Kovalevskaya Top
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Title 
Kovalevskaya Top

Alternative Title 
Kowalewskaja Kreisel

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CC Attribution  NoDerivatives 3.0 Germany:
You are free to use, copy, distribute and transmit the work or content in unchanged form for any legal purpose as long as the work is attributed to the author in the manner specified by the author or licensor. 
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C 1961

Publisher 

Release Date 
1997

Language 
English

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Production Year 
1996

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Subject Area  
Abstract 
The dynamics of this third integrable case of classical rigid body theory is presented on several levels of abstraction, starting from real motion of a physical model and an analogous computer simulation of the Kovalevskaya equations of motion. The first abstraction involves the separation of a cyclic angular variable, i. e., the transition to a reduced description, and the introduction of a sixdimensional (gammaiota)phase with two Casimir constants. In the next step, relative equilibria are used to identify bifurcations between different topological types of threedimensional energy surfaces. The third level is concerned with the foliation of these energy surfaces by invariant tori, and the identification of critical tori which mark bifurcations in the type of foliation. The tori are shown in various 3D projections and in homeomorphic deformations of the energy surfaces. The final step in the analysis uses the technique of Poincaré surfaces of section. A comprehensive survey on all possible motions is given in terms of animation series where all Poincaré sections for a given energy are shown in succession.

