Tautological one form assigns a numerical value to the momentum $p$ for each velocity?

Question

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.

Answer 1

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 $Z_p = p$.