“The Militarization of Knowledge,” the latest special issue of *b*oundary 2, edited by Paul A. Bové,* *is now available.

The growth of the military and its role in producing and controlling knowledge has reordered the entire system of knowledge production and reproduction in advanced societies. The military has had a profound influence on what is thought, on the style of thinking, and the topics developed. This issue addresses the implications of these facts and how one might best think critically about this process.

Articles in this issue address the expanse of militarization and the positive and negative results of state action on knowledge.

The issue concludes with deep reflection on the consequences of such militarization to the exploration of thought problems within the social order and wonders about the results of centering the power over truth so much within the desiring apparatus of the war machine.

Read the introduction, made freely available.

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SOME SPECIFIC STUDIES

Author Horia Orasanu

This gathering offers an opportunity for a small, interdisciplinary group of researchers from around the world to meet colloquium on the theme “Mathematical learning from childhood to adulthood and other studies as doctoral

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CURRICULUM VITAE and themes afferent ; so

ABSTRACT Consider a rigid body for which all of the principal moments of inertia are distinct. Let $I_{z’z’} > I_{y’y’} > I_{x’x’}$. Suppose that the body is rotating freely about one of its principal axes. What happens when the body is slightly disturbed?

Let the body be initially rotating about the $x’$-axis, so that

\begin{displaymath} \mbox{\boldmath$\omega$} = \omega_{x’}\,{\bf e}_{x’}. \end{displaymath} (577)

If we apply a slight perturbation then the angular velocity becomes

\begin{displaymath} \mbox{\boldmath$\omega$} = \omega_{x’}\,{\bf e}_{x’} + \lambda\,{\bf e}_{y’} + \mu\,{\bf e}_{z’}, \end{displaymath} (578)

where $\lambda$ and $\mu$ are both assumed to be small. Euler’s equations (504)-(506) take the form

$\displaystyle I_{x’x’}\,\dot{\omega}_{x’} – (I_{y’y’}-I_{z’z’})\,\lambda\,\mu$ $\textstyle =$ $\displaystyle 0,$ (579)

$\displaystyle I_{y’y’}\,\dot{\lambda} – (I_{z’z’}-I_{x’x’})\,\omega_{x’}\,\mu$ $\textstyle =$ $\displaystyle 0,$ (580)

$\displaystyle I_{z’z’} \,\dot{\mu} – (I_{x’x’}-I_{y’y’})\,\omega_{x’}\,\lambda$ $\textstyle =$ $\displaystyle 0.$ (581)

Since $\lambda\,\mu$ is second-order in small quantities–and, therefore, negligible–the first of the above equations tells us that $\omega_{x’}$ is an approximate constant of the motion. The other two equations can be written

$\displaystyle \dot{\lambda}$ $\textstyle =$ $\displaystyle \left[\frac{(I_{z’z’}-I_{x’x’})\,\omega_{x’}}{I_{y’y’}}\right]\mu,$ (582)

$\displaystyle \dot{\mu}$ $\textstyle =$ $\displaystyle – \left[\frac{(I_{y’y’}-I_{x’x’})\,\omega_{x’}}{I_{z’z’}}\right]\lambda.$ (583)

Differentiating the first equation with respect to time, and then eliminating $\dot{\mu}$, we obtain

\begin{displaymath} \ddot{\lambda} + \left[\frac{(I_{y’y’}-I_{x’x’})\,(I_{z’z’}-… …})}{I_{y’y’}\,I_{z’z’}}\right]\omega_{x’}^{\,2} \,\lambda = 0. \end{displaymath} (584)

It is easily demonstrated that $\mu$ satisfies the same differential equation. Since the term in square brackets in the above equation is positive, the equation takes the form of a simple harmonic equation, and, thus, has the bounded solution:

\begin{displaymath} \lambda = \lambda_0 \,\cos({\mit\Omega}_{x’}\,t’ – \alpha). \end{displaymath} (585)

Here, $\lambda_0$ and $\alpha$ are constants of integration, and

\begin{displaymath} {\mit\Omega}_{x’} = \left[\frac{(I_{y’y’}-I_{x’x’})\,(I_{z’z’}-I_{x’x’})}{I_{y’y’}\,I_{z’z’}}\right]^{1/2}\! \omega_{x’}. \end{displaymath} (586)

Thus, the body oscillates sinusoidally about its initial state with the angular frequency ${\mit\Omega}_{x’}$. It follows that the body is stable to small perturbations when rotating about the $x’$-axis, in the sense that the amplitude of such perturbations does not grow in time.

Suppose that the body is initially rotating about the $z’$-axis, and is subject to a small perturbation. A similar argument to the above allows us to conclude that the body oscillates sinusoidally about its initial state with angular frequency

\begin{displaymath} {\mit\Omega}_{z’} = \left[\frac{(I_{z’z’}-I_{x’x’})\,(I_{z’z’}-I_{y’y’})}{I_{x’x’}\,I_{y’y’}}\right]^{1/2}\! \omega_{z’}. \end{displaymath} (587)

Hence, the body is also stable to small perturbations when rotating about the $z’$-axis.

Suppose, finally, that the body is initially rotating about the $y’$-axis, and is subject to a small perturbation, such that

\begin{displaymath} \mbox{\boldmath$\omega$} = \lambda\,{\bf e}_{x’} + \omega_{y’}\,{\bf e}_{y’} + \mu\,{\bf e}_{z’}. \end{displaymath} (588)

It is easily demonstrated that $\lambda$ satisfies the following differential equation:

\begin{display math} \ddot{\lambda} – \left[\frac{(I_{y’y’}-I_{x’x’})\,(I_{z’z’}-… …})}{I_{x’x’}\,I_{z’z’}}\right]\omega_{y’}^{\,2} \,\lambda = 0. \end{displaymath} (589)

Note that the term in square brackets is positive. Hence, the above equation is not the simple harmonic equation. Indeed its solution takes the form

\begin{displaymath} \lambda = A\,{\rm e}^{\,k\,t’} + B\,{\rm e}^{-k\,t’}. \end{displaymath} (590)

Here, $A$ and $B$ are constants of integration, and

\begin{displaymath} k= \left[\frac{(I_{y’y’}-I_{x’x’})\,(I_{z’z’}-I_{y’y’})}{I_{x’x’}\,I_{z’z’}}\right]^{1/2}\omega_{y’}. \end{displaymath} (591)

In this case, the amplitude of the perturbation grows exponentially in time. Hence, the body is unstable to small perturbations when rotating about the $y’$-axis.

In conclusion, a rigid body with three distinct principal moments of inertia is stable to small perturbations when rotating about the principal axes with the largest and smallest moments, but is unstable when rotating about the axis with the intermediate moment.

Finally, if two of the principal moments are the same then it can be shown that the body is only stable to small perturbations when rotating about the principal axis whose moment is distinct from the other two.

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