jacobs university feb. 23, 2011 1 the complex dynamics of spinning tops physics colloquium jacobs...
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Jacobs University Feb. 23, 2011 1
The complex dynamics of spinning tops
Physics ColloquiumJacobs University BremenFebruary 23, 2011
Peter H. Richter University of Bremen
Jacobs University Feb. 23, 2011 2
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
Jacobs University Feb. 23, 2011 3
Rigid bodies in SO(3)
two moments of inertia
two angles for the center of gravity s1, s2, s3
4 essential parameters after scaling of lengths, time, energy:
One point fixed in space, the rest free to move
3 principal axes with respect to fixed pointcenter of gravity anywhere relative to that point
planar
planar
plan
ar
linear
linearlinea
r
Lagrange GeneralEuler
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Rigid bodies in T3
two moments of inertia
two angles for the center of gravity
at least one independent moment of inertia for the Cardan frame
angle between the frame‘s axis and the direction of gravity
6 essential parameters after scaling of lengths, time, energy:
Cardan angles ( )
a little more than 2 SO(3)
→ classical spin?
Lagrange: up – Integr
horiz – Chaos
Euler: symm up – Integr
tilted – Chaos
General: horiz – Interm
asymm up – Chaos
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SO(3)-Dynamics: Euler-Poisson equations
sAA
321 ,,
321 ,,
Al 332211 ,,
coordinates
angular velocity
angular momentum
1 ll
Casimir constants
sAh 2
1energy constant
→ four-dimensional reduced phase space with parameter l
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Relative equilibria: Staude solutions
0
0 sAA
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) s = (0.8, 0.4, 0.3)h32h22
h12
l
h
hstability?
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Integrable cases
)0,0,0(s
)1,0,0(21 sAA
Lagrange: „heavy“, symmetric
Kovalevskaya:
Euler: „gravity-free“
E
L
K
A
)0,0,1(2 321 sAAA
4 integrals
3 integrals
3 integrals
P
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Euler‘s case
-motion decouples from -motion
Poisson sphere potential (h,l)-bifurcation diagram
B
iso-energy surfaces in reduced phase space: , S3, S1xS2, RP3
foliation by 1D invariant tori
S3
S1xS2
RP3
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Lagrange‘s case
¾ < < 1
RP3
S3
2S3
S1xS2
cigar:
S1xS2
S3 RP3
disk: ½ < < ¾
Poisson sphere potentials
B
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Kovalevskaya‘s case
Tori in phase space and Poincaré surface of section
Action integral:
B
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Euler Lagrange Kovalevskaya
Energy surfaces in action representation
B
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Poincaré section 0cos dt
dS
E3h,l
P2h,lU2
h,l V2h,l
S2() R3()
Poisson sphere accessible velocities
S = 0
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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
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
P
S
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Examples
(s1,s2,s3) = (1,0,0)
(s1,s2,s3) = (1,0,0)
integrable non-integrable
black: in
dark: out
light: –
black: out
dark: in
light: –
black: in
dark: out
light: –
black: out
dark: in
light: –
In both cases is the surface of section a torus:
part of the PP torus, outermost circles glued together B
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SummarySummary
• 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 toThanks 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)
0
0
0
12
13
23
0
0
0
12
13
23
ll
ll
ll
0
0
0
12
13
23
ss
ss
ss
S
Al relative equilibrium:
lll variation:
lJ
lAS
A
l
1
1
variational equations:
J: a 6x6 matrix with rank 4 and characteristic polynomial g06 + g14 + g22
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Stability analysis: eigenvalues
2 eigenvalues = 0
4 eigenvalues obtained from g04 + g12 + g2
The two 2 are either real or complex conjugate.
If the 2 form a complex pair, two have positive real part → instability
If one 2 is positive, then one of its roots is positive → instability
Linear stability requires both solutions 2 to be negative: then all are imaginary
We distinguish singly and doubly unstable branches of the bifurcation diagram depending on whether one or two 2 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
1
stable hanging rotation about 1-axis (yellow) connects to upright carrousel motion (red)
3
stable upright rotation about 3-axis (blue) connects to hanging carrousel motion (green)
2
unstable carrousel motion about 2-axis (red and green) connects to stable branches
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Same center of gravity, but permutation of moments of inertia
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M
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