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The complex dynamics of spinning tops

Physics Colloquium

Jacobs University Bremen

February 23, 2011

Peter H. RichterUniversity of Bremen

Jacobs University Feb. 23, 2011

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

linear

linear

planar

planar

linear

Rigid bodies in SO(3)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

4 essential parameters after scaling of lengths, time, energy:

two moments of inertiaa, b (g = 1- a- b)

two angles s,t for the center of gravity s1, s2, s3

Euler

Lagrange

General

Jacobs University Feb. 23, 2011

(j + p, 2p - q, y + p)

Rigid bodies in T3a little more than 2 SO(3)

→ classical spin?

6 essential parameters after scaling of lengths, time, energy:

Euler: symm up – Integr

two moments of inertiaa, b (g = 1-a-b)

asymm up – Chaos

at least one independent moment of inertia r for the Cardan frame

Lagrange: up – Integr

two angles s,t for the center of gravity

tilted – Chaos

angledbetween the frame‘s axis and the direction of gravity

General: horiz – Interm

horiz – Chaos

Jacobs University Feb. 23, 2011

angular velocity

angularmomentum

Casimir constants

energy constant

SO(3)-Dynamics: Euler-Poisson equations→ four-dimensional reduced phase space with parameter l

Jacobs University Feb. 23, 2011

Relative equilibria: Staude solutions

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

Jacobs University Feb. 23, 2011

l2

h

h

stability?

Typical bifurcation diagramA = (1.0,1.5, 2.0) s = (0.8, 0.4, 0.3)

Jacobs University Feb. 23, 2011

Integrable cases

P

Euler: „gravity-free“

E

4 integrals

Lagrange: „heavy“, symmetric

L

3 integrals

Kovalevskaya:

K

3 integrals

Jacobs University Feb. 23, 2011

Poisson sphere potential

Euler‘s casew-motiondecouples from g-motion

S3

S1xS2

RP3

iso-energy surfaces in reduced phase space: , S3, S1xS2, RP3

foliation by 1D invariant tori

B

Jacobs University Feb. 23, 2011

Lagrange‘s case

Poisson sphere potentials

disk: ½ < a < ¾

2S3

cigar:a > 1

S3

¾ < a < 1

S1xS2

B

RP3

RP3

S3

S1xS2

Jacobs University Feb. 23, 2011

Kovalevskaya‘s case

Tori in phase space and Poincaré surface of section

Action integral:

B

Jacobs University Feb. 23, 2011

Poincaré section

E3h,l

S = 0

P2h,l

U2h,l

V2h,l

R3(w)

S2(g)

Poissonsphere

accessible velocities

Jacobs University Feb. 23, 2011

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

Jacobs University Feb. 23, 2011

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

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

Jacobs University Feb. 23, 2011

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

Jacobs University Feb. 23, 2011

Thanks to

- Emil Horozov
- Mikhail Kharlamov
- Igor Gashenenko
- Alexey Bolsinov
- Alexander Veselov
- Victor Enolskii

- Nadia Juhnke
- Andreas Wittek
- Holger Dullin
- Sven Schmidt
- Dennis Lorek
- Konstantin Finke
- Nils Keller
- Andreas Krut

Jacobs University Feb. 23, 2011

variation:

variational equations:

Stability analysis: variational equations (Grammel 1920)J: a 6x6 matrix with rank 4 and characteristic polynomial g0l6 + g1l4 + g2l2

Jacobs University Feb. 23, 2011

Stability analysis: eigenvalues

2 eigenvalues l = 0

4 eigenvalues obtained fromg0l4 + 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

Jacobs University Feb. 23, 2011

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

Jacobs University Feb. 23, 2011

g1

g2

stable hanging rotation about 1-axis (yellow) connects to upright carrousel motion (red)

unstable carrousel motion about 2-axis (red and green) connects to stable branches

stable upright rotation about 3-axis (blue) connects to hanging carrousel motion (green)

w

Orientation of axes, and angular velocitiesJacobs University Feb. 23, 2011

Same center of gravity, but permutation of moments of inertia

Jacobs University Feb. 23, 2011

Jacobs University Feb. 23, 2011

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