For given time-independent Newtonian system of equations of motion and given Poisson Brackets allowed by these equations, it is proven that locally a Lagrangian exists that gives these equations of motion as its regular Euler-Lagrange equations, and gives these Poisson Brackets in a regular process of obtaining the Hamiltonian and its Poisson Brackets. However, this Lagrangian may be using generalized position-velocity variables instead of the original position-velocity variables from the original equations of motion. Also, Darboux Box variables are introduced to prove that all Newtonian dynamical systems are locally isomorphic, meaning that locally there exists a one-to-one function relating the variables describing both systems that preserves equations of motion, Poisson Brackets, the Hamiltonian and, in a sense, also the Lagrangian.
Hebda, Piotr W. and Hebda, Beata, "Darboux-Box Variables and the Existence of a Complete Set of Lagrangians for Given n-Dimensional Newtonian Equations of Motion. The Isomorphism of n-Dimensional Newtonian Dynamical Systems" (2018). Faculty Publications. 7.
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