classical mechanics Classical mechanics is a Theoretical physics, physical theory describing the motion of objects such as projectiles, parts of Machine (mechanical), machinery, spacecraft, planets, stars, and galaxies. The development of classical mechanics inv ...

, the

rotation Rotation or rotational/rotary motion is the circular movement of an object around a central line, known as an ''axis of rotation''. A plane figure can rotate in either a clockwise or counterclockwise sense around a perpendicular axis intersect ...

of a

rigid body In physics, a rigid body, also known as a rigid object, is a solid body in which deformation is zero or negligible, when a deforming pressure or deforming force is applied on it. The distance between any two given points on a rigid body rema ...

such as a

spinning top A spinning top, or simply a top, is a toy with a squat body and a sharp point at the bottom, designed to be rotation, spun on its vertical Axis of rotation, axis, balancing on the tip due to the gyroscopic effect. Once set in motion, a top will ...

under the influence of

gravity In physics, gravity (), also known as gravitation or a gravitational interaction, is a fundamental interaction, a mutual attraction between all massive particles. On Earth, gravity takes a slightly different meaning: the observed force b ...

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 classical mechanics, holonomic constraints are relations between the position variables (and possibly time) that can be expressed in the following form: f(u_1, u_2, u_3,\ldots, u_n, t) = 0 where \ are generalized coordinates that describe t ...

.. In addition to the energy, each of these tops involves two additional

constants of motion In mechanics, a constant of motion is a physical quantity conserved throughout the motion, imposing in effect a constraint on the motion. However, it is a ''mathematical'' constraint, the natural consequence of the equations of motion, rather tha ...

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 physics and mechanics, torque is the rotational analogue of linear force. It is also referred to as the moment of force (also abbreviated to moment). The symbol for torque is typically \boldsymbol\tau, the lowercase Greek letter ''tau''. Wh ...

, and for which the fixed point is the

center of gravity In physics, the center of mass of a distribution of mass in space (sometimes referred to as the barycenter or balance point) is the unique point at any given time where the weighted relative position of the distributed mass sums to zero. For ...

. The Lagrange top is a symmetric top, in which two moments of

inertia Inertia is the natural tendency of objects in motion to stay in motion and objects at rest to stay at rest, unless a force causes the velocity to change. It is one of the fundamental principles in classical physics, and described by Isaac Newto ...

are the same and the center of gravity lies on the

symmetry axis Axial symmetry is symmetry around an axis or line (geometry). An object is said to be ''axially symmetric'' if its appearance is unchanged if transformed around an axis. The main types of axial symmetry are ''reflection symmetry'' and ''rotatio ...

. The Kovalevskaya topPerelemov, A. M. (2002). ''Teoret. Mat. Fiz.'', Volume 131, Number 2, pp. 197–205. 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,

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

Herbert Goldstein Herbert Goldstein (June 26, 1922 – January 12, 2005) was an American physicist and the author of the standard graduate textbook ''Classical Mechanics (Goldstein book), Classical Mechanics''. Life and work Goldstein, long recognized for his s ...

, Charles P. Poole, and John L. Safko (2002). ''Classical Mechanics'' (3rd Edition), Addison-Wesley. . is described at time

t

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

orthogonal In mathematics, orthogonality (mathematics), orthogonality is the generalization of the geometric notion of ''perpendicularity''. Although many authors use the two terms ''perpendicular'' and ''orthogonal'' interchangeably, the term ''perpendic ...

vectors

\hat^1

\hat ^2

and

\hat^3

with corresponding moments of inertia

I_1

I_2

and

I_3

and the

angular velocity In physics, angular velocity (symbol or \vec, the lowercase Greek letter omega), also known as the angular frequency vector,(UP1) is a pseudovector representation of how the angular position or orientation of an object changes with time, i ...

about those axes. In a

Hamiltonian Hamiltonian may refer to: * Hamiltonian mechanics, a function that represents the total energy of a system * Hamiltonian (quantum mechanics), an operator corresponding to the total energy of that system ** Dyall Hamiltonian, a modified Hamiltonian ...

formulation of classical tops, the conjugate dynamical variables are the components of the

angular momentum Angular momentum (sometimes called moment of momentum or rotational momentum) is the rotational analog of Momentum, linear momentum. It is an important physical quantity because it is a Conservation law, conserved quantity – the total ang ...

vector

\bf

along the principal axes :

(\ell_1, \ell_2, \ell_3)= (\mathbf\cdot \hat ^1,\bf\cdot \hat ^2,\bf\cdot \hat ^3)

and the ''z''-components of the three principal axes, :

(n_1, n_2, n_3)= (\mathbf\cdot \hat ^1,\mathbf\cdot \hat ^2,\mathbf\cdot \hat ^3)

The

Poisson bracket In mathematics and classical mechanics, the Poisson bracket is an important binary operation in Hamiltonian mechanics, playing a central role in Hamilton's equations of motion, which govern the time evolution of a Hamiltonian dynamical system. Th ...

relations of these variables is given by :

\ = \varepsilon_ \ell_c, \  \ = \varepsilon_ n_c, \ \ = 0

If the position of the center of mass is given by

\vec_ = (a \mathbf^1 + b \mathbf^2 + c\mathbf^3)

, then the Hamiltonian of a top is given by :

H = \frac+\frac+\frac + mg (a n_1 + bn_2 + cn_3) = \frac+\frac+\frac + mg \vec_\cdot \mathbf,

The equations of motion are then determined by :

\dot_a = \, \dot_a = \.

