Chandrasekhar's variational principle

In astrophysics, Chandrasekhar's variational principle provides the stability criterion for a static barotropic star, subjected to radial perturbation, named after the Indian American astrophysicist Subrahmanyan Chandrasekhar.

Statement
A baratropic star with $$\frac{d\rho}{dr}<0$$ and $$\rho(R)=0$$ is stable if the quantity


 * $$\mathcal{E}(\rho') = \int_V \left| \frac{d\Phi}{d\rho}\right|_0 \rho'^2 d \mathbf{x} - G \int_V\int_V \frac{\rho'(\mathbf{x})\rho'(\mathbf{x'})}{|\mathbf{x}-\mathbf{x'}|} d\mathbf{x}d\mathbf{x'} \quad \text{where} \quad \Phi = -G\int_V \frac{\rho(\mathbf{x'})}{|\mathbf{x}-\mathbf{x'}|}d\mathbf{x},$$

is non-negative for all real functions $$\rho'(\mathbf{x})$$ that conserve the total mass of the star $$\int_V \rho' d\mathbf{x} = 0$$.

where
 * $$\mathbf{x}$$ is the coordinate system fixed to the center of the star
 * $$R$$ is the radius of the star
 * $$V$$ is the volume of the star
 * $$\rho(\mathbf{x})$$ is the unperturbed density
 * $$\rho'(\mathbf{x})$$ is the small perturbed density such that in the perturbed state, the total density is $$\rho+\rho'$$
 * $$\Phi$$ is the self-gravitating potential from Newton's law of gravity
 * $$G$$ is the Gravitational constant