# Equivalences and derivations in Newtonian/Classical Mechanics

In Newtonian mechanics there are several "laws" and axioms:

• Newton's Laws
• Conservation of: Mass, Energy, Momentum, Angular Momentum

I know some are equivalent (e.g., conservation of momentum and Newton's second law) and some are derived from others (e.g. first Newton's law is a special case of the second).

What is the entire directed graph of derivations here, and as a result, what are the minimal sets of assumptions for Newtonian Mechanics?

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The first Newton law isn't a special case of the second, because it derermines the existence of inertial frames. Without that we can't assume the second Newton law. –  PhysiXxx Oct 12 '13 at 13:36
For example, you may start from the Newton laws and then come to energy, impulse and angular momentum conservation laws. –  PhysiXxx Oct 12 '13 at 13:38
As originally formulated by Newton, the first law was a special case of the second. Later, influenced by Mach, textbook authors started presenting it as a statement about the existence of inertial frames. See physics.stackexchange.com/questions/13557/… . –  Ben Crowell Oct 12 '13 at 15:55

This is a good question, but not one with a simple answer. The question also has no generic answer because there are many different and inequivalent statements of Newton's laws floating around. Mach 1893 was the big step forward in clearing up the interpretation of Newton's laws. As far as I can tell, modern textbook presentations tend to take a little of Newton's original formulation and mix in a little Mach.

If you look at Newton's original formulation of the laws, in context, it's pretty clear that although he was self-consciously imitating the style of Euclid, he wasn't actually doing that style of axiomatic reasoning. The first law is written as a special case of the first, which you wouldn't do if you really had in mind a formal axiomatic system. The examples he gives of the first law are not in fact valid examples of the first law.

Based on Newton's original formulation, and ignoring the important foundational issues pointed out by Mach, the first law is a consequence of the second. Conservation of momentum is a consequence of the third law. Conservation of mass is logically independent. Conservation of energy is logically independent (since there is nothing in the laws of motion that restricts fields of force to being conservative). Conservation of angular momentum is logically independent (since there is nothing in the laws of motion that prohibits forces that act at a distance and along directions different from the radius vector).

Going in the opposite direction, Newton's laws can all be proved from conservation of energy and momentum (or from conservation of energy and Galilean relativity). For a treatment in this style, see my book Simple Nature.

From a modern point of view, mechanics is only one tiny branch of physics, and the conservation laws are much more general than mechanics.

Reference

Mach, The Science of Mechanics, 1893; 1902 English Edition, tr. McCormack, http://www.archive.org/details/sciencemechanic00machgoog

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Thanks. I thought conservation of momentum is a derivation of the second law (which just state the sum of all momenta doesn't change), and that conservation of angular momentum is a consequence of the third law (e.g. the proof in wikiproof). –  R S Oct 12 '13 at 16:31
@RS: You need both the 2nd law and the 3rd law to prove conservation of momentum. Can you give a link to the proof you're referring to re angular momentum? It's either wrong or it invokes additional assumptions. –  Ben Crowell Oct 12 '13 at 16:37
I saw this in: proofwiki.org/wiki/Conservation_of_Angular_Momentum –  R S Oct 12 '13 at 16:46
So, what about the other way around? How does one show if the Second and Third laws a consequence of the conservation laws? –  R S Oct 12 '13 at 16:47
The proof on proofwiki says, "The final part of the third law is that these conjugate forces act on through the line that connects the two bodies in question." This is wrong. For an English translation of what Newton's original formulation of the third law says, see archive.org/stream/newtonspmathema00newtrich#page/n87/mode/2up , p. 83. –  Ben Crowell Oct 12 '13 at 17:00