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This paper deals with stability analysis of anisotropic-charged cylindrical geometry coupled with $f(R,T)$ modified gravity theory (${ℳ}{𝒢}{𝒯}$). In this ${ℳ}{𝒢}{𝒯}$, the Lagrangian is a varying function of traces of Ricci and stress tensors. In order to examine the stable behavior of the anisotropic cylindrical cosmic object under the electromagnetic effects, we calculated the modified equations of motion and equations of continuity. in the background of $f(R,T)$ theory. By utilizing the perturbation technique, we constitute a modified collapse equation (${ℳ}{𝒞}{ℰ}$), which contains geometrical and physical variables in the charged scenario. Our investigation continues under the nonminimally coupled function of ${ℳ}{𝒢}{𝒯}$. Furthermore, we investigate the dynamically stable phases of charged geometry at Newtonian $({𝒩})$ and post-Newtonian ($p{𝒩}$) epochs with the help of the rigidity parameter of fluid ($\mathrm{\Gamma }$) by imposing some constraints on physical parameters.

This work is related to the stability of a nonstatic, very restricted class of axially symmetric cosmic matter configuration under the influence of modified gravitational theory (AGT) with a realistic model in anisotropic conditions. We construct dynamically modified equations, continuity equations, mass functions and collapse equations in the scenario of considered AGT. We continue our analysis under the influence of f(R,T) gravity along with a minimally coupled gravitational function of the form $f(R,T)=R+\beta R_c(1-{(1+\frac{R^2}{R_c^2})}^{-n})+\lambda T$, where $R_c$ indicates constant entity. In this gravitational model, f is a function of Ricci scalar invariant R and the trace of the stress–energy tensor T. This gravitational function gives significant findings only for $\beta >0,n>0$, and both belong to the set of positive real numbers and also $\lambda \ll 1$. We establish expressions of an adiabatic index, i.e. stiffness parameter of fluid $(\mathrm{\Gamma })$ utilizing relevant perturbation technique with the help of collapse equation in Newtonian $(\mathbb{N})$ and post-Newtonian $(\mathbb{P}\mathbb{N})$ domains. We consider the particular form of metric as well as material functions in $(\mathbb{N})$ and $(\mathbb{P}\mathbb{N})$ eras, and impose some constraints on physical quantities (i.e. pressure, density, mass, etc.) to carry out the stable behavior of a considered restricted geometry.

This paper aims to investigate the stability constraints under the influence of particular modified gravity theory $𝕄𝔾𝕋$, i.e. $f(R,T)$ gravity in which the Lagrangian is a varying function of $R$ and trace of energy momentum tensor ($T$). We examine stable behavior for compact cylindrical star having anisotropic symmetric configuration. We establish dynamical equations as well as equations of continuity in the background of this particular non-minimal coupled $𝕄𝔾𝕋$. We utilize perturbation technique which will be applied on geometrical as well as material physical quantities to constitute collapse equation. We continue this significant investigation to understand the dynamical behavior of considered cylindrical system under non-minimal coupled $f(R,T)$ functional, i.e. $f(R,T)=R+\xi R_c[1-{\{1+\frac{R^2}{R_c^2}\}}^{-n}]+\lambda RT$. This gravitational function gives compatible findings only for $\xi >0,n\in R^{+}$, also $0<\lambda \ll 1$ and $\lambda RT$ considered in this astrophysical model as coupling entity. This model contains $R_c$ which is constant entity, having the values of order of the effective Ricci scalar $R$. Furthermore, we impose some physical constraints to determine and maintain the stability criteria by establishing the expression of adiabatic index, i.e. $\mathrm{\Gamma }$ for cylindrical anisotropic configuration, in Newtonian $(N)$ and post-Newtonian ($pN$) eras.