  Stars are essentially gravitationally stabilised thermonuclear reactors in
hydrostatic equilibrium. The fundamental differential equation for all
Newtonian 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. This
simple but crucial differential equation for the pressure gradient within any
star, underpins much of astrophysical theory and it can derived by various
methods:via a simple heuristic argument; via the Euler-Poisson equations for a
self-gravitating fluid/gas; via a variational method by taking the 1st
variation of the sum of the thermal and gravitational energies of the star; via
the 2nd variation of the Massiue thermodynamic functional for a
self-gravitating isothermal perfect-gas sphere; from conservation of the virial
tensor; as the non-relativistic limit of the Tolman-Oppenheimer-Volkoff
equation (TOVE). The TOVE for equilibrium of relativistic stars in general
relativity can in turn be derived by various methods: from the energy-momentum
conservation constraint on the Einstein equations applied to a spherically
symmetric perfect fluid/gas; via a constrained optimization method on the mass
and nucleon number; via a maximum entropy variational method for a sphere of
self-gravitating perfect fluid/gas or radiation. An overview is given of all
derivations with emphasis on variational methods. Many important applications
and astrophysical consequences of the Newtonian equilibrium equation are also
reviewed.
