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Given a time-independent system of equations of motion that would not allow the existence of a Lagrangian, generalized position-velocity systems of variables are proposed. It is shown that some of these variable systems will allow the existence of a Lagrangian. It is shown that if a Lagrangian exist, then the Hamiltonian and Poisson Brackets associated with that Lagrangian also exist. A way of constructing an explicit generalized position-velocity system of variables, an explicit Lagrangian, and an associated explicit Hamiltonian and Poisson Brackets is shown for the case when general solutions of the original equations of motion are explicitly known.