Supergravity solution-generating techniques and canonical transformations of σ-models from O(D, D)

Within the framework of the flux formulation of Double Field Theory (DFT) we employ a generalised Scherk-Schwarz ansatz and discuss the classification of the twists that in the presence of the strong constraint give rise to constant generalised fluxes interpreted as gaugings. We analyse the various possibilities of turning on the fluxes Hijk, Fijk, Qijk and Rijk, and the solutions for the twists allowed in each case. While we do not impose the DFT (or equivalently supergravity) equations of motion, our results provide solution-generating techniques in supergravity when applied to a background that does solve the DFT equations. At the same time, our results give rise also to canonical transformations of 2-dimensional σ-models, a fact which is interesting especially because these are integrability-preserving transformations on the worldsheet. Both the solution-generating techniques of supergravity and the canonical transformations of 2-dimensional σ-models arise as maps that leave the generalised fluxes of DFT and their flat derivatives invariant. These maps include the known abelian/non-abelian/Poisson-Lie T-duality transformations, Yang-Baxter deformations, as well as novel generalisations of them.

The Damped Harmonic Oscillator at the Classical Limit of the Snyder-de Sitter Space

Valtancoli in his paper entitled (P. Valtancoli, Canonical transformations and minimal length, J. Math. Phys. 56, 122107 2015) has shown how the deformation of the canonical transformations can be made compatible with the deformed Poisson brackets. Based on this work and through an appropriate canonical transformation, we solve the problem of one dimensional (1D) damped harmonic oscillator at the classical limit of the Snyder-de Sitter (SdS) space. We show that the equations of the motion can be described by trigonometric functions with frequency and period depending on the deformed and the damped parameters. We eventually discuss the influences of these parameters on the motion of the system.

Invariant quadratic operator associated with Linear Canonical Transformations

The main purpose of this work is to identify the general quadratic operator which is invariant under the action of Linear Canonical Transformations (LCTs). LCTs are known in signal processing and optics as the transformations which generalize certain useful integral transforms. In quantum theory, they can be identified as the linear transformations which keep invariant the canonical commutation relations characterizing the coordinates and momenta operators. In this paper, LCTs corresponding to a general pseudo-Euclidian space are considered. Explicit calculations are performed for the monodimensional case to identify the corresponding LCT invariant operator then multidimensional generalizations of the obtained results are deduced. It was noticed that the introduction of a variance-covariance matrix, of coordinate and momenta operators, and a pseudo-orthogonal representation of LCTs facilitate the identification of the invariant quadratic operator. According to the calculations carried out, the LCT invariant operator is a second order polynomial of the coordinates and momenta operators. The coefficients of this polynomial depend on the mean values and the statistical variances-covariances of these coordinates and momenta operators themselves. The eigenstates of the LCT invariant operator are also identified with it and some of the main possible applications of the obtained results are discussed.

The Formulations of Classical Mechanics with Foucault’s Pendulum

Since the pioneering works of Newton (1643–1727), mechanics has been constantly reinventing itself: reformulated in particular by Lagrange (1736–1813) then Hamilton (1805–1865), it now offers powerful conceptual and mathematical tools for the exploration of dynamical systems, essentially via the action-angle variables formulation and more generally through the theory of canonical transformations. We propose to the (graduate) reader an overview of these different formulations through the well-known example of Foucault’s pendulum, a device created by Foucault (1819–1868) and first installed in the Panthéon (Paris, France) in 1851 to display the Earth’s rotation. The apparent simplicity of Foucault’s pendulum is indeed an open door to the most contemporary ramifications of classical mechanics. We stress that adopting the formalism of action-angle variables is not necessary to understand the dynamics of Foucault’s pendulum. The latter is simply taken as well-known and simple dynamical system used to exemplify and illustrate modern concepts that are crucial in order to understand more complicated dynamical systems. The Foucault’s pendulum first installed in 2005 in the collegiate church of Sainte-Waudru (Mons, Belgium) will allow us to numerically estimate the different quantities introduced.

Canonical transformations

This chapter addresses the canonical transformation defined by the given generating function, the rotation in the phase space as a canonical transformation, and themovement of the system as a canonical transformation. The chapter also discusses using the canonical transformations for solving the problems of the anharmonic oscillations and using the canonical transformation to diagonalize the Hamiltonian function of an anisotropic charged harmonic oscillator in a magnetic field. Finally, the chapter addresses the canonical variables which reduce the Hamiltonian function of the harmonic oscillator to zero and using them for consideration of the system of the harmonic oscillators with the weak nonlinear coupling.

Canonical transformations

This chapter addresses the canonical transformation defined by the given generating function, the rotation in the phase space as a canonical transformation, and themovement of the system as a canonical transformation. The chapter also discusses using the canonical transformations for solving the problems of the anharmonic oscillations and using the canonical transformation to diagonalize the Hamiltonian function of an anisotropic charged harmonic oscillator in a magnetic field. Finally, the chapter addresses the canonical variables which reduce the Hamiltonian function of the harmonic oscillator to zero and using them for consideration of the system of the harmonic oscillators with the weak nonlinear coupling.