Lagrange, Euler, and Kovalevskaya tops

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.

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 is a special symmetric top with a unique ratio of the moments of inertia which satisfy the relation

I

1

=

I

2

=

2

I

3

,

{\displaystyle I_{1}=I_{2}=2I_{3},}

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 principal axes).

Hamiltonian formulation of classical tops

The configuration of a classical top is described at time

t

{\displaystyle t}

by three time-dependent principal axes, defined by the three orthogonal vectors

e

^

1

{\displaystyle {\hat {\mathbf {e} }}^{1}}

,

e

^

2

{\displaystyle {\hat {\mathbf {e} }}^{2}}

and

e

^

3

{\displaystyle {\hat {\mathbf {e} }}^{3}}

with corresponding moments of inertia

I

1

{\displaystyle I_{1}}

,

I

2

{\displaystyle I_{2}}

and

I

3

{\displaystyle I_{3}}

and the angular velocity about those axes. In a Hamiltonian formulation of classical tops, the conjugate dynamical variables are the components of the angular momentum vector

L

{\displaystyle {\bf {L}}}

along the principal axes

(

ℓ

1

,

ℓ

2

,

ℓ

3

)

=

(

L

⋅

e

^

1

,

L

⋅

e

^

2

,

L

⋅

e

^

3

)

{\displaystyle (\ell _{1},\ell _{2},\ell _{3})=(\mathbf {L} \cdot {\hat {\bf {e}}}^{1},{\bf {{L}\cdot {\hat {\mathbf {e} }}^{2},{\bf {{L}\cdot {\hat {\mathbf {e} }}^{3})}}}}}

and the z-components of the three principal axes,

(

n

1

,

n

2

,

n

3

)

=

(

z

^

⋅

e

^

1

,

z

^

⋅

e

^

2

,

z

^

⋅

e

^

3

)

{\displaystyle (n_{1},n_{2},n_{3})=(\mathbf {\hat {z}} \cdot {\hat {\mathbf {e} }}^{1},\mathbf {\hat {z}} \cdot {\hat {\mathbf {e} }}^{2},\mathbf {\hat {z}} \cdot {\hat {\mathbf {e} }}^{3})}

The Poisson bracket relations of these variables is given by

{

ℓ

a

,

ℓ

b

}

=

ε

a

b

c

ℓ

c

,

{

ℓ

a

,

n

b

}

=

ε

a

b

c

n

c

,

{

n

a

,

n

b

}

=

0

{\displaystyle \{\ell _{a},\ell _{b}\}=\varepsilon _{abc}\ell _{c},\ \{\ell _{a},n_{b}\}=\varepsilon _{abc}n_{c},\ \{n_{a},n_{b}\}=0}

If the position of the center of mass is given by

R

→

c

m

=

(

a

e

^

1

+

b

e

^

2

+

c

e

^

3

)

{\displaystyle {\vec {R}}_{cm}=(a\mathbf {\hat {e}} ^{1}+b\mathbf {\hat {e}} ^{2}+c\mathbf {\hat {e}} ^{3})}

, then the Hamiltonian of a top is given by

H

=

(

ℓ

1

)

2

2

I

1

+

(

ℓ

2

)

2

2

I

2

+

(

ℓ

3

)

2

2

I

3

+

m

g

(

a

n

1

+

b

n

2

+

c

n

3

)

=

(

ℓ

1

)

2

2

I

1

+

(

ℓ

2

)

2

2

I

2

+

(

ℓ

3

)

2

2

I

3

+

m

g

R

→

c

m

⋅

z

^

,

{\displaystyle H={\frac {(\ell _{1})^{2}}{2I_{1}}}+{\frac {(\ell _{2})^{2}}{2I_{2}}}+{\frac {(\ell _{3})^{2}}{2I_{3}}}+mg(an_{1}+bn_{2}+cn_{3})={\frac {(\ell _{1})^{2}}{2I_{1}}}+{\frac {(\ell _{2})^{2}}{2I_{2}}}+{\frac {(\ell _{3})^{2}}{2I_{3}}}+mg{\vec {R}}_{cm}\cdot \mathbf {\hat {z}} ,}

The equations of motion are then determined by

ℓ

˙

a

=

{

H

,

ℓ

a

}

,

n

˙

a

=

{

H

,

n

a

}

.

{\displaystyle {\dot {\ell }}_{a}=\{H,\ell _{a}\},{\dot {n}}_{a}=\{H,n_{a}\}.}

Explicitly, these are

ℓ

˙

1

=

(

1

I

3

−

1

I

2

)

ℓ

2

ℓ

3

+

m

g

(

c

n

2

−

b

n

3

)

{\displaystyle {\dot {\ell }}_{1}=\left({\frac {1}{I_{3}}}-{\frac {1}{I_{2}}}\right)\ell _{2}\ell _{3}+mg(cn_{2}-bn_{3})}

n

˙

1

=

ℓ

3

I

3

n

2

−

ℓ

2

I

2

n

3

{\displaystyle {\dot {n}}_{1}={\frac {\ell _{3}}{I_{3}}}n_{2}-{\frac {\ell _{2}}{I_{2}}}n_{3}}

and cyclic permutations of the indices.

Mathematical description of phase space

In mathematical terms, the spatial configuration of the body is described by a point on the Lie group

S

O

(

3

)

{\displaystyle SO(3)}

, the three-dimensional rotation group, which is the rotation matrix from the lab frame to the body frame.