Holger R. Dullin's page on the Kovalevskaya Top

During my PhD thesis we produced a movie about the Kovalevskaya top (german version), and a booklet that explains the theory behind the movie.

Most of the colour pictures from my thesis are shown below, many of them also appeared in [3]

Poincare Sections

There are 10 topologically different Poincaré Sections a-j. Same colour and same lighness gives the invariant curve obtained from the intersection of the reduced two-torus with the surface of section. Each hue (red, green, blue, etc.) represents a family of tori which can be deformed into each other at fixed energy and angular momentum. Colour jumps indicate separatrices, while the light or dark centres are stable reduced periodic orbits.

There is an additional discrete symmetry when the angular momentum vanishes (c0, d0).

Each pictures has a particular value of energy and angular momentum. The ten regions of different foliations of the reduced energy surface are indicated in the phase diagram. The disks show where the parameters for the picture of the respective region were chosen.

phase diagram

The corresponding Fomenko graphs (with the same colour scheme)

fomenko graphs

Medium Size

a case a b case b c case c d case d e case e f case f

a-f cases a-f

g cas g h case h i case i j case j c0 case k=c0 d0 case l=d0

g-j,c0,d0cases g-j,c0,d0


Here all sections for fixed energy and with all possible values of the angular momentum are stacked and then the stack is cut along the middle. This is an attempt to visualise the full system with three degrees of freedom by taking a section of a section. Each 3-Torus of the full system appears as either 2 or 4 points in a stack. The black lines indicate families of 2-tori.

stack 1 stack 2 stack 3 stack 4 stack 5 stack 6 stack 7 stack 8 stack 9 stack 10

Energy Surfaces in Action Space

Without symmetry reduction, for energy 12.

Energy Surface

The original figures of [3] and my thesis

Figure 3 of [3], Fig 4.5 of thesis

Fig3 left Fig3 right

Figure 7 and 10 of [3], Fig 4.6 and 4.7 of thesis

In Fig 10 symmetry reduced energy surfaces in action space are shown for 6 different energies.

Fig7 Fig10

Figure 8 and 11 of [3]


Large Size

Case a Case b Case c Case d Case e Case f Case g Case h Case i Case j Case c0 Case d0

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