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Stars are essentially gravitationally stabilised thermonuclear reactors inhydrostatic equilibrium. The fundamental differential equation for allNewtonian gaseous stars in equilibrium is \begin{align}\frac{dp(r)}{dr}=-\frac{\mathscr{G}\mathcal{M}(r)\rho(r)}{r^{2}}\nonumber\end{align} where $p(r),\rho(r)$ are the pressure, density at radius $r$ and$\mathcal{M}(r)$ is the mass contained within a shell of radius $r$ given by$\mathcal{M}(r)=\int_{0}^{r}4\pi \overline{r}^{2}\rho(\overline{r})d\overline{r}$, and $\mathscr{G}$ is Newton's constant. Thissimple but crucial differential equation for the pressure gradient within anystar, underpins much of astrophysical theory and it can derived by variousmethods:via a simple heuristic argument; via the Euler-Poisson equations for aself-gravitating fluid/gas; via a variational method by taking the 1stvariation of the sum of the thermal and gravitational energies of the star; viathe 2nd variation of the Massiue thermodynamic functional for aself-gravitating isothermal perfect-gas sphere; from conservation of the virialtensor; as the non-relativistic limit of the Tolman-Oppenheimer-Volkoffequation (TOVE). The TOVE for equilibrium of relativistic stars in generalrelativity can in turn be derived by various methods: from the energy-momentumconservation constraint on the Einstein equations applied to a sphericallysymmetric perfect fluid/gas; via a constrained optimization method on the massand nucleon number; via a maximum entropy variational method for a sphere ofself-gravitating perfect fluid/gas or radiation. An overview is given of allderivations with emphasis on variational methods. Many important applicationsand astrophysical consequences of the Newtonian equilibrium equation are alsoreviewed.

The Fundamental Equilibrium Equation For Gaseous Stars And The  Tolman-Oppenheimer-Volkoff Equation -- Derivations And Applications With  Emphasis On Optimisational-Variational Methods