Abstract: Motivated by the axiomatic approach to quantum mechanics, we introduce a certain type of non-linear functionals on the space of observables in classical mechanics, called quasi-states. Quasi-states exist on many symplectic manifolds (these are the phase spaces of classical mechanics). It turns out that a slightly more general kind of functionals, called partial quasi-states, exist on yet more symplectic manifolds and are at the intersection of numerous mathematical disciplines - functional analysis, optimization theory, symplectic geometry, Riemannian geometry, and others. We explain how the methods of symplectic geometry are used in order to construct such functionals and present a few of their applications.

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