The cotangent-lift moment map has a plus sign under the library fundamental-field convention

Given: ${\mathrm{A}\mathrm{C}}_{\omega }$, the manifold $Q=\mathbb{R}$ with the translation action of $G=\mathbb{R}$, the lifted action on $T^{\ast }Q$ with canonical coordinates $(q,p)$, and the two candidate component functions $p({\xi }_Q(q))$ and $-p({\xi }_Q(q))$ for $\xi =1$.

${\mathrm{A}\mathrm{C}}_{\omega }$ is countable choice; it is used only through the fundamental-field and cotangent suppliers.

The library fundamental field of the lifted action is ${\xi }_{T^{\ast }Q}={\frac{d}{dt}\mid }_0\widehat{a_{\exp (-t\xi )}}$. Fundamental vector fields for a left action.

For the translation action on $\mathbb{R}$ the fundamental field of $\xi =1$ is the constant vector field ${\xi }_Q=-{\mathrm{\partial }}_q$; the lifted action satisfies $\widehat{t}(q,p)=(q+t,p)$, so its fundamental field is ${\xi }_{T^{\ast }Q}=-{\mathrm{\partial }}_q$ as well. Fundamental vector fields for a left action, The cotangent lift of an action is Hamiltonian with the tautological moment map.

The canonical form is ${\omega }_{\mathrm{c}\mathrm{a}\mathrm{n}}=dq\wedge dp$ in cotangent coordinates, and the component equation of the library convention is $d{\mu }^{\xi }=-{\iota }_{{\xi }_{T^{\ast }Q}}{\omega }_{\mathrm{c}\mathrm{a}\mathrm{n}}$. Tautological one-form on a cotangent bundle, Hamiltonian vector field and Hamiltonian function, The cotangent lift of an action is Hamiltonian with the tautological moment map.

The tautological moment map of The cotangent lift of an action is Hamiltonian with the tautological moment map is $⟨\mu (q,p),\xi ⟩=-p({\xi }_Q(q))$. [given]