Knowlpedia

Theorem

Canonical moment map for a cotangent-lifted action

The cotangent lift of a Lie group action is Hamiltonian with moment map given by pairing covectors with infinitesimal generators.

Statement

Let GG act on QQ by a , and let it act on T∗QT^*Q by the . With ωcan=−dθ\omega_{\mathrm{can}}=-d\theta and ξQ(q)=ddt∣0exp⁡(tξ)⋅q\xi_Q(q)=\left.\frac{d}{dt}\right|_0\exp(t\xi)\cdot q, define

⟨J(αq),ξ⟩=αq(ξQ(q)).\langle J(\alpha_q),\xi\rangle =\alpha_q\bigl(\xi_Q(q)\bigr).

Then J:T∗Q→g∗J:T^*Q\to\mathfrak g^* is an :

d⟨J,ξ⟩=ιξT∗Qωcan,J(g⋅αq)=Ad⁡g∗J(αq).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 . No connection, metric, or trivialization is involved: the pairing defining JJ 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

θαq(ξT∗Q)=αq(ξQ(q))=⟨J(αq),ξ⟩.\theta_{\alpha_q}(\xi_{T^*Q})=\alpha_q(\xi_Q(q)) =\langle J(\alpha_q),\xi\rangle.

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

ιξT∗Q(−dθ)=d(θ(ξT∗Q))=dJξ,\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 proves coadjoint equivariance. The construction is developed in Marsden and Ratiu, §12.1.

Coordinate example

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

⟨J(q,p),ξ⟩=p⋅ξ.\langle J(q,p),\xi\rangle=p\cdot\xi.

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

Conventions and scope
References
  1. Ralph Abraham and Jerrold E. Marsden, Foundations of Mechanics, 2nd ed., AMS Chelsea, 2008. DOI record. Relevant: §4.2, lifted actions and momentum mappings.
  2. 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.