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Register a new userVariationally equivalent problems and variations of Noether currents
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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.
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We consider Sturm-Liouville problems on the finite interval. We show that spectral data for the case of Dirichlet boundary conditions are equivalent to spectral data for Neumann boundary conditions. In particular, the solution of the inverse problem for the first one is equivalent to the solution of the inverse problem for the second one. Moreover, we discuss similar results for other Sturm-Liouville problems, including a periodic case.
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 $omega (lambda, mathfrak{K})$.
In this communication, we show that both infinite-dimension
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.
A theorem is proved which determines the first integrals of the form $I=K_{ab}(t,q)dot{q}^{a}dot{q}^{b}+K_{a}(t,q)dot{q}^{a}+K(t,q)$ of autonomous holonomic systems using only the collineations of the kinetic metric which is defined by the kinetic energy or the Lagrangian of the system. It is shown how these first integrals can be associated via the inverse Noether theorem to a gauged weak Noether symmetry which admits the given first integral as a Noether integral. It is shown also that the associated Noether symmetry is possible to satisfy the conditions for a Hojman or a form-invariance symmetry therefore the so-called non-Noetherian first integrals are gauged weak Noether integrals. The application of the theorem requires a certain algorithm due to the complexity of the special conditions involved. We demonstrate this algorithm by a number of solved examples. We choose examples from published works in order to show that our approach produces new first integrals not found before with the standard methods.
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