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In the context of classical mechanics, a classical anomaly is a central extension of a Noether current algebra (e.g. Toppan 01).
In terms of symplectic geometry this means the following (Arnold 78, appendix 5.A). Given a Lie group $G$ acting by symplectomorphisms on a phase space symplectic manifold $(X,\omega)$, this symmetry has a classical anomaly if it does not lift to a genuine $G$-Hamiltonian action, but only to a projective $G$-Hamiltonian action, hence to a Hamiltonian action of a central extension $\widehat G$ of $G$. Specifically, the classical anomaly of the original symplectic $G$-action is the 2-cocycle which classifies this extension.
If the original $G$-action is by flows of Hamiltonian vector fields (just not with explicitly chosen Hamiltonians), then there is a universal classical anomaly given by the pullback of the quantomorphism group extension
This $\widehat G$ is the Heisenberg group of the given $G$-action (See also Fiorenza-Rogers-Schreiber 13 for discussion in higher prequantum geometry).
Notice that on the infinitesimal level of Lie algebras, using that the Lie algebra of the quantomorphism group is the Poisson Lie algebra $\mathfrak{pois}(X,\omega)$, this means that an infinitesimal action of a Lie algebra $\mathfrak{g}$ via Hamiltonian vector fields on $X$ has a classical anomaly if it lifts to an action with consistently chosen Hamiltonians – also called a moment map – only after passing to a central Lie algebra extension
The canonical Galileo group-action on the phase space of non-relativistic classical mechanics has a classical anomaly, given by a group 2-cocycle proportional to the mass of the system, the Galileo 2-cocycle (e.g. Chen-Shaw-Yen 85, Azcárraga-Izquierdo 95 Marle 14).
higher and integrated Kostant-Souriau extensions:
(∞-group extension of ∞-group of bisections of higher Atiyah groupoid for $\mathbb{G}$-principal ∞-connection)
(extension are listed for sufficiently connected $X$)
A textbook discussion of the concept is (without the terminology yet) is in appendix 5.A of
A discussion under the term “classical central charge” is in
For a list of some examples and further pointers to the (historical) literature, see
See also
Discussion in terms of Heisenberg group extensions and generalization to higher symplectic geometry is in
Discussion in the context of formalization of classical field theory in cohesive homotopy theory is in
The example of the Galileo 2-cocycle is discussed for instance in
Chen, Shaw, Yen, An example of a 2-cocycle, pdf
Charles-Michel Marle, The manifold of Motions and the total mass of a mechanical system, 2014 (pdf)
and in the broader context of WZW model terms in
following