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
Famous quotes containing the words potential and/or equation:
“The germ of violence is laid bare in the child abuser by the sheer accident of his individual experience ... in a word, to a greater degree than we like to admit, we are all potential child abusers.”
—F. Gonzalez-Crussi, Mexican professor of pathology, author. Reflections on Child Abuse, Notes of an Anatomist (1985)
“A nation fights well in proportion to the amount of men and materials it has. And the other equation is that the individual soldier in that army is a more effective soldier the poorer his standard of living has been in the past.”
—Norman Mailer (b. 1923)