Classify non-Abelian kinematic symplectic structures

Determine whether imposing the Legendre demand for every energy on phase spaces whose twists take values in an adjoint bundle yields a classification of kinematic structures analogous to the classification by closed real two-forms on cotangent bundles.

Background

The manuscript classifies symplectic structures on an ordinary cotangent bundle subject to the requirement that, for every Hamiltonian, the projected Hamiltonian velocity equal the fibre derivative of the Hamiltonian. The result is the canonical symplectic form plus the pullback of a closed real-valued two-form.

The authors note that Feynman’s bracket argument has been extended to non-Abelian gauge theories and that principal-bundle reduction provides a natural setting for adjoint-bundle-valued twists. They explicitly leave unresolved whether the same all-energies Legendre condition produces an analogous classification in that non-Abelian setting.