Explicitly, these are

\dot \ell_1 = \left(\frac - \frac\right)\ell_2 \ell_3 + mg(c n_2 - b n_3)

\dot n_1 = \frac n_2 - \frac 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 In mathematics, a Lie group (pronounced ) is a group (mathematics), group that is also a differentiable manifold, such that group multiplication and taking inverses are both differentiable. A manifold is a space that locally resembles Eucli ...

SO(3)

, the three-dimensional

rotation group In mathematics, the orthogonal group in dimension , denoted , is the group of distance-preserving transformations of a Euclidean space of dimension that preserve a fixed point, where the group operation is given by composing transformations. ...

, which is the rotation matrix from the lab frame to the body frame. The full configuration space or phase space is the

cotangent bundle In mathematics, especially differential geometry, the cotangent bundle of a smooth manifold is the vector bundle of all the cotangent spaces at every point in the manifold. It may be described also as the dual bundle to the tangent bundle. This m ...

T^*SO(3)

, with the fibers

T^*_RSO(3)

parametrizing the angular momentum at spatial configuration

R

. The Hamiltonian is a function on this phase space.

Euler top

The Euler top, named after

Leonhard Euler Leonhard Euler ( ; ; ; 15 April 170718 September 1783) was a Swiss polymath who was active as a mathematician, physicist, astronomer, logician, geographer, and engineer. He founded the studies of graph theory and topology and made influential ...

, is an untorqued top (for example, a top in free fall), with Hamiltonian :

H_ = \frac+\frac+\frac,

The four constants of motion are the energy

H_

and the three components of angular momentum in the lab frame, :

(L_1,L_2,L_3) = \ell_1 \mathbf^1 +\ell_2\mathbf^2+ \ell_3 \mathbf^3.

Lagrange top

The Lagrange top, named after

Joseph-Louis Lagrange Joseph-Louis Lagrange (born Giuseppe Luigi Lagrangia is a symmetric top in which

I_1=I_2=2I

I_3=I

and the center of mass lies in the plane perpendicular to the symmetry axis

\mathbf _ = h \mathbf^1

. It was discovered by

Sofia Kovalevskaya Sofya Vasilyevna Kovalevskaya (; born Korvin-Krukovskaya; – 10 February 1891) was a Russian mathematician who made noteworthy contributions to analysis, partial differential equations and mechanics. She was a pioneer for women in mathematics a ...

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 The French Academy of Sciences (, ) is a learned society, founded in 1666 by Louis XIV at the suggestion of Jean-Baptiste Colbert, to encourage and protect the spirit of French Scientific method, scientific research. It was at the forefron ...

in 1888. The Hamiltonian is :

H_= \frac+ mgh n_1.

The four constants of motion are the energy

H_

, the Kovalevskaya invariant :

K = \xi_+ \xi_-

where the variables

\xi_\pm

are defined by :

\xi_\pm = (\ell_1\pm i \ell_2 )^2- 2 mgh I(n_1\pm i n_2),

the angular momentum component in the ''z''-direction, :

L_z = \ell_1n_1+\ell_2n_2+\ell_3n_3,

and the magnitude of the ''n''-vector :

n^2 = n_1^2 + n_2^2 + n_3^2.

Nonholonomic constraints

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 and integrated by

Sergey Chaplygin Sergey Alexeyevich Chaplygin (; 5 April 1869 – 8 October 1942) was a Russian and Soviet physicist, mathematician, and mechanical engineer. He is known for mathematical formulas such as Chaplygin's equation and for a hypothetical substa ...

in 1948) is also integrable (

I_1=I_2=4I_3

). Its center of gravity lies in the

equatorial plane The celestial equator is the great circle of the imaginary celestial sphere on the same plane as the equator of Earth. By extension, it is also a plane of reference in the equatorial coordinate system. Due to Earth's axial tilt, the celestial e ...

.Hazewinkel, Michiel; ed. (2012).
Encyclopaedia of Mathematics
', pp. 271–2. Springer. .

References

{{reflist

External links

Kovalevskaya Top – from Eric Weisstein's World of Physics

Kovalevskaya Top
Spinning tops Hamiltonian mechanics