This page will briefly describe how to use this tool to propagate the accelerations for a symmetric top with a fixed point in gravity. Rotations of rigid bodies such as the symmetric top can be uniquely described by the Euler angles φ, θ and ψ. A simple way picture the coordinate system for our top is to consider the image below:
If we fix the top to spin at a constant rate (d2ψ/d2t = 0), our problem reduces to only two dimensions, and we would like to follow the time evolution of just φ and θ. Like the double pendulum example, there are two angular coordinates. φ will be our x coordinate and θ will be our y coordinate. If you have not already done so, launch the Newton Tool.
Acceleration Equations
Using the drop-down menu, select 2 dimensions in the Acceleration(s) panel. Two fields for the accelerations will appear. Notice the variables used are x, y, vx, vy, ax, and ay.
The acceleration equations (see, for example, Theoretical Mechanics of Particles and Continua by Fetter and Walecka) for this example are expressed in φ and θ.
![\begin{equation}
\ddot{\phi} = [ \frac{2 \cos \theta}{I_{1} \sin^{3} \theta} ( p_{\psi} \cos \theta - p_{\phi} ) + \frac{p_{\psi}}{I_{1} \sin \theta} ]
\dot{\theta} \end{equation}
\begin{equation}
\ddot{\theta} = \frac{\cos \theta}{I_{1}^{2} \sin^{3} \theta}
( p_{\phi}^{2} - 2 p_{\phi}p_{\psi} \cos \theta + p_{\psi}^{2}) -
\frac{p_{\phi}p_{\psi}}{I_{1}^{2} \sin
\theta} + \frac{Mgl}{I_{1}}\sin \theta
\end{equation}](images/top_acceleration.jpg "\begin{equation}
\ddot{\phi} = [ \frac{2 \cos \theta}{I_{1} \sin^{3} \theta} ( p_{\psi}
\cos \theta - p_{\phi} ) + \frac{p_{\psi}}{I_{1} \sin \theta} ]
\dot{\theta} \end{equation}
\begin{equation}
\ddot{\theta} = \frac{\cos \theta}{I_{1}^{2} \sin^{3} \theta}
( p_{\phi}^{2} - 2 p_{\phi}p_{\psi} \cos \theta + p_{\psi}^{2}) -
\frac{p_{\phi}p_{\psi}}{I_{1}^{2} \sin
\theta} + \frac{Mgl}{I_{1}}\sin \theta
\end{equation}")
where pφ and pψ are constants of the motion (i.e., independent of time), I1 is the moment of inertia about the axis of symmetry, M is the mass of the top, l is the length and g is the acceleration due to gravity. In order for the system to have "top-like" behavior, small amplitude arguments suggest that θ should deviate slightly from its initial value, θo. Therefore, as a function of time, we expect θ = θo + η(t), where η(t) is a small perturbation. It can be shown that to a good degree of accuracy that the stable top behaves like a harmonic oscillator, namely, d2η/d2t = - Ω2η. The top has oscillatory (stable) solutions if Ω2 > 0. The functional form of Ω2 is given in terms of the initial angle θo:
We can meet the above condition on Ω2 (Ω2 > 0) with careful selection of the constants above. One such selection is given by the following ratio choices:
With these assumptions, the accelerations reduce to the ones below. Carefully cut and paste these into the fields in the Acceleration Panel then click next.
x acceleration
(2*COS(y)*(2*COS(y)-1)/SIN(y)^3 + 2/SIN(y))*vy
y acceleration
COS(y)*(1-4*COS(y)+4)/SIN(y)^3 - 2/SIN(y) + SIN(y)
It's very important to notice that x and y are no longer the Cartesian coordinates x and y, but rather are two variables that represent Euler rotation angles of the symmetric top. These coordinates are consequently measured in radians.
Initial Conditions
For simplicity, we would like to choose an initial θo that makes the initial time rate of change of φ equal to zero. To do this, we note that
Setting the above to 0 and solving for θ gives θo=arccos(0.5)=1.047197. We will also set φo=0 as well. On the initial conditions panel set the initial x to be 0. and the initial y to be 1.047197. These values are in radians. Also set the initial velocities to zero.
Plots
Below is a plot of θ vs. φ. Notice that φ increases roughly linearly with time and that the top is precessing with angle θ. The precession is not at all constant in time and the top "nutates" or wobbles significantly. To understand the dynamics, one can imagine the "top" of the top sketching out a pattern that looks very much like this graph on an imaginary sphere as it precesses (and nutates) about its incident angle θo.
y = θ vs. x = φ
Let's see what happens for a small deviation from θo. If we let θo = θo - 0.1 = 0.947197 we obtain dφo/dt = -0.254815. Go back to the Initial Conditions panel and enter the changes above. The resulting graph of θ vs. φ shows that the top now has a more pronounced "wobble". Notice also that the value of y (that is, θ) exceeds π/2=1.57 which means that the top dips below the horizontal plane periodically. This is allowed since the point is fixed in gravity and θ has not been constrained to be less than π/2. (This is evident in the movie of the second top below.)
y = θ vs. x = φ
Movies
One may use the data generated from the Newton Tool and IDL (or other software) to generate movies of the motion of the first top or of the motion of the second top. In these movies, the thick line represents the physical top while the thin line is the projection onto the xy (horizontal) plane.
Challenge
In the examples above, plot φ as a function of time to show that it is roughly linear with time. Try to increase θo slightly to see what happens to the amplitude of nutation. Can you find a selection of initial conditions that would cause a constant precession? Also, can you change the ratio of the constants in the equations above to produce an unstable system? What happens?
