1. Show that for some suitably smooth function
, the integral
Take the delta function to be the limit as of a step function of width and height and take “suitably smooth” to mean “has a Taylor series”.
Then show that
by evaluating the integral
for some smooth function
2. Determine the orbit
by changing variables from
See these notes.
3. Calculate the deflection of “light corpuscles” with mass
by the sun.
See Newtonian gravitational deflection of light revisited for a full treatment or Bending light for one that assumes the speed of light is a constant and neglects acceleration towards the earth.
4. Prove Newton’s first superb theorem: the gravitational force exerted by a spherical mass distribution acts as if the entire mass were concentrated in a point at the center of the distribution.
5. Prove Newton’s second superb theorem.
Wikipedia calls these the shell theorem.
6. Find a formula for the transit time of a gravity train.