View source for Kowalewski top

''Kovalevskaya top''

In 1889, S. Kovalevskaya [[#References|[a11]]] solved the following problem: find all rigid bodies, rotating about a fixed point in the presence of gravity, such that the equations of motion are integrable in the sense of Kowalewski. The latter means that the system admits solutions, expressible as [[Laurent series|Laurent series]] in time <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110160/k1101601.png" />, which contain a number of free parameters equal to the number of degrees of freedom minus one; the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110160/k1101602.png" />-dimensional phase space for such a rigid body is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110160/k1101603.png" />. After the standard reduction to the body coordinates (reduction to the [[Lie algebra|Lie algebra]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110160/k1101604.png" />), for the system to be integrable in the sense of Kowalewski, the Jacobian matrix of the indicial locus (of the equations of motion) must have integer spectrum. This strong condition leads to the following three cases of integrable rotating bodies: i) the Euler top, for which the fixed point and the centre of mass coincide; ii) the Lagrange top, which is an axially-symmetric rigid body, with centre of mass on the axis of symmetry; and iii) the so-called Kowalewski top, which is a solid body rotating about a fixed point such that:

a) its principal moments of inertia <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110160/k1101605.png" /> with regard to the fixed point satisfy the relation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110160/k1101606.png" />;

b) its centre of mass belongs to the equatorial plane (corresponding to the moment <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110160/k1101607.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110160/k1101608.png" />) through the fixed point. The motion (in body coordinates) is governed by the equations

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110160/k1101609.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110160/k11016010.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110160/k11016011.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110160/k11016012.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110160/k11016013.png" /> denote, respectively, the angular momentum, the centre of mass and the unit vector in the direction of gravity; also, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110160/k11016014.png" />. Upon appropriate rescaling and picking the axes of inertia in the equatorial plane, one can achieve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110160/k11016015.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110160/k11016016.png" />. Besides the Casimir invariants, the angular momentum in the direction of gravity <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110160/k11016017.png" />, the length of the squared gravity axis <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110160/k11016018.png" />, and the conservation of energy <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110160/k11016019.png" />, Kovalevskaya exhibited a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110160/k11016020.png" />th invariant, which is quartic, upon giving weight <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110160/k11016021.png" /> to the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110160/k11016022.png" />'s and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110160/k11016023.png" /> to the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110160/k11016024.png" />'s:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110160/k11016025.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110160/k11016026.png" /></td> </tr></table>

==Algebraic integrability.==
The motion is "algebraically integrable" in a sense defined by M. Adler and P. van Moerbeke [[#References|[a3]]], [[#References|[a5]]]: for fixed but arbitrary values of the constants of motion, the level surfaces are affine parts <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110160/k11016027.png" /> of Abelian varieties <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110160/k11016028.png" /> (complex algebraic tori; cf. [[Algebraic torus|Algebraic torus]]). For the Kowalewski top, the level surfaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110160/k11016029.png" /> are affine parts <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110160/k11016030.png" /> of Abelian surfaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110160/k11016031.png" /> of polarization <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110160/k11016032.png" /> (cf. [[Polarized algebraic variety|Polarized algebraic variety]]), and the complement <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110160/k11016033.png" /> is a union of two ample divisors (curves) of genus <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110160/k11016034.png" />, each of them being a doubly ramified (in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110160/k11016035.png" /> points) cover of an [[Elliptic curve|elliptic curve]]. The Kowalewski system then "linearizes" on these Abelian varieties, and the motion can be written in terms of Abelian integrals (quadratures; cf. also [[Abelian integral|Abelian integral]]), involving an intricate change of variables, governed by the algebraic geometry above; see [[#References|[a13]]] and for the quadratures [[#References|[a9]]]. A birational mapping transforms Kowalewski's motion into a [[Geodesic flow|geodesic flow]] on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110160/k11016036.png" /> for the Manakov metric [[#References|[a4]]].

==Lax pairs.==
The Kowalewski top equations can be expressed as a Lax pair (see [[#References|[a8]]]) of the following form:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110160/k11016037.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a1)</td></tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110160/k11016038.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110160/k11016039.png" /> is an indeterminate, and where the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110160/k11016040.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110160/k11016041.png" /> are <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110160/k11016042.png" /> matrices, defined below. The motion then linearizes on the Jacobian of the corresponding spectral curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110160/k11016043.png" />, which is a double cover (of genus <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110160/k11016044.png" />) of a [[Hyper-elliptic curve|hyper-elliptic curve]] of genus <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110160/k11016045.png" />, ramified at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110160/k11016046.png" /> points. The matrices in (a1) are given by:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110160/k11016047.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110160/k11016048.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110160/k11016049.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110160/k11016050.png" /></td> </tr></table>

where

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110160/k11016051.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110160/k11016052.png" /></td> </tr></table>

