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

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.

• 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. – Max Jun 12 at 9:08
• 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? – roi_saumon Jun 12 at 11:43
• 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. – Max Jun 12 at 13:00

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 $$Z_p = p$$.