andreas Wittek
• Holger Dullin
• Sven Schmidt
• Dennis Lorek
• Konstantin Finke
• Nils Keller
• Andreas Krut
•
Emil Horozov
• Mikhail Kharlamov
• Igor Gashenenko
• Alexey Bolsinov
• Alexander Veselov
• Victor Enolskii
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Stability analysis: variational equations (Grammel 1920)
relative equilibrium:

 = g

variation:


l =  Ag



l  l +d l
g  g + dg
variational equations:



-1
 dg   -  
 dg 
A
 dg 
   = 
   = J   
-1
 dl
dl   S

A


dl 
 
  
 0

 =  g3
-g
 2
-g3
0
g1
g2 

- g1 
0 
 0

 =  l3
-l
 2
- l3
0
l1
l2 

- l1 
0 
 0

S =  s3
- s
 2
- s3
0
s1
s2 

- s1 
0 
J: a 6x6 matrix with rank 4 and characteristic polynomial g0l6 + g1l4 + g2l2
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Stability analysis: eigenvalues
2 eigenvalues l = 0
4 eigenvalues obtained from g0l4 + g1l2 + g2 = 0
The two l2 are either real or complex conjugate.
If the l2 form a complex pair, two l have positive real part → instability
If one l2 is positive, then one of its roots l is positive → instability
Linear stability requires both solutions l2 to be negative: then all l are imaginary
We distinguish singly and doubly unstable branches of the bifurcation
diagram depending on whether one or two l2 are non-negative
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Typical scenario
•
•
•
hanging top starts with two pendulum
motions and develops into rotation about
axis with highest moment of inertia (yellow)
upright top starts with two unstable modes,
then develops oscillatory behaviour and
finally becomes doubly stable (blue)
2 carrousel motions appear in saddle node
bifurcations, each with one stable and one
singly unstable branch. The stable
branches join with the rotations about axes
of largest (red) and smallest (green)
moments of inertia. The unstable branches
join each other and the unstable Euler
rotation
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Orientation of axes, and angular velocities
g1
stable hanging rotation
about 1-axis (yellow)
connects to upright
carrousel motion (red)
g3
g2
unstable carrousel motion
about
 2-axis (red and
green) connects to stable
branches
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stable upright rotation
about 3-axis (blue)
connects to hanging
carrousel motion (green)
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Same center of gravity, but permutation of moments of inertia
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M
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