# Search results

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...

17. ### I Cordinates on a manifold

Maybe I'm starting to see the problem with my definition: the coordinates would just be a local parametrization of the curve but maybe I'd lack enough structure to do regular calculus over them. The cleanest way is to define $\phi$ from $M$ to $\mathbb{R}^n$ and then the coordinates on...
18. ### I Cordinates on a manifold

It says: --- starts --- Let $(U,\phi)$ be a coordinate chart with $p \in U$ and suppose that $\phi(p)=q$. If $x^1,\ldots,x^n$ are the standard coordinate functions on $\mathbb{R}^n$ then $q$ has coordinates $(x^1(q),\ldots,x^n(q))$. Thus we can write  \phi(p) =...
19. ### I Cordinates on a manifold

Whether something is an abuse or not depends on the definitions you choose. You decided to define the $x_i$ the way you did, which makes my notation an abuse. But the point is this: what do you gain by using your definition? Or in other words, what do I lose by not using it?
20. ### I Cordinates on a manifold

We're not talking about tangent spaces here. The manifold could be non-smooth. We're talking about the charts $(U,\phi)$ where $\phi:U\to\mathbb{R}^n$ is an homeomorphism. The point is whether we need to introduce coordinates $x_1,\ldots,x_n$ explicitly or if $x_i$ is just $\phi_i$...
21. ### I Cordinates on a manifold

An element of $\mathbb{R}^n$ has the form $(x_1,\ldots,x_n)$ by definition, so we must have $\phi(p) = (\phi_1(p),\ldots,\phi_n(p))$. Why should we define a coordinate system for $\mathbb{R}^n$ when $\mathbb{R}^n$ is already a cartesian product? I would agree with you if we were...
22. ### A On Newton's first and second laws

... or your reference frame is not inertial, as in this case.
23. ### I Cordinates on a manifold

Let $M$ be an $n$-dimensional (smooth) manifold and $(U,\phi)$ a chart for it. Then $\phi$ is a function from an open of $M$ to an open of $\mathbb{R}^n$. The book I'm reading claims that coordinates, say, $x^1,\ldots,x^n$ are not really functions from $U$ to $\mathbb{R}$, but...
24. ### A On Newton's first and second laws

That's because of gravity and friction. If you remove all forces, my acceleration won't be 0 in that frame.
25. ### A On Newton's first and second laws

Thank you all for your answers. As for QM I don't think I'll ever learn about it as I'm learning Mechanics to better understand underactuated robotics and locomotion in particular.

Any example?
27. ### A On Newton's first and second laws

According to Scheck's definition, inertial frames are frames with respect to which Newton's first law has analytic form $\ddot{\pmb{r}}(t)=0$.
28. ### A On Newton's first and second laws

I'm reading Scheck's book about Mechanics and it says that Newton's first law is not redundant as it defines what an inertial system is. My problem is that we could say the same about Newton's second law. Indeed, Newton's second law is only valid, in general, for inertial systems, so it also...
29. ### A Definition of Tensor and... Cotensor?

Thank you both!
30. ### A Definition of Tensor and... Cotensor?

Why are there (at least) two definitions of a tensor? For some people a tensor is a product of vectors and covectors, but for others it's a functional. While it's true that the two points of view are equivalent (there's an isomorphism) I find having to switch between them confusing, as a...