Euler's Angles

Michael Fowler

Introduction

So far, our analysis of rotational motion has been of essentially one dimensional, or more precisely one angular parameter, motion: rotating about an axis, rolling, precessing, etc. But this leaves out many interesting phenomena: the wobbling of a slowing down top, nutation, and so on. We need a well-defined set of parameters for the orientation of a rigid body in space to make further progress in analyzing the dynamics.

The standard set is Euler's Angles. What you actually observe as you watch a child's top beginning to wobble as it slows down is the changing direction of the axis -- this direction is given by the first two of Euler's angles: which are just the usual spherical coordinates, the angle  measured from the vertical direction and the azimuthal angle  about that vertical axis. Euler's third angle, specifies the orientation of the top about its own axis, completing the description of the precise positioning of the top. To describe the motion of the wobbling top as we see it, we evidently need to cast the equations of motion in terms of these angles.

Definition

The rotational motion of a rigid body is completely defined by tracking the set of body principal axes , with origin at the center of mass, as they turn relative to a set of fixed-in-space axes  The principal axes can be completely defined relative to the fixed set by three angles: the two angles  fix the direction of  but that leaves the pair free to turn in the plane perpendicular to the angle  fixes their orientation.

To see these angles, start with the fixed axes, draw a circle centered at the origin in the horizontal plane. Now draw a circle of the same size, also centered at the same origin, but in the principal axes  plane. Landau calls the line of intersection of these circles (or discs) the line of nodes. It goes through the common origin, and is a diameter of both circles.

The angle between these two planes, which is also the angle between  (since they're the perpendiculars to the planes) is labeled 

The angle between this line of nodes and the  axis is  It should be clear that  together fix the direction of  then the other axes are fixed by giving  the angle between  and the line of nodes  The direction of measurement of  around  are given by the right-hand or corkscrew rule.

Angular Velocity and Energy in Terms of Euler's Angles

Since the position is uniquely defined by Euler's angles, angular velocity is expressible in terms of these angles and their derivatives.

The strategy here is to find the angular velocity components along the body axes  of  in turn. Once we have the angular velocity components along the principal axes, the kinetic energy is easy.

From the diagram,  is along the line  and therefore in the  plane: notice it is at an angle  with respect to  Its components are therefore 

Now  is about the axis. The principal axis  is at angle  to the axis, so  has component about  and  in the  plane, that latter component along a line perpendicular to , and therefore at angle  from the  axis. Hence 

The angular velocity  is already along a principal axis, 

To summarize, the Euler angle angular velocities (components along the body's principal axes) are:

from which, the angular velocity components along those in-body axes  are:

For a symmetric top, meaning  the rotational kinetic energy is therefore

For this symmetrical case, as Landau points out, we could have taken the  axis momentarily along the line of nodes  giving

Free Motion of a Symmetrical Top

As a warm up in using Euler's angles, we'll redo the free symmetric top covered in the last lecture. With no external torques acting the top will have constant angular momentum 

We'll take  in the fixeddirection. The axis of the top is along 

Taking the  axis along the line of nodes  (in the figure on the previous page) at the instant considered, the constant angular momentum

Remember, this newaxis (see diagram!) is perpendicular to the  axis we've taken  along, so  and  is constant, meaning that the principal axis  describes a cone around the constant angular momentum vector  The rate of precession follows from the constancy of  Writing the absolute magnitude of the angular momentum as   (remember  is in the direction, and is momentarily along ) so the rate of precession  Finally, the component of  along the  axis of symmetry of the top is  so the top's spin along its own axis is 

Motion of a Symmetrical Top around a Fixed Base with Gravity: Nutation

Denoting the distance of the center of mass from the fixed bottom point as  (along the axis) the moment of inertia about a line perpendicular to the axis at the base point is

( being usual center of mass moment.)

The Lagrangian is (being the origin, in direction )

Notice that the coordinates  do not appear explicitly, so there are two constants of motion:

That is, the angular momentum about  is conserved, because the two forces acting on the top, the gravitational pull at the center of mass and the floor reaction at the bottom point, both act along lines intersecting the axis, so never have torque about  The angular momentum about  is conserved because the gravitational torque acts perpendicular to this line.

We have two linear equations in  with coefficients depending on  and the two constants of motion  The solution is straightforward, giving

The (conserved) energy

Using the constants of motion to express  in terms of  and the constants  , then subtracting a -independent term to reduce clutter,  we have

The range of motion in  is given by 

For  goes to infinity at . It has a single minimum between these points. (This isn't completely obvious -- one way to see it is to change variable to , following Goldstein. Multiplying throughout by , and writing  gives a one dimensional particle in a potential problem, and the potential is a cubic in  Of course some roots of  could be in the unphysical region  In any case, there are at most three roots, so since the potential is positive and infinite at it has at most two roots in the physical range.)

From the one-dimensional particle in a potential analogy, it's clear that  oscillates between these two points and  This oscillation is called nutation. Now  could change sign during this oscillation, depending on whether or not the angle  is in the range. Visualizing the path of the top center point on a spherical surface centered at the fixed point, as it goes around it oscillates up and down, but if there is this sign change, it will "loop the loop", going backwards on the top part of the loop.

Steady Precession

Under what conditions will a top, spinning under gravity, precess at a steady rate? The constancy of mean that  are constants.

The  Lagrange equation is

For constant  so, with

Since this is a quadratic equation for the precession rate, there are two solutions in general: on staring at a precessing top, this is a bit surprising! We know that for the top, when it's precessing nicely, the spin rate  far exceeds the precession rate  Assuming  to be of similar size, this means the first term in the quadratic is much smaller than the second. If we just drop the first term, we get the precession rate

Note that this is independent of angle -- the torque varies as  but so does the horizontal component of the angular momentum, which is what's changing.

This is the familiar solution for a child's fast-spinning top precessing slowly. But this is a quadratic equation, there's another possibility: in this large  limit, this other possibility is that  is itself of order , so now in the equation the last term, the gravitational one, is negligible, and

This is just the nutation of a free top! In fact, of course, both of these are approximate solutions, only exact in the limit of infinite spin (where one goes to zero, the other to infinity), and a more precise treatment will give corrections to each arising from the other. Landau indicates the leading order gravitational correction to the free body nutation mode.

Stability of Top Spinning about Vertical Axis

(from Landau) For  Near 

The vertical position is stable against small oscillations provided  or 

Exercise: suppose you set the top vertical, but spinning at less than , the value at which it is just stable. It will fall away, but bounce back, and so on. Show the maximum angle it reaches is given by