Finding Derivations of Newton's Law of Universal Gravitation online

Question: was Newton's law of universal gravitation found empirically (from measurements) or can it be derived mathematically? The short answer is yes -- both. The historical process is separate from what can be done now.

Empirical (historical) route

The second paragraph of the Wikipedia article on Newton's Law of Universal Gravitation says, "This is a general physical law derived from empirical observations by what Isaac Newton called inductive reasoning." There's a video on YouTube describing that Newton started with

• acceleration of gravity on earth, g = 9.8 m/s^2
• distance between Earth and Moon = 60*(radius of earth)
• radius of Earth = 
• orbital period of moon = 27.32 days

Newton figured out how fast the Moon is circling the Earth using

 velocity = distance/time

where, in the case of the moon circling the Earth,

 distance the moon travels around the Earth = circumference = 2*pi*r_{orbit of Moon}

Plugging in numbers, velocity of Moon around Earth = 1022 m/s. That can then be plugged into the centripetal acceleration,

a_{centripetal} = v^2/r

How does a_{centripetal} compare to the gravitational acceleration g?

g/a_{centripetal} = 60^2

Noticing the 60 is common to the ratio and the distance between the Earth and the Moon, Newton figures that gravitation follows an inverse square law. Newton then checked this against data from observational studies of planets.

That's a big leap to F = G*(M*m)/(r^2), so are there more mathematical routes?

Mathematical Derivations

From nasa.gov

I first found NASA's classroom materials. There are some leaps that I wasn't able to follow. The same content of that page is duplicated on a related NASA page. The derivation starts with F=ma and a_{centripetal} = v^2/r. The author mentions Kepler's Third Law but says T^2 \approx R^3 (which is dimenionally inconsistent) when they mean T^2 \propto R^3. The misuse of \approx and \propto continues throughout the rest of the derivation.

velocity = distance/time

and the distance here is the circumfrence, 2*pi*r, so

period T = (2*pi*R)/v

Drop the 2*pi to get

period v \approx R/T

Square both sides and apply Kepler's Third law

T^2 \propto R^3

to get

v^2 \propto 1/R

The second source of my confusion is subscripting versus superscripting -- v_2 versus v^2.

F = (m*(v^2))/R

I tried submitting a correction to NASA's feedback page but couldn't since the Captcha is missing. :(

From a tutorial

Next I found another student-oriented page that has a derivation which is even less helpful than NASA's. The derivation presented starts from angular speed and F=mr\omega^2.

YouTube tutorial

Happily I found this helpful video from Daniel M, "Deriving Newton's Law of Universal Gravitation".

Steps to convert a published scientific paper into the Physics Derivation Graph

This post documents the steps taken to get the derivations in the paper Speed of sound from fundamental physical constants into the Physics Derivation Graph. 

1) Read the paper. I need to understand the point of the paper and where content is in the text. 

2) Identify the primary result of the derivation. Which equation is the output? Is there more than one? In this paper, equation 1 is the primary result.

Tangent: I tested whether the most popular equation is the most important.
This is measurable using the command

cat sound1.tex | tr " " "\n" |\          # replace spaces with newlines -- word tokenization
   grep "ref{" |\                        # find internal references to labeled expressions
   sed 's/.*\\ref{//' | sed 's/}.*//' |\ # remove the irrelevant latex "\ref{}" to isolate the expression label ID
   sort | uniq -c | sort -n              # rank the popularity of expression labels
2 v001
3 abinitio
3 all
3 ratio
3 v0
3 v01
3 v1
5 elemental
6 bohr
6 rydberg
11 a
15 v00
15 v3

Decoding those latex labels to equations in the PDF, "v3" is equation 9, which is tied for most popular with "v00" -- equation 4. The second most popular label "a" is equation 10. The equation I'm claiming as the primary result, equation 1, is labeled as "v0" and is referenced 3 times.

3) What sections of the paper contain the derivation?
There are actually two derivations of the same result.

• one derivation is on page 2 column 1
• another derivation is on page 2 column 2

4) Determine whether the derivation already exist in the Physics Derivation Graph database. Check

• https://derivationmap.net/list_all_expressions
• https://derivationmap.net/list_all_symbols
• https://derivationmap.net/list_all_derivations

5) On paper, write out equations from the paper. Here I'm limiting the scope to one derivation on page 2 column 1. Fill in additional equations that are missing from the text. Use arrows to denote flow of derivation.

Often I'll make math mistakes or go down paths that turn out to be irrelevant. If I were a typical researcher, this is where the reconstruction process would stop.

6) On a separate paper, rewrite the graph and label the edges in the directed graph to describe the operation being carried out. 

7) On a separate paper, rewrite the content from the previous step to refine the "operation being carried out" to a directed graph with inference rules available in the Physics Derivation Graph. 

8) Identify all symbols used in the derivation: \rho, V, x, a, m, \alpha, M, K, \hbar, c, \pi, \epsilon_0

9) Identify which symbols in the derivation already exist in the Physics Derivation Graph and which need to be added. Record the symbol ID of each for later use.

10) Add missing symbols into the the Physics Derivation Graph database using  https://derivationmap.net/list_all_symbols#add%20symbol 

11) Identify which inference rules the derivation uses. 

12) Identify which inference rules for this derivation exist in the Physics Derivation Graph and which need to be added. 

13) If novel inference rules are needed for this derivation, enter those in https://derivationmap.net/list_all_inference_rules#add_inf_rule

14) Using https://derivationmap.net/start_new_derivation/, enter the expressions and inference rules for the steps of the derivation. For each step, 

• ensure the symbols are referenced correctly in each expression
• ensure the SymPy version of the expression is correct

The output of this step is https://derivationmap.net/review_derivation/608598/

15) Offline, in the JSON file containing the derivation, edit the symbol IDs and SymPy representations as needed to get the step validations to be correct. 

Review the modifications on https://derivationmap.net/review_derivation/608598/

TODO: The extensive linking of variables to definitions is not made accessible in the output, so there is no way for the user to leverage the linked information. 

• expressions in the PDF/HTML output contain variables which have no link to their definitions
• variables in the text of the PDF/HTML output are not linked to the expressions or their definitions

As a result of the above observation, https://github.com/allofphysicsgraph/proofofconcept/issues/217 was opened.