Keywords  Kovalevskaya motion Kovalevskaya top circular dynamics 
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The dynamics of spinning tops is one of the major themes in classical mechanics. The problem is to understand the motion of a rigid body one point of which is kept fixed  here: at the center of the Cardan suspension. Euler and Lagrange were able to solve the equations of motion for two special cases which have since been associated with their names: the Euler case of a torquefree top, where the fixed point is the center of gravity, and the Lagrange case of a symmetric heavy top, where two moments of inertia are the same, and the center of gravity is on the symmetry axis. In general, the motion of a spinning top is nonintegrable, chaotic. But Sonya Kovalevskaya found a third and last case that can be integrated. The Russian
01:18
mathematician Sonya Kovalevskaya was awarded the Prix Bordin of the Paris Academy of Science, in 1888. In her contribution she presented the complete solution for the problem of a spinning top with two equal moments of inertia, the third one being half as large, and with the center of gravity not on the symmetry axis but in the plane of the two equal moments of inertia. This rigid body has
01:46
been built to fulfill Kovalevskaya´s requirements. However, in order to fix a point, a frame has been added. As a result, the motion is chaotic. Our aim is to understand and classify the various types of motion of the integrable Kovalevskaya top, i. e., the mathematical idealization of this real model, without frame, and without friction. This computer model behaves according to the solutions of Kovalevskaya equations of motion. At low energy, gravity dominates. The three degrees of freedom correspond to three pure types of motion. Rotation in stable hanging position, about the axis with the center of gravity. Pendulum motion about the second major axis. The moment of inertia is the same as about the first axis. Pendulum motion about the third major axis.  The moment of inertia is half as big. A superposition of the two pendulum motions. The typical motion is a combination of all three types. At high energy, gravity is no longer important. Rotation about the axis with the small moment of inertia is stable. Its orientation in space is arbitrary. In case the body rotates about its first major axis, the residual gravity causes precessional motion. If the center of gravity is too high, this rotation is unstable. The mathematical analysis requires a choice of suitable coordinates. The configuration space is SO3. It is convenient to specify the location of the three major axes of inertia  in a space fixed coordinate system. The equations of motion are invariant against rotation about the direction of gravity. The vertical component of the angular moment is a constant of motion. The rotiation is separated, the configuration space reduced. Viewed from the body fixed frame of reference, the direction of the vertical defines a reduced position. It moves on the surface of a sphere  the reduced configuration space S2.
04:46
A complete description in the reduced phase space requires, in addition, specification of the angular momenta.Relative equilibria are periodic motions that appear as points in the reduced phase space. They exist only for special combinations of the parameters energy h and vertical component l of the angular momentum. The simplest equilibrium solutions are those where all energy is in the rotation about the vertical. This is the sleeping top in hanging position. The motion is stable. If there is enough energy to lift the center of gravity to the upright position, the top may be sleeping again. These motions fall on a line which is shifted by the potential energy of the unstable equilibrium. But note there are two different types of this motion, separated by a third line of relative equilibria. In both cases the motion is unstable. In the lower part, the relative equilibrium is doubly hyperbolic. The body escapes from equilibrium in varying directions. On the upper part of the line, the equilibrium is elliptic hyperbolic. The body leaves it along regular spiral trajectories. A characteristic feature are the equilibrium solutions on the third line: the merrygoround motion. The long branch indicates stable equillibria. None of the major axes coincides with the direction of gravity. The unstable merrygoround motion exists on the small piece, near the cusp. The center of gravity is higher than in the stable case. The relative equilibria mark bifurcations of the energy surface of the reduced system. In each of the four phases of the bifurcation diagram, the energy surface is a different threedimensional manifold, with a topology depending on the accessible region of configuration space. There an orbit fills a twodimensional area. All such areas  at given values of energy and angular momentum  are bounded by level lines of the effective potential. The sleeping top in hanging position corresponds to a potential minimum at the south pole of the sphere. The north pole corresponds to the upright sleeping top. The critical points of the effective potential correspond to the relative equilibria. They all lie in a plane that later shall be used as a Poincaré surface of section. At low energy, the level lines enclose a disk. At high energy, the entire surface of the sphere becomes accessible. These are the only two possibilities at small angular momenta. The energy surfaces are S3 for the disks, and RP3 for the sphere. At large angular momenta, there exists another type of energy surface, S1 cross S2. For energies between saddle and maxima, the accessible area of configuration space has two holes. The maxima are stable merrygoround motion. At intermediate angular momenta, there is still a fourth type of energy surface, K3. There the accessible area has three holes. The saddle points correspond to unstable merrygoround motion. The bifurcation diagram associates each combination of parameters with an energy surface. As the Kovalevskaya top is an integrable system, the energy surfaces are foliated by tori. We choose a certain combination of parameters. A typical trajectory densely fills a torus in phase space. Selfintersections cannot be avoided in the projection onto the space of angular momenta.  The inside of the torus is dark, the outside light. Tori may be characterized by a third constant of motion, the Kovalevskaya constant. The maximum value of the Kovalevskaya constant corresponds to a simple motion. The trajectory is periodic in the reduced phase space. In projection onto the space of angular momenta it is but a line. Upon lowering the Kovalevskaya constant, we obtain the generic tori of the system. At a critical value of the constant, the tori degenerate to form a separatrix. Its center is formed by an unstable periodic orbit. The motion leaves it along the separatrix. On the other side of the separatrix, there exists a symmetric pair of tori. A bifurcation has taken place. After that, the tori change continuously again. At the minimum value of the constant, they degenerate and become stable periodic orbits. The corresponding motion in full phase space is quasiperiodic. The animation gives a survey on the foliation of the energy surface by tori. As long as they change continuously, they
11:17
