So I've been reading this about white dwarves, and various other sites about white dwarves. In all of them, they say that we can find the radius of a white dwarf by minimizing its total energy. I know that for almost all physics problems we do this, but why is it so? Does it have something to do with variational calculus or something else entirely...?

Grateful for any help!

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A missing bit of reasoning here is that the "minimum energy" state of a white dwarf is almost reached before it has cooled down, because the internal energy of a degenerate gas (that dominates the equation of state) is independent of temperature (the minimum internal temperatures of any WDs is about $10^{6}$ K!). In fact the radii of new-born white dwarfs are larger than the radii of older white dwarfs precisely because they have yet to get into a (nearly) completely degenerate state. @georginaelston – Rob Jeffries Oct 15 '15 at 12:58