Local existence of Dynamically Allowed Brackets and a local existence of a Hamiltonian associated with these Brackets, for a given, possibly time dependent, N-dimensional Equations of Motion that may include Constraints

Dynamically Allowed Brackets are defined for given equations of motion for an N-dimensional mechanical system. The equations of motion may be explicitly time dependent, and may include explicitly time dependent constraints. Local existence of the Dynamically Allowed Brackets is shown. The local existence of a Hamiltonian reproducing the given equations of motion with the use of these Dynamically Allowed Brackets is proven

Spontaneous Dimension Reduction and the Existence of a local Lagrange-Hamilton Formalism for Given n-Dimensional Newtonian Equations of Motion

A partially explicit construction of a Lagrange-Hamiltonian formalism for an arbitrary n -dimensional Newtonian system of equations of motion is given. Additional variables used in the construction are spontaneously reduced by the Dirac’s constraints resulting from degeneracy of the proposed Lagrangian, so that only the variables that appear in the original system of equations remain. A Hamiltonian and dynamical Dirac’s brackets are calculated

An example of Lagrangian for a non-holonomic system

An adjustable two-mass-point Chaplygin Sleigh is used as an example of a non-holonomic system. Newtonian equations of motion based the assumption of zero virtual work done by constraints are calculated. A Lagrangian that reproduces these equations as its unmodified Euler-Lagrange equations is then explicitly given. The Lagrangian uses variables that are present in the Chaplygin Sleigh equations of motion, as well as some additional variables. Some of the Euler-Lagrange equations of that Lagrangian are non-differential. These non-differential equations automatically and completely reduce out all of these additional variables, so that only the variables that appear in the original equations of motion remain in the final dynamics of the system

Spontaneous Dimension Reduction and the Existence of a Local Lagrangian for Given n-Dimensional Newtonian Equations of Motion

A partially explicit construction of a Lagrangian for an n -dimensional Newtonian system of equations of motion is given. Extra variables used in the construction are spontaneously reduced by the constraints resulting from degeneracy of the proposed Lagrangian, so that only the variables that appear in the original system of equations remain. An explicit example of a Lagrangian for a system not satisfying Helmholtz conditions is given