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In the wiki, it is written that :

the tautological one-form assigns a numerical value to the momentum $p$ for each velocity $\dot {q}$, and more: it does so such that they point "in the same direction", and linearly, such that the magnitudes grow in proportion

But if I take the definition, the 1-form on $T^*Q$ is such that for $(x,\alpha) \in T^*Q$, we have :

$\theta_{(x,\alpha)}(v) = \alpha(d\pi v)$ which assigns to a tangent vector in T*Q (which I don't even know how to interpret), a number [and not a numerical value of the moment to a velocity (which is in the tangent bundle $TQ$) no?]. I don't understand how this work.

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    $\begingroup$ Did you mean "assigns a number to a tangent vector in $T^*Q$"? One could say that $\theta$ assigns "a numerical value" to the pair $p$ (a momentum covector) and velocity vector $d\pi v$ (this projected vector is in $TQ$) bilinearly, and is tautological because "of course" covectors pair with vectors (momenta with velocities). The wiki writeup is very confusing indeed. $\endgroup$ – Max Jun 12 at 9:08
  • $\begingroup$ Oh, it is more clear indeed. Also, I understand that $d\pi v$ is a velocity. But what is the element $v$? How do you think of it? $\endgroup$ – roi_saumon Jun 12 at 11:43
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    $\begingroup$ Here $v$ is a tangent to $T^*M$, and element of $TT^*M$. I do not think about it too much :) More seriously, knowing that it can be projected (to $d\pi v$) and otherwise treating it as a tangent vector to a symplectic manifold (which just happens to be $T^*M$, but whatever) seems to be enough for the most part. $\endgroup$ – Max Jun 12 at 13:00
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It is just playing with words but I'll give it a try and interpret wikipedia word by word:

Consider $E:= T^*Q$ purely as a vector bundle, i.e., remember the data of the projection $\pi: E\rightarrow Q$, and forget the relation to $Q$. Let's call the elements $p\in E$ momenta. We will work with $p\in E_q$ for a point $q\in Q$. A horizontal $1$-form $\theta\in \Omega^1(E)$ associates a numerical value $Z_p(\dot{q})$ to the momentum $p$ for each velocity $\dot{q}\in T_qQ$ by defining $Z_p(\dot{q}) := \theta_p(v)$, where $v\in T_p E$ is any lift of $\dot{q}$ under $d\pi: TE\rightarrow TQ$. This defines a linear functional $Z_p : T_q Q \rightarrow \mathbb{R}$. This functional might be completely unrelated to the linear functional $p: T_qQ\rightarrow{R}$. However, in the case when $\theta$ is the tautological $1$-form, the covectors $Z_p$ and $p$ "point in the same direction", where the direction of a covector might be defined as the gradient with respect to some metric. Moreover, the assignment $p\mapsto Z_p$ is linear, such that the magnitudes grow in proportion. In fact, for the canonical $1$-form, we have $Z_p = p$.

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