Talk:Tautological one-form

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The canonical 1-form is always a symplectic potential, but the converse is not true, as a symplectic potential is a local object that exists on a contractible subset of a symplectic manifold such that d theta = omega. Thus they only coincide if the symplectic manifold is contractible and only up to the addition of an exact 1-form. 93.222.57.53 (talk) 12:46, 22 March 2013 (UTC)

I added the above remark to the article. 67.198.37.16 (talk) 04:44, 3 May 2019 (UTC)

Assessment comment

The comment(s) below were originally left at Talk:Tautological one-form/Comments, and are posted here for posterity. Following several discussions in past years, these subpages are now deprecated. The comments may be irrelevant or outdated; if so, please feel free to remove this section.

The canonical one form cannot be expressed in coordinates by the formula appearing in the article (and in most references) because a linear combination of differentials of scalar functions on a manifold is a one-form on the manifold and NOT on a one-form the tangent bundle! Please read my observation and answer me.

Last edited at 17:37, 6 August 2009 (UTC). Substituted at 02:38, 5 May 2016 (UTC)

Deleting the "physical interpretation" section

The "Physical interperation" section seems very misleading to me, because the claim that "the tautological one-form is a device that converts velocities into momenta" seems to imply that the tautological one-form establishes a canonical isomorphism between $TQ$ and $T^{*}Q$ that allows one to convert between vectors and one-forms on Q. But this is not true; instead, the tautological one-form establishes a canonical isomorphism between $TM$ and $T^{*}M$, where M is the cotangent bundle $T^{*}Q$ of Q, rather than Q itself. The tautological one-form by itself is not enough to convert between velocities (vectors) and momenta (one-forms) on the configuration space; for that, one needs to define either a Lagrangian function on $TQ$ or a Hamiltonian function on $T^{*}Q$. I've deleted this entire section, because it seems fundamentally misleading. Ted.tem.parker (talk) 19:57, 18 February 2023 (UTC)