# reperiendi

## 1.1. Newton’s laws

1. Show that for some suitably smooth function $f(x)$, the integral $\int_{-\infty}^{\infty} \delta(x)~f(x)~dx = f(0)$.

Take the delta function to be the limit as $w\to 0$ of a step function of width $w$ and height $1/w$ and take “suitably smooth” to mean “has a Taylor series”.

$\displaystyle \begin{array}{rl}\int_{-\infty}^{\infty} \delta(x)~f(x)~dx &= \lim_{w\to 0} \left( \int_{-\infty}^{-w/2} 0~f(x)~dx + \int_{-w/2}^{w/2} \frac{1}{w}~f(x)~dx + \int_{w/2}^{\infty} 0~f(x)~dx \right) \\ &= \lim_{w\to 0} \frac{1}{w} \int_{-w/2}^{w/2} f(x)~dx \\ &= \lim_{w\to 0} \frac{1}{w} \int_{-w/2}^{w/2} \sum_{n=0}^{\infty} \frac{f^{(n)}(0)x^n}{n!} dx \\ &= \lim_{w\to 0} \frac{1}{w} \left.\left(\sum_{n=0}^{\infty} \frac{f^{(n)}(0)x^{n+1}}{(n+1)!} \right)\right|_{-w/2}^{w/2} \\ &= \lim_{w\to 0} \frac{1}{w} \left(f(0)w + \mbox{terms involving higher powers of }w\right) \\ &= f(0) \end{array}$

Then show that $\delta(ax)=\delta(x)/|a|$ by evaluating the integral $\int_{-\infty}^{\infty} \delta(x)~f(x)~dx$ for some smooth function $f(x)$.

$\displaystyle \begin{array}{rl}\int_{-\infty}^{\infty} \delta(ax)~f(x)~dx &= \lim_{w\to 0} \frac{1}{w} \int_{-w/2|a|}^{w/2|a|} f(x)~dx \\ &= \lim_{w\to 0} \frac{1}{w} \left(f(0)w/|a| + \mbox{terms involving higher powers of }w\right) \\ &= f(0)/|a| \end{array}$

2. Determine the orbit $r(\theta)$ by changing variables from $r$ to $u = \frac{1}{r}$

See these notes.

3. Calculate the deflection of “light corpuscles” with mass $\epsilon > 0$ 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.