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Recently we found that canonical gauge-natural superpotentials are obtained as global sections of the {em reduced} $(n-2)$-degree and $(2s-1)$-order quotient sheaf on the fibered manifold $bY_{zet} times_{bX} mathfrak{K}$, where $mathfrak{K}$ is an appropriate subbundle of the vector bundle of (prolongations of) infinitesimal right-invariant automorphisms $bar{Xi}$. In this paper, we provide an alternative proof of the fact that the naturality property $cL_{j_{s}bar{Xi}_{H}}omega (lambda, mathfrak{K})=0$ holds true for the {em new} Lagrangian $omega (lambda, mathfrak{K})$ obtained contracting the Euler--Lagrange form of the original Lagrangian with $bar{Xi}_{V}in mathfrak{K}$. We use as fundamental tools an invariant decomposition formula of vertical morphisms due to Kolav{r} and the theory of iterated Lie derivatives of sections of fibered bundles. As a consequence, we recover the existence of a canonical generalized energy--momentum conserved tensor density associated with $omega (lambda, mathfrak{K})$.
We consider systems of local variational problems defining non vanishing cohomolgy classes. In particular, we prove that the conserved current associated with a generalized symmetry, assumed to be also a symmetry of the variation of the corresponding local inverse problem, is variationally equivalent to the variation of the strong Noether current for the corresponding local system of Lagrangians. This current is conserved and a sufficient condition will be identified in order such a current be global.
We extend the concept of implementability of semigroups of evolution operators associated with dynamical systems to quantum case. We show that such an extension can be properly formulated in terms of Jordan morphisms and isometries on non-commutative $L^p$ spaces. We focus our attention on a non-commutative analog of the Banach-Lamperti theorem.
The classical quantization of a Lienard-type nonlinear oscillator is achieved by a quantization scheme (M.C. Nucci. Theor. Math. Phys., 168:997--1004, 2011) that preserves the Noether point symmetries of the underlying Lagrangian in order to construct the Schrodinger equation. This method straightforwardly yields the correct Schrodinger equation in the momentum space (V. Chithiika Ruby, M. Senthilvelan, and M. Lakshmanan. J. Phys. A: Math. Gen., 45:382002, 2012), and sheds light into the apparently remarkable connection with the linear harmonic oscillator.
We will read, through the Emmy Noether paper and the two concepts of `proper and `improper conservation laws, the problem, posed by Hilbert, of the nature of the law of conservation of energy in the theory of General Relativity. Epistemological issues involved with the two kind of conservation laws will be enucleate.
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