Courant bracket twisted both by a 2-form B and by a bi-vector $$\theta $$

We obtain the Courant bracket twisted simultaneously by a 2-form B and a bi-vector $$\theta $$ θ by calculating the Poisson bracket algebra of the symmetry generator in the basis obtained acting with the relevant twisting matrix. It is the extension of the Courant bracket that contains well known Schouten–Nijenhuis and Koszul bracket, as well as some new star brackets. We give interpretation to the star brackets as projections on isotropic subspaces.

Why scalar–tensor equivalent theories are not physically equivalent?

Whether Jordan’s and Einstein’s frame descriptions of [Formula: see text] theory of gravity are physically equivalent, is a long standing debate. However, practically none questioned on true mathematical equivalence, since classical field equations may be translated from one frame to the other following a transformation relation. Here, we show that, neither Noether symmetries, Noether equations, nor may quantum equations be translated from one to the other. The reason being, — conformal transformation results in a completely different system, with a different Lagrangian. Field equations match only due to the presence of diffeomorphic invariance. Unless a symmetry generator is found which involves Hamiltonian constraint equation, mathematical equivalence between the two frames appears to be vulnerable. In any case, in quantum domain, mathematical and therefore physical equivalence cannot be established.

Higher dimensional systems of differential equations obtainable by iterative use of complex methods

A procedure had been developed to solve systems of two ordinary and partial differential equations (ODEs and PDEs) that could be obtained from scalar complex ODEs by splitting into their real and imaginary parts. The procedure was extended to four dimensional systems obtainable by splitting complex systems of two ODEs into their real and imaginary parts. As it stood, this procedure could be extended to any even dimension but not to odd dimensional systems. In this paper, the complex splitting is used iteratively to obtain three and four dimensional systems of ODEs and four dimensional systems of PDEs for four functions of two and four variables that correspond to a scalar base equation. We also provide characterization criteria for such systems to correspond to the base equation and a clear procedure to construct the base equation. The new systems of four ODEs are distinct from the class obtained by the single split of a two dimensional system. The previous complex methods split each infinitesimal symmetry generator into a pair of operators such that the entire set of operators do not form a Lie algebra. The iterative procedure sheds some light on the emergence of these "Lie-like" operators. In this procedure the higher dimensional system may not have any or the required symmetry for being directly solvable by symmetry and other methods although the base equation can have sufficient symmetry properties. Illustrative examples are provided.

Potential vorticity in magnetohydrodynamics

A version of Noether's second theorem using Lagrange multipliers is used to investigate fluid relabelling symmetries conservation laws in magnetohydrodynamics (MHD). We obtain a new generalized potential vorticity type conservation equation for MHD which takes into account entropy gradients and the J × B force on the plasma due to the current J and magnetic induction B. This new conservation law for MHD is derived by using Noether's second theorem in conjunction with a class of fluid relabelling symmetries in which the symmetry generator for the Lagrange label transformations is non-parallel to the magnetic field induction in Lagrange label space. This is associated with an Abelian Lie pseudo algebra and a foliated phase space in Lagrange label space. It contains as a special case Ertel's theorem in ideal fluid mechanics. An independent derivation shows that the new conservation law is also valid for more general physical situations.

Symmetries, Associated First Integrals, and Double Reduction of Difference Equations

We determine the symmetry generators of some ordinary difference equations and proceeded to find the first integral and reduce the order of the difference equations. We show that, in some cases, the symmetry generator and first integral areassociatedvia the “invariance condition.” That is, the first integral may be invariant under the symmetry of the original difference equation. When this condition is satisfied, we may proceed to double reduction of the difference equation.

Finding symmetries by incorporating initial conditions as side conditions

It is generally believed that in order to solve initial value problems using Lie symmetry methods, the initial condition needs to be left invariant by the infinitesimal symmetry generator that admits the invariant solution. This is not so. In this paper we incorporate the imposed initial value as a side condition to find ‘infinitesimals’ from which solutions satisfying the initial value can be recovered, along with the corresponding symmetry generator.

EXPLICITLY SPACE- AND TIME-DEPENDENT D'ALEMBERT EQUATIONS WITH SYMMETRIES

The general d'Alembert equation □u + f (x0, x1, u) = 0 is considered, where □ is the two-dimensional d'Alembert operator. We classify the equation for functions f by which it admits several Lie symmetry algebras, which include the Lorentz symmetry generator. The corresponding symmetry reductions are listed.