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...
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...
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...
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...
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...
@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...
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...
Speaking of push-forward, one book says that it's also called differential, but another book defines the differential differently: ##df(X_p) = X_p(f)##. Which is it?
I also noticed that your definition of pull-back is somewhat different from mine. Your definition is ##(f^\star w)(X_p) :=...
Basically, you have defined the tangent space and the cotangent space by the push-forward and pull-back induced by a map ##F:M\to N.## One can also note that the matrix associated with ##F_*## is just the Jacobian matrix of ##F## (which is more or less equivalent to your remark about the Chain...
I understand that a covector is just a vector, but can we say that a cotangent space is just a tangent space? They're both vector spaces but are they both tangent to the manifold at a point? To me "tangent" means that it has to do with derivations, whereas cotangent means it's related to...
In the exercises on differential forms I often find expressions such as $$
\omega = 3xz\;dx - 7y^2z\;dy + 2x^2y\;dz
$$ but this is only correct if we're in "flat" space, right?
In general, a differential ##1##-form associates a covector with each point of ##M##. If we use some coordinates...
In my head I was identifying ##x^i## with ##\phi^i## and so the ##x^i## were local coordinates directly on ##U##.
In the meantime I'd like to thank you all for your patience!