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.

One solution.

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