Canonical moment map for a cotangent-lifted action

Statement

Let $G$ act on $Q$ by a smooth Lie group action, and let it act on $T^*Q$ by the cotangent lift. With $\omega_{\mathrm{can}}=-d\theta$ and $\xi_Q(q)=\left.\frac{d}{dt}\right|_0\exp(t\xi)\cdot q$ , define

\langle J(\alpha_q),\xi\rangle =\alpha_q\bigl(\xi_Q(q)\bigr).

Then $J:T^*Q\to\mathfrak g^*$ is an equivariant moment map:

d\langle J,\xi\rangle=\iota_{\xi_{T^*Q}}\omega_{\mathrm{can}}, \qquad J(g\cdot\alpha_q)=\operatorname{Ad}_g^*J(\alpha_q).

Consequently the cotangent-lifted action is a Hamiltonian Lie group action. No connection, metric, or trivialization is involved: the pairing defining $J$ is determined entirely by the original action and the tautological cotangent geometry. The construction is canonical.

Why the formula is canonical

The tautological one-form satisfies

\theta_{\alpha_q}(\xi_{T^*Q})=\alpha_q(\xi_Q(q)) =\langle J(\alpha_q),\xi\rangle.

Every cotangent lift preserves $\theta$ . Cartan's formula therefore gives

\iota_{\xi_{T^*Q}}(-d\theta) =d\bigl(\theta(\xi_{T^*Q})\bigr)=dJ^\xi,

which proves the moment-map identity without coordinates. Naturality of the infinitesimal generators under the group action proves coadjoint equivariance.

Coordinate example

For translations of $Q=\mathbb R^n$ by $G=\mathbb R^n$ , identify $T^*\mathbb R^n$ with pairs $(q,p)$ . The infinitesimal generator of $\xi\in\mathbb R^n$ is the constant vector $\xi$ , so

\langle J(q,p),\xi\rangle=p\cdot\xi.

Thus $J(q,p)=p$ : linear momentum is the moment map for translations.

Conventions and scope

References

Ralph Abraham and Jerrold E. Marsden, Foundations of Mechanics, 2nd ed., AMS Chelsea, 2008. DOI record. Relevant: §4.2, lifted actions and momentum mappings.
Jerrold E. Marsden and Tudor S. Ratiu, Introduction to Mechanics and Symmetry, 2nd ed., Texts in Applied Mathematics 17, Springer, 1999. DOI record. Relevant: §12.1, momentum maps for cotangent-lifted actions.