Abstract
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.
Highlights
The search of quantum gravity is one the active field of research that has attracted much attention in the last decades
We first studied the properties of the 1D deformed Poisson brackets which are regarded as the classical limit of the generalized Heisenberg commutators
We reviewed the innovative concept of β-canonical transformation introduced by Valtancoli (Valtancoli; 2015) appointed in this work as the deformed canonical transformation
Summary
The search of quantum gravity is one the active field of research that has attracted much attention in the last decades. At the classical limit of the SdS space the solutions of free particle and harmonic oscillator systems had been obtained by substituting the generalized commutations with the deformed Poisson brackets ( Mignemi, 2012). In this paper we are interesting in the study of a one dimensional (1D) damped harmonic oscillator (Kanai, 1948; Caldirola, 1914), in the deformed Poisson brackets. We explicitly solve in section (4) the 1D damped harmonic oscillator in the classical limit of SdS model.
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