In classical mechanics, the rotation of a rigid body such as a spinning top under the influence of gravity is not, in general, an integrable problem. There are however three famous cases that are integrable, the Euler, the Lagrange, and the Kovalevskaya top, which are in fact the only integrable cases when the system is subject to holonomic constraints.[1] [2] [3] In addition to the energy, each of these tops involves two additional constants of motion that give rise to the integrability.
The Euler top describes a free top without any particular symmetry moving in the absence of any external torque, and for which the fixed point is the center of gravity. The Lagrange top is a symmetric top, in which two moments of inertia are the same and the center of gravity lies on the symmetry axis. The Kovalevskaya top[4] is a special symmetric top with a unique ratio of the moments of inertia which satisfy the relation
I1=I2=2I3,
That is, two moments of inertia are equal, the third is half as large, and the center of gravity is located in the plane perpendicular to the symmetry axis (parallel to the plane of the two degenerate principle axes).
The configuration of a classical top[5] is described at time
t
\hat{e
\hat{e
\hat{e
I1
I2
I3
\bf{L}
(\ell1,\ell2,\ell3)=(L ⋅ \hat{\bf{e}}1,\bf{L} ⋅ \hat{e
and the z-components of the three principal axes,
(n1,n2,n3)=(\hat{z
The Poisson bracket relations of these variables is given by
\{\ella,\ellb\}=\varepsilonabc\ellc, \{\ella,nb\}=\varepsilonabcnc, \{na,nb\}=0
If the position of the center of mass is given by
\vec{R}cm=(a\hate1+b\hate2+c\hate3)
H=
| + | |||||||
2I1 |
| + | |||||||
2I2 |
| |||||||
2I3 |
+mg(an1+bn2+cn3)=
| + | |||||||
2I1 |
| + | |||||||
2I2 |
| |||||||
2I3 |
+mg\vec{R}cm ⋅ \hat{z
The equations of motion are then determined by
\ell |
a=\{H,\ella\},
n |
a=\{H,na\}.
SO(3)
T*SO(3)
* | |
T | |
RSO(3) |
R
The Euler top, named after Leonhard Euler, is an untorqued top (for example, a top in free fall), with Hamiltonian
H\rm=
| + | |||||||
2I1 |
| + | |||||||
2I2 |
| |||||||
2I3 |
,
The four constants of motion are the energy
H\rm
(L1,L2,L3)=\ell1\hate1
2+ | |
+\ell | |
2\hate |
\ell3\hate3.
The Lagrange top,[6] named after Joseph-Louis Lagrange, is a symmetric top with the center of mass along the symmetry axis at location,
R\rm=h\hate3
H\rm=
| + | |||||||
2I |
| |||||||
2I3 |
+mghn3.
The four constants of motion are the energy
H\rm
\ell3
Lz=\ell1n1+\ell2n2+\ell3n3,
and the magnitude of the n-vector
n2=
2 | |
n | |
1 |
+
2 | |
n | |
2 |
+
2 | |
n | |
3 |
The Kovalevskaya top[4] is a symmetric top in which
I1=I2=2I
I3=I
R\rm=h\hate1
H\rm=
| ||||||||||||||||
2I |
+mghn1.
The four constants of motion are the energy
H\rm
K=\xi+\xi-
where the variables
\xi\pm
\xi\pm=(\ell1\pmi\ell2)2-2mghI(n1\pmin2),
the angular momentum component in the z-direction,
Lz=\ell1n1+\ell2n2+\ell3n3,
and the magnitude of the n-vector
n2=
2 | |
n | |
1 |
+
2 | |
n | |
2 |
+
2. | |
n | |
3 |
If the constraints are relaxed to allow nonholonomic constraints, there are other possible integrable tops besides the three well-known cases. The nonholonomic Goryachev–Chaplygin top (introduced by D. Goryachev in 1900[7] and integrated by Sergey Chaplygin in 1948[8]) is also integrable (
I1=I2=4I3