Bifurcations and foliations of the Kovalevskaya System Peter H. Richter - Institut für Theoretische...

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Bifurcations and foliations of the Kovalevskaya System

Peter H. Richter - Institut für Theoretische PhysikPeter H. Richter - Institut für Theoretische Physik

International Conference International Conference

Classical Problems of Rigid Body Dynamics Classical Problems of Rigid Body Dynamics

Donetsk, June 23-25, 2004Donetsk, June 23-25, 2004

Survey

• Mathematical vs. physical definition • Equations and constants of motion • Analysis: → VB program

– Kovalevskaya, Kötter, Zhukovsky, Appelrot

• Topology: → IWF video

– Kharlamov, Fomenko, Oshemkov, Bolsinov

• Physics: – Periods, actions

Equations of motion and integrals

Euler-Poisson equations

Poisson sphere radius

angular momentum

energy

Kovalevskaya constant

first integrals

Solving the

equations

The amazing richness of the Kovalevskaya System

4 topologically different (h,l)-level sets

4 Appelrot classes of critical toriwith 2+2+4+7 subclasses

5 Liouville families of invariant tori

5 Kharlamov types of foliations

10 kinds of Poincaré surfaces of section

10 types of iso-energy surfaces h=const

Poincaré surfaces of section

S3 → S2

RP3 → T2

IK3 → 3S2 G

S1xS2 → 2S2

Fomenko stacks

- 0.5 0.9 1.2 1.45 1.7

1.8 1.9 2.3 4.0 9.0

Periods and Actions

periods

H flow K flow

actions

DRV1998

Action representation of iso-energy surfaces

- 0.5 0.9 1.2

1.45 2.3 9.0

Acknowledgements

• Andreas Wittek• Maria Juhnke • Emil Horosov• Holger R. Dullin

• Mikhail P. Kharlamov• Alexey V. Bolsinov• Igor Gashenenko • Alexander P. Veselov

Bifurcations and foliations of the Kovalevskaya System Peter H. Richter - Institut für Theoretische...

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