Lagrange, Euler, and Kovalevskaya tops

In classical mechanics, the precession of a rigid body such as a 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.^[1]^[2] In addition to the energy, each of these tops involves three 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 in 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^[3]^[4] is special symmetric top with a unique ratio of the moments of inertia satisfy the relation

$I_{1}=I_{2}=2I_{3}$ ,

and in which 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 equal points). The nonholonomic Goryachev-Chaplygin top (introduced by D. Goryachev in 1900^[5] and integrated by Sergey Chaplygin in 1948^[6]^[7]) is also integrable ( $I_{1}=I_{2}=4I_{3}$ ). Its center of gravity lies in the equatorial plane.^[8]

Hamiltonian formulation of classical tops

A classical top^[9] is defined by three principal axes, defined by the three orthogonal vectors ${\hat {\mathbf {e} }}^{1}$ , ${\hat {\mathbf {e} }}^{2}$ and ${\hat {\mathbf {e} }}^{3}$ with corresponding moments of inertia $I_{1}$ , $I_{2}$ and $I_{3}$ . In a Hamiltonian formulation of classical tops, the conjugate dynamical variables are the components of the angular momentum vector ${\vec {L}}$ along the principal axes

$(l_{1},l_{2},l_{3})=({\vec {L}}\cdot {\hat {\mathbf {e} }}^{1},{\vec {L}}\cdot {\hat {\mathbf {e} }}^{2},{\vec {L}}\cdot {\hat {\mathbf {e} }}^{3})$

and the z-components of the three principal axes,

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

The Poisson algebra of these variables is given by

$\{l_{a},l_{b}\}=\epsilon _{abc}l_{c},\ \{l_{a},n_{b}\}=\epsilon _{abc}n_{c},\ \{n_{a},n_{b}\}=0$

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

$H={\frac {(l_{1})^{2}}{2I_{1}}}+{\frac {(l_{2})^{2}}{2I_{2}}}+{\frac {(l_{3})^{2}}{2I_{3}}}+mg(an_{1}+bn_{2}+cn_{3}),$

The equations of motion are then determined by

${\dot {l}}_{a}=\{H,l_{a}\},{\dot {n}}_{a}=\{H,n_{a}\}$

Euler top

The Euler top is an untorqued top, with Hamiltonian

$H_{E}={\frac {(l_{1})^{2}}{2I_{1}}}+{\frac {(l_{2})^{2}}{2I_{2}}}+{\frac {(l_{3})^{2}}{2I_{3}}},$

The four constants of motion are the energy $H_E$ and the three components of angular momentum in the lab frame,

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

Lagrange top

The Lagrange top is a symmetric top with the center of mass along the symmetry axis at location, ${\vec {R}}_{cm}=h\mathbf {\hat {e}} ^{3}$ , with Hamiltonian

$H_{L}={\frac {(l_{1})^{2}+(l_{2})^{2}}{2I}}+{\frac {(l_{3})^{2}}{2I_{3}}}+mghn_{3}.$

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

$L_{z}=l_{1}n_{1}+l_{2}n_{2}+l_{3}n_{3},$

and the magnitude of the n-vector

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

Kovalevskaya top

The Kovalevskaya top^[3]^[4] is a symmetric top in which $I_{1}=I_{2}=2I_{3}=I$ and the center of mass lies in the plane perpendicular to the symmetry axis ${\vec {R}}_{cm}=h\mathbf {\hat {e}} ^{1}$ . It was discovered by Sofia Kovalevskaya in 1888 and presented in her paper 'Sur Le Probleme 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

$H_{K}={\frac {(l_{1})^{2}+(l_{2})^{2}+2(l_{3})^{2}}{2I}}+mghn_{1}.$

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

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

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

$\xi _{\pm }=(l_{1}\pm il_{2})^{2}-2mghI(n_{1}\pm in_{2}),$

the angular momentum component in the z-direction,

$L_{z}=l_{1}n_{1}+l_{2}n_{2}+l_{3}n_{3},$

and the magnitude of the n-vector

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

References

↑ Audin, M. (1996). Spinning Tops: A Course on Integrable Systems. New York: Cambridge University Press. ISBN 9780521779197.
↑ Whittaker, E. T. (1952). A Treatise on the Analytical Dynamics of Particles and Rigid Bodies. Cambridge University Press. ISBN 9780521358835.
1 2 S. Kovalevskaya (1889). "Sur Le Probleme De La Rotation D'Un Corps Solide Autour D'Un Point Fixe", Acta Math 12, p.177–232. (French)
1 2 Perelemov, A. M. (2002). Teoret. Mat. Fiz., Volume 131, Number 2, p.197–205. (French)
↑ Goryachev, D. (1900). "On the motion of a rigid material body about a fixed point in the case A = B = C'", Mat. Sb., 21. (Russian). Cited in Bechlivanidis & van Moerbek (1987) and Hazewinkel (2012).
↑ Chaplygin, S.A. (1948). "A new case of rotation of a rigid body, supported at one point", Collected Works, Vol. I, pp.118-124. Moscow: Gostekhizdat. (Russian). Cited in Bechlivanidis & van Moerbek (1987) and Hazewinkel (2012).
↑ Bechlivanidis, C. and van Moerbek, P. (1987). "The Goryachev-Chaplygin Top and the Toda Lattice", ProjectEuclid.org. Accessed: November 30 2016.
↑ Hazewinkel, Michiel; ed. (2012). Encyclopaedia of Mathematics, p.271-2. Springer. ISBN 9789401512886.
↑ Herbert Goldstein, Charles P. Poole, and John L. Safko (2002). Classical Mechanics (3rd Edition), Addison-Wesley. ISBN 9780201657029.

External links

Kovalevskaya Top -- from Eric Weisstein's World of Physics

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