Some variational principles associated with ODEs of maximal symmetry. Part 2: The general case

Variational and divergence symmetries are studied in this paper for the whole class of linear and nonlinear equations of maximal symmetry, and the associated first integrals are given in explicit form. All the main results obtained are formulated as theorems or conjectures for equations of a general order. A discussion of the existence of variational symmetries with respect to a different Lagrangian, which turns out to be the most common and most readily available one, is also carried out. This leads to significantly different results when compared with the former case of the transformed Lagrangian. The latter analysis also gives rise to more general results concerning the variational symmetry algebra of any linear or nonlinear equations.

On the Variational Problems without Having Desired Variational Symmetries

We will have an attempt to present a method for constructing variational problems without having a desired one-parameter transformation group as a variational symmetry. For this, we use the notation ofμ-symmetry which was introduced by Giuseppe Gaeta and Paola Morando in 2004. Moreover, our given method enabled us to solve those constructed variational problems usingμ-symmetries.

A NOTE ON THE IMPLEMENTATION OF POINCARÉ SYMMETRY IN NONCOMMUTATIVE FIELD THEORY

We argue that Poincaré symmetry can be implemented in noncommutative field theory (NCFT) if we allow the parameter of noncommutative deformation θμν to change as a two-tensor under the corresponding space–time symmetry. The implementation is consistent with the definition of θμν in terms of space–time coordinates and with the Moyal star product. Inspired by the standard definition of a variational symmetry we found a universal way to correct the implementation of the Poincaré symmetry by a term proportional to the variation of θμν in such a way that the new transformation define a symmetry of the theory. Finally we present as an example the case of NCYM theory and comment about the obstructions to implement generalized space–time symmetries in NCFT-like conformal or diffeomorphism transformations.