Such Lax pairs arise in the context of Kostant–Kirillov–Souriau orbits, for Lie algebra splittings into two Lie subalgebras, as summarized by the Adler–Kostant–Symes theorem [[#References|[a1]]], [[#References|[a2]]]. A variation of the latter theorem was applied in [[#References|[a8]]] to a loop algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110160/k11016053.png" /> (cf. also [[Loop|Loop]]) with a new bracket <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110160/k11016054.png" />, constructed as follows: consider the [[Lie algebra|Lie algebra]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110160/k11016055.png" /> and its decomposition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110160/k11016056.png" /> into even and odd subspaces for the automorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110160/k11016057.png" />. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110160/k11016058.png" />. From this, construct the twisted loop algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110160/k11016059.png" />, together with its splitting and dual splitting, for a non-degenerate ad-invariant pairing:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110160/k11016060.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110160/k11016061.png" /></td> </tr></table>

For the Kowalewski top, pick <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110160/k11016062.png" />, viewed as a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110160/k11016063.png" />-dimensional subspace of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110160/k11016064.png" />-dimensional <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110160/k11016065.png" />. After conjugation by a constant matrix, the Lax pair has the form (a1).

Another Lax pair mentioned in [[#References|[a4]]] and, in a slight variation, in [[#References|[a12]]] is based on the following idea: the birational mapping, mentioned above, from the Kowalewski top to the geodesic flow on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110160/k11016066.png" /> for the Manakov metric leads to the following Lax pair:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110160/k11016067.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a2)</td></tr></table>

and the motion linearizes on the Jacobian of the spectral curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110160/k11016068.png" />, which is Kowalewski's original hyper-elliptic curve. In (a2),

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110160/k11016069.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110160/k11016070.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110160/k11016071.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110160/k11016072.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110160/k11016073.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110160/k11016074.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110160/k11016075.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110160/k11016076.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110160/k11016077.png" /></td> </tr></table>

See also [[Goryachev–Chaplygin top|Goryachev–Chaplygin top]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> M. Adler, "On a trace functional for formal pseudodifferential operators and the symplectic structure for the Korteweg–de Vries type equations" ''Invent. Math.'' , '''50''' (1979) pp. 451–500</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> M. Adler, P. van Moerbeke, "Completely integrable systems, Euclidean Lie algebras and curves; Linearization of Hamiltonian Systems, Jacobi varieties and representation theory" ''Adv. Math.'' , '''38''' (1980) pp. 267–317;318–379 {{MR|0597729}} {{ZBL|0455.58017}} </TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> M. Adler, P. van Moerbeke, "The algebraic integrability of geodesic flow on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110160/k11016078.png" />" ''Invent. Math.'' , '''67''' (1982) pp. 297–331 {{MR|0665159}} {{ZBL|0539.58012}} </TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top"> M. Adler, P. van Moerbeke, "The Kowalevski and Hénon–Heiles motions as Manakov geodesic flows on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110160/k11016079.png" /> - a two-dimensional family of Lax pairs" ''Comm. Math. Phys.'' , '''113''' (1988) pp. 659–700</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top"> M. Adler, P. van Moerbeke, "The complex geometry of the Kowalevski–Painlevé analysis" ''Invent. Math.'' , '''97''' (1989) pp. 3–51</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top"> A.G. Reyman, M.A. Semenov-Tian-Shanski, "Reduction of Hamiltonian systems, affine Lie algebras and Lax equations I" ''Invent. Math.'' , '''54''' (1979) pp. 81–100 {{MR|549548}} {{ZBL|}} </TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top"> A.G. Reyman, M.A. Semenov-Tian-Shanski, "Reduction of Hamiltonian systems, affine Lie algebras and Lax equations II" ''Invent. Math.'' , '''63''' (1981) pp. 423–432 {{MR|0620678}} {{ZBL|}} </TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top"> A.I. Bobenko, A.G. Reyman, M.A. Semenov-Tian-Shansky, "The Kowalevski top 9 years later: A Lax pair, generalizations and explicit solutions" ''Comm. Math. Phys.'' , '''122''' (1989) pp. 321–354</TD></TR><TR><TD valign="top">[a9]</TD> <TD valign="top"> P. Vanhaecke, "Linearising two-dimensional integrable systems and the construction of angle-action variables" ''Math. Z.'' , '''211''' (1992) pp. 265–313</TD></TR><TR><TD valign="top">[a10]</TD> <TD valign="top"> V.V. Golubev, "Lectures on the integration of equations of motions of a heavy solid about a fixed point" , Gostekhizdat (1953) (English translation by the Israeli information agency)</TD></TR><TR><TD valign="top">[a11]</TD> <TD valign="top"> S. Kowalewski, "Sur le problème de la rotation d'un corps solide autour d'un point fixe" ''Acta Math.'' , '''14''' (1889) pp. 81–93</TD></TR><TR><TD valign="top">[a12]</TD> <TD valign="top"> L. Haine, E. Horozov, "A Lax pair for Kowalevski's top" ''Physica D'' , '''29''' (1987) pp. 173–180 {{MR|923889}} {{ZBL|}} </TD></TR><TR><TD valign="top">[a13]</TD> <TD valign="top"> E. Horozov, P. van Moerbeke, "The full geometry of Kowalevski's top and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110160/k11016080.png" />-abelian surfaces" ''Comm. Pure Appl. Math'' , '''42''' (1989) pp. 357–407</TD></TR></table>