Jacobs University Feb. 23, 2011 1 The complex dynamics of spinning tops Physics Colloquium Jacobs...

Jacobs University Feb. 23, 2011 1

The complex dynamics of spinning tops

Physics ColloquiumJacobs University BremenFebruary 23, 2011

Peter H. Richter University of Bremen


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


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


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


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


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


Typical bifurcation diagram

A = (1.0,1.5, 2.0) s = (0.8, 0.4, 0.3)h32h22

h12

l

h

hstability?


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


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


Lagrange‘s case

¾ < < 1

RP3

S3

2S3

S1xS2

cigar:

S1xS2

S3 RP3

disk: ½ < < ¾

Poisson sphere potentials

B


Kovalevskaya‘s case

Tori in phase space and Poincaré surface of section

Action integral:

B


Euler Lagrange Kovalevskaya

Energy surfaces in action representation

B


Poincaré section 0cos dt

dS

E3h,l

P2h,lU2

h,l V2h,l

S2() R3()

Poisson sphere accessible velocities

S = 0


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


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


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


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


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


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


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


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


Same center of gravity, but permutation of moments of inertia


M

Jacobs University Feb. 23, 2011 1 The complex dynamics of spinning tops Physics Colloquium Jacobs...

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