# Recent content by kiuhnm

1. ### A Virtual work in Atwood's machine

OK, thanks. I was confused by the remark "This trivial problem emphasizes that the forces of constraint--here the tension in the rope--appear nowhere in the Lagrangian formulation." Let's say I want to be extremely formal. How would I proceed? The constraint is $x_1+x_2=l$, where $x_i$ is...
2. ### A Virtual work in Atwood's machine

The first chapter in Goldstein's Classical Mechanics ends with 3 examples about how to apply Lagrange's eqs. to simple problems. The second example is about the Atwood's machine. The book says that the tension of the rope can be ignored, but I don't understand why. The two masses can move...
3. ### A Euler's Principal Axis

You're probably thinking about the eigendecomposition of the inertia matrix. This is something unrelated to that. Here's the lecture: It turns out we're assuming that $\boldsymbol\omega$ is parallel to the principal axis $\hat{\boldsymbol e}$ so, by the transport theorem, the inertial...
4. ### A Euler's Principal Axis

When we solve Euler's differential equations for rigid bodies we find the angular acceleration $\dot{\boldsymbol\omega}$ and then the angular velocity $\boldsymbol\omega$. Integrating $\boldsymbol\omega$ is less straightforward, so we start from a representation of the attitude, take its...
5. ### Integral of a differential form

Is my solution correct?
6. ### Integral of a differential form

1. Homework Statement Suppose that a smooth differential $n-1$-form $\omega$ on $\mathbb{R}^n$ is $0$ outside of a ball of radius $R$. Show that $$\int_{\mathbb{R}^n} d\omega = 0.$$ 2. Homework Equations $$\oint_{\partial K} \omega = \int_K d\omega$$ 3. The Attempt at a...
7. ### I Differential forms and bases

Yes, I get it now. See my reply to @fresh_42. Thank you too!
8. ### I Differential forms and bases

@fresh_42 I see it now. Thank you so much for your very detailed post! The book I'm reading does define the pullback of maps on manifolds. I got confused because it doesn't give an explicit formula for the pullback of forms. Instead, it says that the pullback can be extended to differential...
9. ### I Differential forms and bases

It seems to me fresh_42 gave the same exact definition I'm using: $(\phi^* \nu)(p) = \nu(\phi(p)) = (\nu\circ\phi)(p)$. His expression for differential forms is just a property of the $d$ operator, according to my book. In $(f^*(w))(X_p) := w(f_* X_p)$ you do the pullback on $w$ by...