[Paper Review] Lagrangian and Noncommutativity

Ignacio Cortese, J. Antonio Garcı́a|ArXiv.org|May 7, 2003

Noncommutative and Quantum Gravity Theories14 references3 citations

TL;DR

This paper addresses the inverse problem of constructing a second-order Lagrangian from a first-order system with noncommutative configuration space variables, using auxiliary variables and Darboux transformations. It derives a formal non-local Lagrangian that reproduces the projected second-order dynamics and constructs an equivalent local second-order Lagrangian for the noncommutative harmonic oscillator, demonstrating consistency via symmetry and boundary condition matching.

ABSTRACT

We analyze the relation between the concept of auxiliary variables and the Inverse problem of the calculus of variations to construct a Lagrangian from a given set of equations of motion. The problem of the construction of a consistent second order dynamics from a given first order dynamics is investigated. At the level of equations of motion we find that this reduction process is consistent provided that the mapping of the boundary data be taken properly into account. At the level of the variational principle we analyze the obstructions to construct a second order Lagrangian from a first order one and give an explicit formal non-local Lagrangian that reproduce the second order projected dynamics. Finally we apply our ideas to the so called ``Noncommutative classical dynamics''.

Motivation & Objective

• To resolve the inverse problem of constructing a second-order Lagrangian from a first-order system with noncommutative configuration space coordinates.
• To investigate the consistency of reducing first-order dynamics to second-order dynamics at the level of equations of motion and variational principles.
• To address the obstruction in constructing a local second-order Lagrangian when momenta are not auxiliary variables in noncommutative systems.
• To provide a formal non-local Lagrangian that reproduces the second-order projected dynamics for general noncommutative systems.
• To construct an equivalent local second-order Lagrangian for the noncommutative harmonic oscillator using a Darboux transformation and auxiliary variable technique.

Proposed method

• Applies the inverse problem of the calculus of variations to determine conditions under which a given set of first-order equations of motion arise from a Lagrangian.
• Identifies auxiliary variables in first-order systems and uses their elimination to derive second-order equations in configuration space.
• Uses Darboux transformations to map noncommutative momenta into auxiliary variables, enabling direct construction of a second-order Lagrangian.
• Derives a formal non-local Lagrangian that reproduces the second-order equations of motion, valid when boundary conditions are properly accounted for.
• Applies the method to the noncommutative harmonic oscillator by transforming variables to make momenta auxiliary, then eliminating them to obtain a closed-form second-order Lagrangian.
• Verifies equivalence of the resulting second-order equations to the original system via symmetry and explicit solution matching.

Experimental results

Research questions

• RQ1 Can a consistent second-order Lagrangian be constructed from a first-order noncommutative dynamical system when momenta are not auxiliary variables?
• RQ2 What are the obstructions to constructing a local second-order Lagrangian from a first-order system with noncommutative configuration space coordinates?
• RQ3 How can a formal non-local Lagrangian be derived such that its equations of motion reproduce the projected second-order dynamics?
• RQ4 Can a Darboux transformation be used to render momenta as auxiliary variables in noncommutative systems, enabling second-order Lagrangian construction?
• RQ5 Is the second-order dynamics obtained via reduction equivalent to the original first-order dynamics when proper boundary conditions are applied?

Key findings

• The reduction of first-order dynamics to second-order equations is consistent at the level of equations of motion when boundary conditions are properly accounted for, as shown by matching solutions of the original and reduced systems.
• A formal non-local Lagrangian is constructed that reproduces the second-order projected dynamics, with equations of motion equivalent to the original first-order system.
• For the noncommutative harmonic oscillator, a local second-order Lagrangian is explicitly derived using a Darboux transformation that renders momenta auxiliary.
• The resulting second-order Lagrangian yields equations of motion proportional to the original second-order system, confirming equivalence via symmetry.
• The solution of the noncommutative harmonic oscillator depends on the parameter θ due to the boundary conditions, demonstrating nontrivial noncommutative dependence.
• The Darboux transformation (88) is shown to be a symmetry of the equations of motion, validating the equivalence of the original and transformed second-order systems.