The 3+1 formalism in teleparallel and symmetric teleparallel gravity

Teleparallel and symmetric teleparallel gravity offer platforms in which gravity can be formulated in interesting geometric approaches, respectively given by torsion and nonmetricity. In this vein, general relativity can be expressed in three dynamically equivalent ways which may offer insights into the different properties of these decompositions such as their Hamiltonian structure, the efficiency of numerical analyses, as well as the classification of gravitational field degrees of freedom. In this work, we take a $$3+1$$ 3 + 1 decomposition of the teleparallel equivalent of general relativity and the symmetric teleparallel equivalent of general relativity which are both dynamically equivalent to curvature based general relativity. By splitting the spacetime metric and corresponding tetrad into their spatial and temporal parts as well as through finding the Gauss-like equations, it is possible to set up a general foundation for the different formulations of gravity. Based on these results, general 3-tetrad and 3-metric evolution equations are derived. Finally through the choice of the two respective connections, the metric $$3+1$$ 3 + 1 formulation for general relativity is recovered as well as the tetrad $$3+1$$ 3 + 1 formulation of the teleparallel equivalent of general relativity and the metric $$3+1$$ 3 + 1 formulation of symmetric teleparallel equivalent of general relativity. The approach is capable, in principle, of resolving common features of the various formulations of general relativity at a fundamental level and pointing out characteristics that extensions and alternatives to the various formulations can present.

A generalized integrable hierarchy related to the relativistic Toda lattice: Hamiltonian structure, Darboux transformation, soliton solution and conservation law

Starting from a more generalized discrete [Formula: see text] matrix spectral problem and using the Tu scheme, some integrable lattice hierarchies (ILHs) are presented which include the well-known relativistic Toda lattice hierarchy and some new three-field ILHs. Taking one of the hierarchies as example, the corresponding Hamiltonian structure is constructed and the Liouville integrability is illustrated. For the first nontrivial lattice equation in the hierarchy, the [Formula: see text]-fold Darboux transformation (DT) of the system is established basing on its Lax pair. By using the obtained DT, we generate the discrete [Formula: see text]-soliton solutions in determinant form and plot their figures with proper parameters, from which we get some interesting soliton structures such as kink and anti-bell-shaped two-soliton, kink and anti-kink-shaped two-soliton and so on. These soliton solutions are much stable during the propagation, the solitary waves pass through without change of shapes, amplitudes, wave-lengths and directions. Finally, we derive infinitely many conservation laws of the system and give the corresponding conserved density and associated flux formulaically.

Quantization ambiguities and the robustness of effective descriptions of primordial perturbations in hybrid Loop Quantum Cosmology

We study the imprint that certain quantization ambiguities may leave in effective regimes of the hybrid loop quantum description of cosmological perturbations. More specifically, in the case of scalar perturbations we investigate how to reconstruct the Mukhanov-Sasaki field in the effective regime of Loop Quantum Cosmology, taking as starting point for the quantization a canonical formulation in terms of other perturbative gauge invariants that possess different dynamics. This formulation of the quantum theory, in terms of variables other than the Mukhanov-Sasaki ones, is crucial to arrive at a quantum Hamiltonian with a good behavior, elluding the problems with ill defined Hamiltonian operators typical of quantum field theories. In the reconstruction of the Mukhanov-Sasaki field, we ask that the effective Mukhanov-Sasaki equations adopt a similar form and display the same Hamiltonian structure as the classical ones, a property that has been widely assumed in Loop Quantum Cosmology studies over the last decade. This condition actually restricts the freedom inherent to certain quantization ambiguities. Once these ambiguities are removed, the reconstruction of the Mukhanov-Sasaki field naturally identifies a set of positive-frequency solutions to the effective equations, and hence a choice of initial conditions for the perturbations. Our analysis constitutes an important and necessary test of the robustness of standard effective descriptions in Loop Quantum Cosmology, along with their observational predictions on the primordial power spectrum, taking into account that they should be the consequence of a more fundamental quantum theory with a well-defined Hamiltonian, in the spirit of Dirac’s long-standing ideas.

Dynamics of Dissipative Systems with Hamiltonian Structures

In this work, we study a class of dissipative, nonsmooth [Formula: see text] degree-of-freedom dynamical systems. As the dissipation is assumed to be proportional to the momentum, the dynamics in such systems is conformally symplectic, allowing us to use some of the Hamiltonian structure. We initially show that there exists an integral invariant of the Poincaré–Cartan type in such systems. Then, we prove the existence of a generalized Liouville Formula for conformally symplectic systems with rigid constraints using the integral invariant. A two degree-of-freedom system is analyzed to support the relevance of our results.

Damping Formation Mechanism and Damping Injection of Virtual Synchronous Generator Based on Generalized Hamiltonian Theory

Invertor as a virtual synchronous generator (VSG) to provide virtual inertia and damping can improve the stability of a microgrid, in which the damping is one of the fundamental problems in dynamics. From the view of the Hamiltonian dynamics, this paper researches the damping formation mechanism and damping injection control of VSG. First, based on the energy composition and dynamic characteristics of VSG, the differential equations system of VSG is established and is transformed into the generalized Hamiltonian system. Second, the effects of the three parameters of VSG, the damping coefficient D, active power droop coefficient, and time constant of excitation TE on damping characteristics are researched from a dynamic perspective, and simulation research is carried out with an isolated microgrid. Lastly, the control design method of Hamiltonian structure corrections used to add the damping factor and design the equivalent control inject damping to improve the stability of the isolated microgrid. Research shows that the voltage and frequency stability of the isolated microgrid can be effectively improved by selecting three key parameters of VSG and damping injection control. The innovations of this paper are 1. The Hamiltonian model of the inverter is deduced and established by taking the inverter as a virtual generator. 2. Based on the Hamiltonian model, damping characteristics of inverter in the microgrid are studied. 3. Hamiltonian structure correction method is applied to the inverter, and equivalent damping injection is designed to improve the stability of the microgrid.

A Family of Integrable Differential-Difference Equations: Tri-Hamiltonian Structure and Lie Algebra of Vector Fields

Starting from a novel discrete spectral problem, a family of integrable differential-difference equations is derived through discrete zero curvature equation. And then, tri-Hamiltonian structure of the whole family is established by the discrete trace identity. It is shown that the obtained family is Liouville-integrable. Next, a nonisospectral integrable family associated with the discrete spectral problem is constructed through nonisospectral discrete zero curvature representation. Finally, Lie algebra of isospectral and nonisospectral vector fields is deduced.