Poisson's Equation and Gravitational Potential
Since the gravitational field has zero curl (equivalently, gravity is a conservative force) as mentioned above, it can be written as the gradient of a scalar potential, called the gravitational potential:
Then the differential form of Gauss's law for gravity becomes Poisson's equation:
This provides an alternate means of calculating the gravitational potential and gravitational field. Although computing g via Poisson's equation is mathematically equivalent to computing g directly from Gauss's law, one or the other approach may be an easier computation in a given situation.
In radially symmetric systems, the gravitational potential is a function of only one variable (namely, ), and Poisson's equation becomes (see Del in cylindrical and spherical coordinates):
while the gravitational field is:
When solving the equation it should be taken into account that in the case of finite densities ∂ϕ/∂r has to be continuous at boundaries (discontinuities of the density), and zero for r = 0.
Read more about this topic: Gauss' Law For Gravity
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