Lagrange, Euler, and Kovalevskaya tops

The configuration of a classical top[6] is described at time ${\displaystyle t}$ by three time-dependent principal axes, defined by the three orthogonal vectors ${\displaystyle {\hat {\mathbf {e} }}^{1}}$, ${\displaystyle {\hat {\mathbf {e} }}^{2}}$ and ${\displaystyle {\hat {\mathbf {e} }}^{3}}$ with corresponding moments of inertia ${\displaystyle I_{1}}$, ${\displaystyle I_{2}}$ and ${\displaystyle I_{3}}$ 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 ${\displaystyle {\bf {L}}}$ along the principal axes

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

and the z-components of the three principal axes,

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

The Poisson bracket relations of these variables is given by

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

If the position of the center of mass is given by ${\displaystyle {\vec {R}}_{cm}=(a\mathbf {\hat {e}} ^{1}+b\mathbf {\hat {e}} ^{2}+c\mathbf {\hat {e}} ^{3})}$, then the Hamiltonian of a top is given by

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

The equations of motion are then determined by

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

Explicitly, these are

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

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

and cyclic permutations of the indices.

The Euler top, named after Leonhard Euler, is an untorqued top (for example, a top in free fall), with Hamiltonian

${\displaystyle H_{\rm {E}}={\frac {(\ell _{1})^{2}}{2I_{1}}}+{\frac {(\ell _{2})^{2}}{2I_{2}}}+{\frac {(\ell _{3})^{2}}{2I_{3}}},}$

The four constants of motion are the energy ${\displaystyle H_{\rm {E}}}$ and the three components of angular momentum in the lab frame,

${\displaystyle (L_{1},L_{2},L_{3})=\ell _{1}\mathbf {\hat {e}} ^{1}+\ell _{2}\mathbf {\hat {e}} ^{2}+\ell _{3}\mathbf {\hat {e}} ^{3}.}$

The Lagrange top,[7] named after Joseph-Louis Lagrange, is a symmetric top with the center of mass along the symmetry axis at location, ${\displaystyle \mathbf {R} _{\rm {cm}}=h\mathbf {\hat {e}} ^{3}}$, with Hamiltonian

${\displaystyle H_{\rm {L}}={\frac {(\ell _{1})^{2}+(\ell _{2})^{2}}{2I}}+{\frac {(\ell _{3})^{2}}{2I_{3}}}+mghn_{3}.}$

The four constants of motion are the energy ${\displaystyle H_{\rm {L}}}$, the angular momentum component along the symmetry axis, ${\displaystyle \ell _{3}}$, the angular momentum in the z-direction

${\displaystyle L_{z}=\ell _{1}n_{1}+\ell _{2}n_{2}+\ell _{3}n_{3},}$

and the magnitude of the n-vector

${\displaystyle n^{2}=n_{1}^{2}+n_{2}^{2}+n_{3}^{2}}$

The Kovalevskaya top[4][5] is a symmetric top in which ${\displaystyle I_{1}=I_{2}=2I}$, ${\displaystyle I_{3}=I}$ and the center of mass lies in the plane perpendicular to the symmetry axis ${\displaystyle \mathbf {R} _{\rm {cm}}=h\mathbf {\hat {e}} ^{1}}$. It was discovered by Sofia Kovalevskaya in 1888 and presented in her paper "Sur le problème de la rotation d'un corps solide autour d'un point fixe", which won the Prix Bordin from the French Academy of Sciences in 1888. The Hamiltonian is

${\displaystyle H_{\rm {K}}={\frac {(\ell _{1})^{2}+(\ell _{2})^{2}+2(\ell _{3})^{2}}{2I}}+mghn_{1}.}$

The four constants of motion are the energy ${\displaystyle H_{\rm {K}}}$, the Kovalevskaya invariant

${\displaystyle K=\xi _{+}\xi _{-}}$

where the variables ${\displaystyle \xi _{\pm }}$ are defined by

${\displaystyle \xi _{\pm }=(\ell _{1}\pm i\ell _{2})^{2}-2mghI(n_{1}\pm in_{2}),}$

the angular momentum component in the z-direction,

${\displaystyle L_{z}=\ell _{1}n_{1}+\ell _{2}n_{2}+\ell _{3}n_{3},}$

and the magnitude of the n-vector

${\displaystyle n^{2}=n_{1}^{2}+n_{2}^{2}+n_{3}^{2}.}$

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[8] and integrated by Sergey Chaplygin in 1948[9][10]) is also integrable (${\displaystyle I_{1}=I_{2}=4I_{3}}$). Its center of gravity lies in the equatorial plane.[11]

• Cardan suspension

• Kovalevskaya Top – from Eric Weisstein's World of Physics
• Kovalevskaya Top