Mielke–Baekler gravity consists in the usual Einstein–Hilbert action with a cosmological term and rotational and translational Chern–Simons terms with arbitrary couplings. For a particular choice of these couplings, we can obtain Einstein–Hilbert action, its teleparallel equivalent and the exotic Witten’s gravity. In this work, we use the Chern–Simons formalism to generalize the three dimensional Mielke–Baekler gravity theory in order to introduce spin-3 fields. We study its asymptotic symmetries, black hole solution and also analyse its canonical structure at its singular point.

In this paper, it is proposed a quantization procedure for the one-dimensional harmonic oscillator with time-dependent frequency, time-dependent driven force, and time-dependent dissipative term. The method is based on the construction of dynamical invariants previously proposed by the authors, in which fundamental importance is given to the linear invariants of the oscillator. Keywords Dynamical invariants • Dissipative systems • Quantum damped oscillator • Time-dependent systems 1 Introduction Dynamical invariants were first used by Ermakov to show the connection between solutions of some special differential equations, referred to as Steen-Ermakov equations [1]. These equations were first studied by Steen [2] and then rediscovered by other authors [3, 4]. After that, Ray and Reid used the Ermakov approach to construct invariants for a much broader class of differential equations [5-7]. This purely mathematical interest was the start point of significant developments in classical and quantum dynamics. The importance of the dynamical invariants of a system should not be underrated. In classical mechanics, the dynamical constants of motion are the variables that allow complete M. C. Bertin

In this paper, it is proposed a quantization procedure for the one-dimensional harmonic oscillator with time-dependent frequency, time-dependent driven force, and time-dependent dissipative term. The method is based on the construction of dynamical invariants previously proposed by the authors, in which fundamental importance is given to the linear invariants of the oscillator.

Mielke-Baekler gravity consists in the usual Einstein-Hilbert action with a cosmological term and rotational and translational Chern-Simons terms with arbitrary couplings. For a particular choice of these couplings, we can obtain Einstein-Hilbert action, its teleparallel equivalent, and the exotic Witten's gravity.In this work, we use the Chern-Simons formalism to generalize the three dimensional Mielke-Baekler gravity theory in order to introduce spin-3 fields. We study its asymptotic symmetries, black hole solution, and also analyse its canonical structure at its singular point.