Canonical coordinates and tautological one-form: about a paragraph in the Wikipedia article on the Tautological One-form

Question

As @peek-a-boo wrote in one of his answer, "the word "momentum" gets thrown around more often than candy during Halloween".

I found two definitions of momentum generalized coordinates I want to reconcile one way or another. We go with the usual adapted coordinate charts on a manifold $M$: the ones on $TM$ are noted $(q, \dot{q})$ and the ones on $T^*M$ are noted $(q, p)$.

The first definition is given in the Wikipedia article about Canonical coordinates as a function on the set of vector fields on $M$ to the set of functions on $T^*M$ by $$X\to \mu_X(p)=p(X(\pi(p))$$ where $p\in T^*M$ and $\pi: T^*M\to M$ the projection.

The second definition is a function on the set of functions on $TM$ to the set of one-forms on $TM$ given by

$$f\to\Theta_f=(\text{F}f)^*\alpha$$ where $\alpha$ is the tautological one form defined on $T^*M$, and $\text{F}f$ the Legendre transform of $f$.

We get $\mu_{\frac{\partial}{\partial q}}=p$ and I want to find a way to get $p$ using the second moment map $\Theta_{\dot{q}}$ if possible (here I have edited my question following the correction of @peek-a-boo in his comment). How can I do it?

Answer 1

Actually, after chatting with @peek-a-boo, I discovered that what was bothering me was the Wikipedia glose in the article about the Tautological one-form where it is said 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. It is called "tautological" precisely because, "of course", velocity and momenta are necessarily proportional to one-another. It is a kind of solder form, because it "glues" or "solders" each velocity to a corresponding momentum. The choice of gluing is unique; each momentum vector corresponds to only one velocity vector, by definition. The tautological one-form can be thought of as a device to convert from Lagrangian mechanics to Hamiltonian mechanics.

I was trying to make sense of this paragraph, and finding a formula linking the two. The only formula I found was $$p=\alpha(\frac{\partial}{\partial q})$$ where where $\alpha$ is the tautological one form, and $\frac{\partial}{\partial q}$ is considered as an element of $TT^*M$ using the push forward by the zero section $s$ of the projection $\pi: T^*M\to M$, that is to say we abusively write $\frac{\partial}{\partial q}$ for $s_*(\frac{\partial}{\partial q})$, but it does not link $\dot{q}$ to $p$.

@peek-a-boo convinced me that using only the tautological one form to connect the two should not be possible without using an extra structure like a Legendre transform or a riemannian metric directly connecting the two tangent / cotangent bundles because it would amount to find a natural way to connect the two bundles in general.

Therefore since the evaluation map is the only natural way to connect a vector space and its dual, what Wikipedia meant was the fact that the Tautological one-form was precisely doing the same at the level of the bundle, not less, no more.