are the same color. Such families of tori may be represented as edges of a graph. The height of a point in the graph indicates the value of the Kovalevskaya constant. End points correspond to stable, branch points to unstable isolated periodic orbits. The graphs change not only at bifurcations of the energy surface, but also at bifurcations of periodic orbits. This gives rise to further lines in the bifurcation diagram. Regions with the same kind of foliation are given a letter. At the transition from B to A, the green families disappear in a pitch fork bifurcation of their stable periodic orbits. In A, the energy surface is foliated by a single family of tori. There exist two stable orbits as end points of this one family. Phase C shares with A and B the same energy surface S3. Out of a tangent bifurcation, a new family of tori appears, the blue tori. The
12:28
orange region is bounded by unstable orbits. The familiar green tori end in a minimum. Three edges meet in the upper branch point. The red and blue tori merge in the separatrix, the center of which is formed by a hyperbolic periodic orbit. On the other side of the separatrix, we have the orange tori. The lower branch point is of the same type except we traverse it in the opposite direction. The orange family ends in the separatrix. Its center is again a hyperbolic orbit. The two green tori on the other side are related by a symmetry. The transition from C to D involves a bifurcation of the energy surface. Red and Blue meet here in a different way: two symmetric yellow tori emerge.  At another separatrix,
13:34
they transform into the familiar
13:36
green tori. Four edges meet in the upper branch point. Red and Blue end up in two hyperbolic orbits which are connected by a separatrix. On the other side, there are two symmetric orbits. In the lower part, we observe twice the third type of branch point where two edges meet. The two yellow families end up in two separatrices, each of which contains an inverse hyperbolic orbit. On the other side of each separatrix, there is again only one torus. At the transition to E, the blue family disappears in a pitchfork bifurcation. The red tori transform directly into the yellow, whose transition into the green type is the same as in D.
14:42
At the upper branch point, there are again three edges meeting; the lower has not changed with respect to D. In phase F, the energy surface is S1 cross S2. The stable red torus splits into two symmetric yellow tori at the separatrix. They end up in stable periodic motions. Graph
15:04
F consists of only three families of tori.  Consider now the real motion corresponding to the upper end of the graph. Periodicity in the reduced system means synchrony between rotational and oscillatory motion. The total motion is quasiperiodic due to precession. The branch point corresponds to a similar motion in almost upright position, but this one is unstable. The two lower endpoints correspond to two stable periodic motions. Their only difference is an exchange of the two symmetric weights. There is one point in parameter space where the four different types of energy surface meet. Its neighbourhood is particularly complicated. In region H, the red torus develops into two yellow ones; these are not stable in the end, but split again into two tori each.
16:32
The violet families are closely related to the merrygoround motion. In phase I, the topology of the energy surfaces is the most complicated. The accessible region in configuration space is a sphere with three holes.  The sequence of bifurcations is the same as
16:52
in H.
17:00
The graph of I, except for colors, is like that of H. Poincaré had the idea to intersect the energy surfaces, and thereby to obtain a comprehensive picture of their foliation. We show this for S3 which is represented here as a ball in R3. Its foliation is particularly simple in phase A because there is only a single family of tori to fill the S3 ball . A torus from the neighbourhood of a stable orbit is a thin tube. A second torus from near the other periodic orbit is entangled with the first. The colors serve here to distinguish the tori. Tori further away from the stable orbits enclose the thinner ones; in the central part of the family, they grow fairly big. We get an impression of how the set of all tori fills the ball representing S3 in R3. In contrast to the projection onto the space of angular momenta, this representation of tori is free from selfintersection. All tori are intersected by a twodimensional sphere inside the ball. This surface of section corresponds to the plane, shown earlier, that contains the critical points of the effective potential. The Poincaré section proper is the restriction of this picture to the surface of the sphere, but it is instructive to look inside the sphere to get an impression of the foliation. More complicated foliations of the S3 are found in phases B, C, and J. In B the tori of three families are mutually entangled. Some distance away from the stable periodic orbit: a torus with a large Kovalevskaya constant. The green torus is taken from near the separatrix between red and green. Its symmetric partner is entangled with both. In between there is the unstable periodic orbit. A torus from the red family, closer to the separatrix, embraces the two green tori. To illustrate the transition at the separatrix, the embracing torus is made transparent. We discard half of S3 by means of a Poincaré section, and consider the foliation at the interior. Phase C has four stable and two unstable periodic orbits.We choose tori from the neighbourhoods of the stable orbits, colors corresponding to the graphs.Intersecting the tori, we take a look at the foliation. The essential information is already contained in the surface. The Poincaré section is now shown in planar projection, and with continuous coloring. The colors correspond to the families of tori. Their brightness increases with K. The stable periodic orbits are the light and dark centers. Separatrices are recognizable as color discontinuities. Intersections of such discontinuities are unstable periodic orbits.For constant values of the energy, all Poincaré section are now shown in succession. At low energy, there is only one type of foliation. With increasing angular momentum, the surface of section shrinks to the point of a sleeping top in hanging position. At a larger value of the energy, we have two topologically different energy surfaces, RP3 and S3, and four different types of foliation. In D the surface of section is a torus. Bifurcation of the energy surface. In C: a sphere. Blue disappears in a tangent bifurcation. Phase B. Green disappears in a pitchfork bifurcation. Phase A. At high values of energy we have three different types of energy surface and six phases of of different foliation. Phase D  At zero angular momentum the section is symmetric. Blue disappears in a pitch fork bifurcation. Phase E. Period doubling bifurcation, pink emerges. Phase H. Bifurcaton of the energy surface. The surface of section decays into two spheres. Phase G. green and pink disappear in a pitch fork bifurcation Phase F. Bifurcation of the energy surface. Phase A  the surface of section is only one sphere. In the central part of parameter space, the energy surface changes at each transition. Phase H. In the constriction at the boundary: the upright sleeping top. Phase J. At the constriction in pink: the stable merrygoround motion. Phase I . In the vanishing small spheres: the unstable merrygoround motion. Phase B. Let us now retrace the levels of abstraction  back to the top. The Poincaré surface of section an intersected torus  the same torus as a whole  its projection into the space of angular momenta  a generating trajectory the same trajectory in configuration space  the Kovalevskaya top in the corresponding initial configuration.