Abstract
A reasonable generalization of Hamiltonian theory to $3m$-dimensional phase space suggests a geometrical structure giving the proper characteristic vector field. This structure, however, has only a single integral invariant, and implies no sensible generalization of either Poisson-bracket formalism, or Hamilton-Jacobi theory. Associated statistical mechanics and quantization are unlikely. The algebraic source of the difficulty is the lack of understanding of canonical expressions and classes of closed 3-forms.
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