View source for Goryachev-Chaplygin top

A rigid body rotating about a fixed point, for which:

a) the principal moments of inertia <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110170/g1101701.png" />, with regard to the fixed point, satisfy the relation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110170/g1101702.png" />;

b) the centre of mass belongs to the equatorial plane through the fixed point;

c) the principal angular momentum is perpendicular to the direction of gravity, i.e., <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110170/g1101703.png" />. First introduced by D. Goryachev [[#References|[a4]]] in 1900, the system was later integrated by S.A. Chaplygin [[#References|[a3]]] in terms of hyper-elliptic integrals (cf. also [[Hyper-elliptic integral|Hyper-elliptic integral]]). The system merely satisfying a) and b) is not algebraically integrable, but on the locus, defined by c), it is; namely, it has an extra invariant of homogeneous degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110170/g1101704.png" />:

<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/g/g110/g110170/g1101705.png" /></td> </tr></table>

C. Bechlivanidis and P. van Moerbeke [[#References|[a1]]] have shown that the problem has asymptotic solutions which are meromorphic in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110170/g1101706.png" />; the system linearizes on a double cover of a hyper-elliptic Jacobian (i.e., of the [[Jacobi variety|Jacobi variety]] of a [[Hyper-elliptic curve|hyper-elliptic curve]]; cf. also [[Plane real algebraic curve|Plane real algebraic curve]]), ramified exactly along the two hyper-elliptic curves, where the phase variables blow up; see also [[#References|[a5]]]. An elementary algebraic mapping transforms the Goryachev–Chaplygin equations into equations closely related to the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110170/g1101707.png" />-body Toda lattice. A Lax pair is given in [[#References|[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/g/g110/g110170/g1101708.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/g/g110/g110170/g1101709.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110170/g11017010.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110170/g11017011.png" /> are given by the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110170/g11017012.png" /> right-lower corner of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110170/g11017013.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110170/g11017014.png" /> and 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/g/g110/g110170/g11017015.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/g/g110/g110170/g11017016.png" /></td> </tr></table>

See also [[Kowalewski top|Kowalewski top]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  C. Bechlivanidis,   P. van Moerbeke,   "The Goryachev–Chaplygin top and the Toda lattice"  ''Comm. Math. Phys.'' , '''110'''  (1987)  pp. 317–324</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  A.I. Bobenko,   V.B. Kuznetsov,   "Lax representation and new formulae for the Goryachev–Chaplygin top"  ''J. Phys. A'' , '''21'''  (1988)  pp. 1999–2006</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  S.A. Chaplygin,   "A new case of rotation of a rigid body, supported at one point" , ''Collected works'' , '''I''' , Gostekhizdat  (1948)  pp. 118–124  (In Russian)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  D. Goryachev,   "On the motion of a rigid material body about a fixed point in the case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110170/g11017017.png" />"  ''Mat. Sb.'' , '''21'''  (1900)  (In Russian)</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  L. Piovan,   "Cyclic coverings of Abelian varieties and the Goryachev–Chaplygin top"  ''Math. Ann.'' , '''294'''  (1992)  pp. 755–764</TD></TR></table>