Momentum maps and Noether conservation

Let a Lie group G act smoothly by symplectomorphisms on M. For a Lie-algebra element ξ, let ξ_M be the vector field obtained by differentiating exp(tξ)·x. A momentum map J:M→g* satisfies dJ_ξ=ι_{ξ_M}ω, where J_ξ(x)=⟨J(x),ξ⟩. In our convention this means X_{J_ξ}=ξ_M. Equivariance additionally asks J(g·x)=g·J(x), where the coadjoint action is defined by (g·μ)(ξ)=μ(Ad_{g^{-1}}ξ). We will explicitly assume equivariance when using reduction.

Every ξ_M preserves ω, so its contraction is closed, but it need not be exact. Translation on a symplectic two-torus gives the closed nonexact form dp from the earlier lesson, preventing a global real momentum component for that direction. Even when individual generators exist, arranging their group-level compatibility can introduce additional conditions. Adding a constant covector to J preserves its differential defining equations, but preserves equivariance only when the constant is fixed by the coadjoint action. Thus the generator equation and equivariance should not be treated as identical assertions.

If H is invariant under the group, then ξ_MH=0 for every infinitesimal generator. Since ξ_M=X_{J_ξ}, our observable convention gives ξ_MH={H,J_ξ}. Antisymmetry then gives dJ_ξ/dt={J_ξ,H}=0 along the H flow. This is the complete local conservation calculation. It applies where the trajectories exist and to autonomous generators; an explicitly time-dependent symmetry or Hamiltonian requires its corresponding modified statement. A momentum map supplies candidate quantities, while energy invariance decides whether those quantities are conserved by the selected dynamics.

For a cotangent-lifted action on a configuration space Q, the tautological form is invariant. Cartan’s identity and ω=−dθ give the natural component J_ξ(q,p)=p(ξ_Q(q)). Translation along a configuration axis therefore has its conjugate momentum as generator. Planar rotation has angular momentum q₁p₂−q₂p₁. These formulas follow from covector evaluation and do not require a Euclidean kinetic energy. Whether the mechanical Hamiltonian respects the translation or rotation is a separate test on its potential and other terms.

Fixing a conserved momentum value selects an invariant subset of phase space. Quotienting a symmetry that acts inside that subset can then remove redundant motion, but not every level or quotient is smooth. Equivariance identifies which subgroup preserves a nonzero value, while regularity and freeness determine whether the level and orbit space are manifolds. Those conditions enter the next lesson. Simply counting one conserved scalar and deleting one coordinate is not a proof that the remaining dynamics has a nondegenerate symplectic form.

Several momentum components can also fail to Poisson commute for nonabelian symmetries, so symmetry alone does not automatically produce a Liouville-integrable system. Commuting circle symmetries are easier to combine into independent integrals, but their rank can still drop at fixed points. The useful diagnostic is therefore explicit: compute the infinitesimal generator, integrate its contraction if possible, verify equivariance when required, and then check the bracket with the actual H. This sequence separates topological obstructions, algebraic compatibility and dynamical conservation instead of collapsing them into one appeal to symmetry.