We will ``approximate'' the asymmetric top by a finite set of point masses which are attached to each other by massless, rigid rods which ensure that the motion of the whole ensemble is described by a path in the group of Euclidean motions in space. The exposition given below is similar to that which can be found in the book on Classical Mechanics by V. I. Arnold.

For the *i*-th point mass with mass *m*_{i}, its position **r**_{i} is a
function of time *t* given by the formula

Now let us consider the velocity vectors of individual point masses,

(*t*) = (*t*)^{ . }**u**_{i} + (*t*)

Since
(*t*)×**u** = *g*(*t*)^{-1 . }(*t*)^{ . }**u**

(
(*t*)×**r** = (*t*)^{ . }*g*(*t*)^{-1 . }**r**

(
(*t*) = (*t*)×(**r**_{i} - **R**(*t*)) + (*t*)

The momentum of the system as a whole is given by
(the remaining terms vanish because

= *m*_{i}(**r**_{i} - **R**)×((*t*)×(**r**_{i} - **R**))

is referred to as the angular momentum of the system
= *m*_{i}**u**_{i}×(×**u**_{i})

which is called the angular momentum in the
We can also compute the total kinetic energy
*T*_{tot} of the system

The latter term is the kinetic energy of a point mass concentrated at the centre of mass. The former term

The behaviour of the *i*-th point mass is dependent on the force
**F**_{i} acting on it. Newton's law states that
**F**_{i} = *m*_{i}. From the expression for
and the
identity
= *g*^{ . } we obtain

Let

The former term is called the

We apply

= × +

This expression in the body frame for the derivative of the angular
momentum in terms of the torque and the angular velocity is called
Euler's equation.
To summarise, the motion of an asymmetric top can separated into two
components, the motion of its centre of mass and the motion about the
centre of mass. The centre of mass behaves exactly like a point of mass
*M* with position **R** subject to a force **F** under
Newton's equations. The motion
about the centre of mass is described in the body frame by a path *g*
is the group of rotations, the moment of inertia tensor *I* and torque
to which the body is subjected. The equation of motion in this
case is Euler's equation. Since the motion of a point mass in a
Newtonian system is well understood, we will concentrate on the latter.

The only expression that depends on the distribution of masses is *I*.
Replacing the summation by integration and the masses by a mass density
, we see that another expression for the moment of inertia is

(*x*, *y*, *z*).*I*((*x*, *y*, *z*)) = const.^{ . }((*b*^{2} + *c*^{2})*x*^{2}(*c*^{2} + *a*^{2})*y*^{2}(*a*^{2} + *b*^{2})*z*^{2})

In particular, note that the ``shape'' of the surface
We say that the rotation is *inertial* if the torque vanishes;
i. e.
= 0.
It follows that
= 0 so that the angular
momentum
in the stationary frame is conserved. In
particular, the magnitude
|||| = |||| of the
angular momentum in the body frame is preserved. Moreover, taking the
derivative of the rotational kinetic energy
*T* = . = .
we see (using the symmetry of *I*) that

. = .*I*() = (.*I*() + .*I*()) = = . = .

Thus
. = = 0 and the so kinetic
energy
Poinsot offered a more precise description as follows. Consider the
image *g*^{ . }*E* of the ellipsoid. The vector
= *g*^{ . } lies on this ellipsoid. Moreover, a
vector **w** is tangent to *E* at
if
**w**. = **w**.*I*() = 0. Thus a vector
**x** is tangent to *g*^{ . }*E* at
if
**x**. = 0. Said differently, the tangent plane to
*g*^{ . }*E* at
consists of vectors **y** such that
**y**. = .. We recognise the
latter expression as 2*T*, twice the rotational kinetic energy, which
is a constant of motion. In other words, is also a constant of
motion.

To summarise, the rotational motion *g* is such that the moving
ellipsoid *g*^{ . }*E* remains tangent to a fixed plane and the point of
tangency provides the axis of rotation; such a motion of the ellipsoid
*E* is called rolling without slipping on the plane . To recover
the rotational motion of the original top we note that
(**r**_{i} - **R**) = *g*^{ . }*u*_{i}; so the top is ``affixed'' to the
ellipsoid through its centre of mass with the body frame aligned so
that the eigen-basis of the moment of inertia are the principal axes of
the ellipsoid. *Warning*: the reader should beware that we are * not* describing the inertial rotational motion of a top shaped like
*E*--rather the motion of *E* described gives a nice geometric
description of the motion of the original top.

Kapil Hari Paranjape 2003-08-01