# Can all the theorems of classical mechanics be deduced from Newton's laws?

As above, is the whole edifice of Newtonian mechanics built upon Newton's three laws of motion? Can I deduce all the theorems without referring to further assumptions?

• Yes. One only needs $\textbf{F} = d\textbf{p}/dt$ and $\textbf{F}_{12} = - \textbf{F}_{21}$. One also assumes that all interaction forces between two particles lie on a line joining them. – Ultima Jul 21 '15 at 0:37
• Do you mean the assumption that all forces between particles are central? – Rescy_ Jul 21 '15 at 0:39
• Yes, I've revised my previous comment. Also, the proof of conservation of total angular momentum relies on this fact. – Ultima Jul 21 '15 at 0:39
• The question formulation (v1) seems to be imprecise (at least if OP is hoping for an affirmative answer). E.g. the Parallel axis theorem/Steiner's theorem depends on geometry rather than Newton's laws. – Qmechanic Jul 21 '15 at 12:04
• The title and body of the question are at odds with one another. The title asks about classical mechanics; does classical electromagnetism count? On the other hand, the body of the question asks about Newtonian mechanics. – David Hammen Jul 21 '15 at 12:59

If, by "Newtonian Mechanics" you mean what Newton derived, then yes, by definition.

But if you mean "classical mechanics" including rigid body dynamics then the answer is a resounding "no"[1] and the main reason is that Newton's three laws by themselves are not enough to imply conservation of angular momentum.

For conservation of angular momentum, you need an assumption further to Newton's three laws and the usual one is that the interaction force between two bodies points along the line joining the bodies[2]. Equivalently, one can get this from an assumption of spatial isotropy so that the formula for the force on a body from another must be invariant with respect to a rotation of co-ordinates. Given a body, the only co-ordinate free definition of a force direction that can be derived from the position of the other body relative to the first alone is of a vector along the the relative position vector if the force acts instantaneously[3]. More generally, conservation of angular momentum can be thought of as deriving from Noether's theorem applied to the rotational symmetry of the relevant classical Lagrangian.

Euler's Mechanics, where the three Euler laws are the rotational equivalent of Newton's translational ones for rigid bodies, are thus seen to be a definite broadening of Newton's laws. It is sometimes said that Euler's laws are Newton's with the assumption that everything in a rigid body must undergo only proper isometric transformations (i.e. "move rigidly") but the need for a conservation of angular momentum shows that there is another ingredient needed to get from Newton to Euler.

Footnotes:

[1]: That is even if we assume we're allowed to replace Newton's statement of the third law with its modern version of conservation of momentum (so that we're not bothered by pesky noninstantaneous electromagnetic interactions messing with the original statement).

[2]: I haven't read the Principia so I am not sure whether Newton actually comes up with the interaction force along joining line idea. I would not be surprised if he did, but then it is still further to his three laws.

[3]: If the force doesn't act instantaneously, relative positions change before the force arrives, and so we have a retarded version of this assertion in relativistic mechanics (by retarded, I mean analogous to the Feynman force formula and the Liénard-Weichert potentials).

If "for every action there is an equal and opposite reaction" (F12=-F21) allows friction as a legitimate reaction, then no. Non-conservative forces like friction are not fundamental and would allow non-conserved energy and momentum if additional constraints are not included (like "heat" from thermo). They are not time-reversible. You can't derive Langrangian and Hamiltonian mechanics unless you use the modern mathematical form of Newton's 2nd law that forces conservation of forces. The 4th mathematical expression of mechanics, stationary action, is the only one that can solve certain non-equilibrium problems, according to a paper I saw. So many places on the internet want to say Langrangian and Hamiltonian mechanics can be derived from "Newton's laws" and do not even mention stationary action, but there are some important details not mentioned. See Feynman's "Principle of Least Action" chapter in the red books.

• The Feynman lectures haven't been red for a while now. – Timaeus Aug 1 '15 at 3:21

No.

Even if you include some additional things for angular momentum there are still many things not in Newton's laws of motion.

For instance, the fact that mass doesn't change from one value on Tuesday to a different value on Wednesday.

Newton talks about mass in the Principia but it isn't about motion or forces so it isn't part of his Laws of Motion.

Other basics like how to combine forces or what forces are allowed to depend on are not covered by just the Laws of Motion themselves bare from everything else.

And many definitions are not there at all: energy, external versus internal forces, relativity of observers.

How would you talk about potential energy if you didn't know that was legitimate? And without it how would you get conservation of energy?

Sometimes people argue about what is classical mechanics and what is not. And why some people include everything except quantum so classical includes GR even the most restrictive ideas about what is classical mechanics include energy.

Newton was not the final word on classical mechanics. And he had some words (about mass for instance) before he got to his laws of motion.

So no. You can't get all of Classical Mechanics from three (or even four) Laws of Motion.

And just since some comments were saying you don't need the first law, I will mention that you do lose something without the first law, you lose determinism itself.