Department of Mathematics

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.

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

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.

Spontaneous Dimension Reduction and the Existence of a Local Lagrangian 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.

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

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.

An example of Lagrangian for a non-holonomic system

Given time-independent system of equations of motion and given a local, non-trivial constant of motion for these equations, it is shown that there exists a locally defined system of dynamical brackets, such that this given constant of motion, used together with these dynamical brackets, becomes a local Hamiltonian for the given system of equations of motion.

For a Given, Time Independent, System of Equations of Motion, Any Non-Trivial Constant of Motion is Locally a Hamiltonian

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.

Generalized Position-Velocity Variables and the Existence of a Lagrange-Hamilton Formalism for Given n-Dimensional Newtonian Equations of Motion

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.

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

Assuming two given time-independent Newtonian systems of the same dimensions, each of the two systems including its own given set of time-independent generalized Poisson Brackets and a time-independent Hamiltonian, there always locally exists a one-to-one function from variables of one system to the variables of the other system, such that it transforms equations of motion of the first system into equations of motion of the second system, the Poisson Brackets of the first system into the Poisson Brackets of the second system, and the Hamiltonian of the first system into the Hamiltonian of the second system.
One interpretation of the above is that all mechanical systems of the same dimension are locally identical, and the variety of systems we observe in the real world is due only to the fact that we use different systems of variables when making our observations.

The Fundamental Theorem of Classical Mechanics

Starting with an unconstrained mechanical system that is governed by an initial unconstrained Lagrangian, subsequently modified by nonholonomic, linear in velocities, constraints, an explicit construction for a Lagrangian that will produce mechanically correct equations of motion for that constrained nonholonomic system is given. Obtaining a Hamiltonian from that Lagrangian is briefly discussed.