Leonhard Euler,Joseph-Louis Lagrange, andSofia Vasilyevna Kovalevskaya

Inclassical mechanics, therotation of arigid body such as aspinning top under the influence ofgravity is not, in general, anintegrable problem. There are however three famous cases that are integrable, theEuler, theLagrange, and theKovalevskaya top, which are in fact the only integrable cases when the system is subject toholonomic constraints.^[1][2][3] In addition to the energy, each of these tops involves two additionalconstants of motion that give rise to theintegrability.

The Euler top describes a free top without any particular symmetry moving in the absence of any externaltorque, and for which the fixed point is thecenter of gravity. The Lagrange top is a symmetric top, in which two moments ofinertia are the same and the center of gravity lies on thesymmetry axis. The Kovalevskaya top^[4][5] is a special symmetric top with a unique ratio of themoments of inertia which satisfy the relation

${\displaystyle I_1=I_2=2I_3,}$

That is, two moments of inertia are equal, the third is half as large, and the center of gravity is located in theplane perpendicular to the symmetry axis (parallel to the plane of the two degenerate principle axes).

The configuration of a classical top^[6] is described at time${\displaystyle t}$ by three time-dependentprincipal axes, defined by the threeorthogonal 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 theangular velocity about those axes. In aHamiltonian formulation of classical tops, theconjugate dynamical variables are the components of theangular momentum vector${\displaystyle \mathbf{L}}$ along the principal axes

${\displaystyle ({\ell }_1,{\ell }_2,{\ell }_3)=(\mathbf{L}\cdot {\hat{\mathbf{e}}}^1,\mathbf{L}\cdot {\hat{\mathbf{e}}}^{\mathbf{2}}\mathbf{,}\mathbf{L}\cdot {\hat{\mathbf{e}}}^{\mathbf{3}}\mathbf{)}}$

and thez-components of the three principal axes,

${\displaystyle (n_1,n_2,n_3)=(\hat{\mathbf{z}}\cdot {\hat{\mathbf{e}}}^1,\hat{\mathbf{z}}\cdot {\hat{\mathbf{e}}}^2,\hat{\mathbf{z}}\cdot {\hat{\mathbf{e}}}^3)}$

ThePoisson bracket relations of these variables is given by

${\displaystyle \{{\ell }_a,{\ell }_b\}={\epsilon }_{abc}{\ell }_c,\{{\ell }_a,n_b\}={\epsilon }_{abc}{n}_c,\{n_a,n_b\}=0}$

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

${\displaystyle H=\frac{({\ell }_1{)}^2}{2I_1}+\frac{({\ell }_2{)}^2}{2I_2}+\frac{({\ell }_3{)}^2}{2I_3}+mg(a{n}_1+b{n}_2+c{n}_3)=\frac{({\ell }_1{)}^2}{2I_1}+\frac{({\ell }_2{)}^2}{2I_2}+\frac{({\ell }_3{)}^2}{2I_3}+mg{\overrightarrow{R}}_{cm}\cdot \hat{\mathbf{z}},}$

The equations of motion are then determined by

${\displaystyle {\dot{\ell }}_a=\{H,{\ell }_a\},{\dot{n}}_a=\{H,n_a\}.}$

Explicitly, these are$${\displaystyle {\dot{\ell }}_1=\left(\frac{1}{I_3}-\frac{1}{I_2}\right){\ell }_2{\ell }_3+mg(c{n}_2-b{n}_3)}$$$${\displaystyle {\dot{n}}_1=\frac{{\ell }_3}{I_3}{n}_2-\frac{{\ell }_2}{I_2}{n}_3}$$and cyclic permutations of the indices.

In mathematical terms, the spatial configuration of the body is described by a point on theLie group${\displaystyle SO(3)}$, the three-dimensionalrotation group, which is the rotation matrix from the lab frame to the body frame. The full configuration space or phase space is thecotangent bundle${\displaystyle T^{\ast }SO(3)}$, with the fibers${\displaystyle T_R^{\ast }SO(3)}$ parametrizing the angular momentum at spatial configuration${\displaystyle R}$. The Hamiltonian is a function on this phase space.

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

${\displaystyle H_{\mathrm{E}}=\frac{({\ell }_1{)}^2}{2I_1}+\frac{({\ell }_2{)}^2}{2I_2}+\frac{({\ell }_3{)}^2}{2I_3},}$

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

${\displaystyle (L_1,L_2,L_3)={\ell }_1{\hat{\mathbf{e}}}^1+{\ell }_2{\hat{\mathbf{e}}}^2+{\ell }_3{\hat{\mathbf{e}}}^3.}$

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

${\displaystyle H_{\mathrm{L}}=\frac{({\ell }_1{)}^2+({\ell }_2{)}^2}{2I}+\frac{({\ell }_3{)}^2}{2I_3}+mgh{n}_3.}$

The four constants of motion are the energy${\displaystyle H_{\mathrm{L}}}$, the angular momentum component along the symmetry axis,${\displaystyle {\ell }_3}$, the angular momentum in thez-direction

${\displaystyle L_z={\ell }_1{n}_1+{\ell }_2{n}_2+{\ell }_3{n}_3,}$

and the magnitude of then-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${\displaystyle {\mathbf{R}}_{\mathrm{c}\mathrm{m}}=h{\hat{\mathbf{e}}}^1}$. It was discovered bySofia 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 theFrench Academy of Sciences in 1888. The Hamiltonian is

${\displaystyle H_{\mathrm{K}}=\frac{({\ell }_1{)}^2+({\ell }_2{)}^2+2({\ell }_3{)}^2}{2I}+mgh{n}_1.}$

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

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

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

${\displaystyle {\xi }_{\pm }=({\ell }_1\pm i{\ell }_2{)}^2-2mghI(n_1\pm i{n}_2),}$

the angular momentum component in thez-direction,

${\displaystyle L_z={\ell }_1{n}_1+{\ell }_2{n}_2+{\ell }_3{n}_3,}$

and the magnitude of then-vector

${\displaystyle n^2=n_1^2+n_2^2+n_3^2.}$

If the constraints are relaxed to allownonholonomic constraints, there are other possible integrable tops besides the three well-known cases. The nonholonomicGoryachev–Chaplygin top (introduced by D. Goryachev in 1900^[8] and integrated bySergey Chaplygin in 1948^[9][10]) is also integrable (${\displaystyle I_1=I_2=4I_3}$). Its center of gravity lies in theequatorial plane.^[11]

• Cardan suspension

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

• Spinning tops
• Hamiltonian mechanics

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