We have investigated the accelerating behaviour of the Universe in f(Q, T) gravity in an isotropic and homogeneous space-time. We have initially derive the dynamical parameters in the general form of $$f(Q,T)=\alpha Q^m+\beta T$$f ( Q , T ) = α Q m + β T (Xu et al. in Eur Phys J C 79:708, 2019) and then split it into two cases (i) one with $$m=1$$m = 1 and the (ii) other with $$\beta =0$$β = 0 . In the first case, it reduces to the linear form of the functional f(Q, T) and second case leads to the higher power of the nonmetricity Q. In an assumed form of the hyperbolic scale factor, the models are constructed and its evolutionary behaviours are studied. The geometrical parameters as well the equation of state parameter are obtained and found to be in the preferred range of the cosmological observations. Marginal variation has been noticed in the behaviour of $$\omega $$ω and $$\omega _{eff}$$ω eff at present time. The violation of strong energy conditions in both the cases are shown. The dynamical system analysis for the models has been performed.

In symmetric teleparallel geometry the curvature and torsion tensors are assumed to vanish identically, while the dynamics of gravity is encoded by nonmetricity. Here the spatially homogeneous and isotropic connections that can accompany the flat Friedmann-Lemaître-Robertson-Walker metric come in three sets. As the trivial set has received much attention, we focus on the two alternative sets, which introduce an extra degree of freedom into the equations. Working in the context of symmetric teleparallel scalar-tensor gravity with generic nonminimal coupling and potential, we show that the extra free function in the connection cannot play the role of dark matter nor dark energy, but it drastically alters the scalar field behavior. We determine the restrictions on the model functions which permit the standard cosmological scenario of successive radiation, dust matter, and scalar potential domination eras to be stable. However, the alternative connections also introduce a rather general possibility of the system meeting a singularity in finite time.

We present Hamilton’s equations for the teleparallel equivalent of general relativity (TEGR), which is a reformulation of general relativity based on a curvatureless, metric compatible, and torsionful connection. For this, we consider the Hamiltonian for TEGR obtained through the vector, antisymmetric, symmetric and trace-free, and trace irreducible decomposition of the phase space variables. We present the Hamiltonian for TEGR in the covariant formalism for the first time in the literature, by considering a spin connection depending on Lorentz matrices. We introduce the mathematical formalism necessary to compute Hamilton’s equations in both Weitzenböck gauge and covariant formulation, where for the latter we must introduce new fields: Lorentz matrices and their associated momenta. We also derive explicit relations between the conjugate momenta of the tetrad and the conjugate momenta for the metric that are traditionally defined in GR, which are important to compare both formalisms.