We consider Lagrangians of classical mechanics which are weakly invariant with respect to a given symmetry group, i.e. whose left hand side of the equations of motion are invariant with respect to a given group. Obstructions to corresponding Lagrangian to be invariant with respect to this symmetry group arise because of cohomology of Lie algebra of symmetries and the configuration space which may have non-trivial interplay. We shall discuss physical consequence of this phenomenon which are usually revealed on quantum mechanical level.Examples include non-relativistic particle in a constant magnetic field and an example of free particle on the punctured sphere which leads to Dirac monopole. A beautiful example is a case of non-relativistic free particle considered as a limit of relativistic one, when Bargmann cocycle ceases to be a boundary.