I have a doubt, I hope you can help me. In the case of a spinning top precessing around the $y$-axis, there's a torque $\vec \tau$ which comes from the weight of the toy. This torque is perpendicular to the angular momentum $\vec L$. We have the relation: $$ \frac{d\vec L}{dt} = \vec \tau$$ I understand angular momentum changes its direction, but why not the module?
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This is formally equivalent to the fact, that a magnetic field does not change the magnitude of a velocity. The answer is, because the torque is perpendicular to the angular momentum (as you point out, cast in math $\vec L \cdot \vec \tau = 0$). Then it is a one liner: $$\partial_t \left|\vec L\right| = \frac{2 \vec L \cdot \dot{\vec L}}{2 \sqrt{\vec L \cdot \vec L}} = \frac 1 {\left|\vec L\right|}\vec L \cdot \vec \tau = 0.$$
answered Jul 13, 2015 at 22:44
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$\begingroup$ Just one question: the formula you wrote, i haven't reached that level of math. Is it partial derivative of angular momentum? If so, what should i read to understad the rest of the equation. Thank you $\endgroup$– OmarCommented Jul 14, 2015 at 0:29
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$\begingroup$ You can also understand it geometrically. And yes, it is a partial derivative of the modulus of the angular momentum with respect to time. I guess any maths in physics book will do. $\endgroup$ Commented Jul 15, 2015 at 19:09