Namespaces
Variants
Actions

Difference between revisions of "Goryachev-Chaplygin top"

From Encyclopedia of Mathematics
Jump to: navigation, search
m (tex encoded by computer)
 
Line 1: Line 1:
 +
<!--
 +
g1101701.png
 +
$#A+1 = 16 n = 1
 +
$#C+1 = 16 : ~/encyclopedia/old_files/data/G110/G.1100170 Goryachev\ANDChaplygin top
 +
Automatically converted into TeX, above some diagnostics.
 +
Please remove this comment and the {{TEX|auto}} line below,
 +
if TeX found to be correct.
 +
-->
 +
 +
{{TEX|auto}}
 +
{{TEX|done}}
 +
 
A rigid body rotating about a fixed point, for which:
 
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" />;
+
a) the principal moments of inertia $  \lambda = ( \lambda _ {1} , \lambda _ {2} , \lambda _ {3} ) $,  
 +
with regard to the fixed point, satisfy the relation $  \lambda _ {1} = \lambda _ {2} = 4 \lambda _ {3} $;
  
 
b) the centre of mass belongs to the equatorial plane through the fixed point;
 
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" />:
+
c) the principal angular momentum is perpendicular to the direction of gravity, i.e., $  \langle  {m, \gamma } \rangle = 0 $.  
 +
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 $  3 $:
 +
 
 +
$$
 +
Q _ {4} = ( m  ^ {2} _ {1} + m _ {2}  ^ {2} ) m _ {3} + 2m _ {1} \gamma _ {3} .
 +
$$
 +
 
 +
C. Bechlivanidis and P. van Moerbeke [[#References|[a1]]] have shown that the problem has asymptotic solutions which are meromorphic in  $  \sqrt t $;
 +
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  $  3 $-
 +
body Toda lattice. A Lax pair is given in [[#References|[a2]]]:
 +
 
 +
$$
 +
- {
 +
\frac{i}{2}
 +
} ( { {h _ {- 1 }  } tilde } h ^ {- 1 } + { {L _ {0} } tilde } + { {L _ {1} } tilde } h )  ^  \bullet  =
 +
$$
 +
 
 +
$$
 +
=
 +
[ { {L _ {- 1 }  } tilde } h ^ {- 1 } + { {L _ {0} } tilde } + { {L _ {1} } tilde } h, { {B _ {0} } tilde } - { {L _ {1} } tilde } h ] ,
 +
$$
  
<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>
+
where  $  { {L _ {0} } tilde } $
 +
and  $  { {L _ {1} } tilde } $
 +
are given by the  $  ( 3 \times 3 ) $
 +
right-lower corner of  $  L _ {0} $
 +
and  $  L _ {1} $
 +
and where
  
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]]]:
+
$$
 +
{ {L _ {- 1 }  } tilde } = {
 +
\frac{1}{2}
 +
} \left (
  
<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>
+
\begin{array}{ccc}
 +
0  &- y _ {3}  & 0  \\
 +
y _ {3}  & 0  &y _ {1} - x _ {1}  ^ {2}  \\
 +
0  &- y _ {2} + x _ {2}  ^ {2}  & 0 \\
 +
\end{array}
  
<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>
+
\right ) ,
 +
$$
  
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
+
$$
 +
{ {B _ {0} } tilde } = \left (
  
<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>
+
\begin{array}{ccc}
 +
{
 +
\frac{3}{2}
 +
} x _ {3}  & 0  &- x _ {1}  \\
 +
0 &{
 +
\frac{3}{2}
 +
} x _ {3}  & 0  \\
 +
- x _ {2}  & 0  &- x _ {3}  \\
 +
\end{array}
  
<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>
+
\right ) .
 +
$$
  
 
See also [[Kowalewski top|Kowalewski top]].
 
See also [[Kowalewski top|Kowalewski top]].

Latest revision as of 19:42, 5 June 2020


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

a) the principal moments of inertia $ \lambda = ( \lambda _ {1} , \lambda _ {2} , \lambda _ {3} ) $, with regard to the fixed point, satisfy the relation $ \lambda _ {1} = \lambda _ {2} = 4 \lambda _ {3} $;

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., $ \langle {m, \gamma } \rangle = 0 $. First introduced by D. Goryachev [a4] in 1900, the system was later integrated by S.A. Chaplygin [a3] in terms of hyper-elliptic integrals (cf. also 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 $ 3 $:

$$ Q _ {4} = ( m ^ {2} _ {1} + m _ {2} ^ {2} ) m _ {3} + 2m _ {1} \gamma _ {3} . $$

C. Bechlivanidis and P. van Moerbeke [a1] have shown that the problem has asymptotic solutions which are meromorphic in $ \sqrt t $; the system linearizes on a double cover of a hyper-elliptic Jacobian (i.e., of the Jacobi variety of a hyper-elliptic curve; cf. also Plane real algebraic curve), ramified exactly along the two hyper-elliptic curves, where the phase variables blow up; see also [a5]. An elementary algebraic mapping transforms the Goryachev–Chaplygin equations into equations closely related to the $ 3 $- body Toda lattice. A Lax pair is given in [a2]:

$$ - { \frac{i}{2} } ( { {h _ {- 1 } } tilde } h ^ {- 1 } + { {L _ {0} } tilde } + { {L _ {1} } tilde } h ) ^ \bullet = $$

$$ = [ { {L _ {- 1 } } tilde } h ^ {- 1 } + { {L _ {0} } tilde } + { {L _ {1} } tilde } h, { {B _ {0} } tilde } - { {L _ {1} } tilde } h ] , $$

where $ { {L _ {0} } tilde } $ and $ { {L _ {1} } tilde } $ are given by the $ ( 3 \times 3 ) $ right-lower corner of $ L _ {0} $ and $ L _ {1} $ and where

$$ { {L _ {- 1 } } tilde } = { \frac{1}{2} } \left ( \begin{array}{ccc} 0 &- y _ {3} & 0 \\ y _ {3} & 0 &y _ {1} - x _ {1} ^ {2} \\ 0 &- y _ {2} + x _ {2} ^ {2} & 0 \\ \end{array} \right ) , $$

$$ { {B _ {0} } tilde } = \left ( \begin{array}{ccc} { \frac{3}{2} } x _ {3} & 0 &- x _ {1} \\ 0 &{ \frac{3}{2} } x _ {3} & 0 \\ - x _ {2} & 0 &- x _ {3} \\ \end{array} \right ) . $$

See also Kowalewski top.

References

[a1] C. Bechlivanidis, P. van Moerbeke, "The Goryachev–Chaplygin top and the Toda lattice" Comm. Math. Phys. , 110 (1987) pp. 317–324
[a2] A.I. Bobenko, V.B. Kuznetsov, "Lax representation and new formulae for the Goryachev–Chaplygin top" J. Phys. A , 21 (1988) pp. 1999–2006
[a3] 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)
[a4] D. Goryachev, "On the motion of a rigid material body about a fixed point in the case " Mat. Sb. , 21 (1900) (In Russian)
[a5] L. Piovan, "Cyclic coverings of Abelian varieties and the Goryachev–Chaplygin top" Math. Ann. , 294 (1992) pp. 755–764
How to Cite This Entry:
Goryachev-Chaplygin top. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Goryachev-Chaplygin_top&oldid=22523
This article was adapted from an original article by P. van Moerbeke (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article