# Can Newton's First Law be derived from Newton's Second Law?

Before I start my discussion here I would like to state the first two laws of Newton in their most correct forms: 1)1st Law: There exist frames of reference with respect to which every body continues to be at rest or in uniform motion in a straight line until compelled by an external force to change that state. Such frames are called inertial frames. 2)2nd Law: In an inertial frame, if external force be applied on a body to change its dynamic state, then rate of change of momentum of the body is directly proportional to the applied force. This constant of proportionality can be taken as 1, 2, 3 or anything as the laws are not experimental but axiomatic in the purview of classical mechanics. Newton took it as 1 reducing the proportionality to equality. Now many textbooks on physics derive Newton's First Law from his Second Law in the following manner: " From Newton's second law, considering mass of the body to be a non-zero constant:

$\overrightarrow { F } =\frac { d\overrightarrow { p } }{ dt } =\frac { d(m\overrightarrow { v } ) }{ dt } =m\frac { d\overrightarrow { v } }{ dt } =m\overrightarrow { a }$ Putting $\overrightarrow { F } =\overrightarrow { 0 }$: $m\overrightarrow { a } =m\frac { d\overrightarrow { v } }{ dt } =\overrightarrow { 0 } \\ \Longrightarrow \frac { d\overrightarrow { v } }{ dt } =\overrightarrow { 0 } \\ \Longrightarrow \overrightarrow { v } =constant$

Which is our first law, no force means no change of dynamic state." But had it been so, Newton wouldn't have enunciated his first law as a separate and independent one (though Newton's works, mathematically speaking, were absolutely non-rigorous and not precise at all, and I view them as a concoction of vague definitions, nice intuitions and distorted mathematics). Newton's three laws are form the axiomatic basis of classical mechanics and axioms cannot be proved from axioms within the same formal system (thanks to Godel for the proof of this statement). Many books justify this fact by saying that during that time three was a favorite number for the physicists. Galileo's three laws of falling bodies, Kepler's three laws of planetary motion etc. The books argue that Newton's laws were three in number for the same reason. But this is absolutely not a healthy explanation. Notice that Newton's First Law asserts the existence of Inertial Frames. It is these frames where the Second Law holds. So the first law talks of the inertial frames. What is an inertial frame? Well, many books say that it is a frame where Newton's laws hold. But this somewhat cyclic. If instead, we define inertial frames as those which are at relative rest with respect to a free particle (i.e. a particle with no real forces being acted upon) then the Second Law axiomatically holds in such frames. However, using the Second Law, we cannot actually prove the EXISTENCE of inertial frames as talked of in the First Law. And hence using Newton's Second Law we CANNOT prove his First Law.

Note by Kuldeep Guha Mazumder
6 years ago

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I kind of think it's more like this:

1) 1st Law defines what "inertial frames" are, and then goes to say that either a) such inertial frames move [with respect to others moving similarly] with uniform [vectorial] velocity, or b) move with changing [vectorial] velocity when external "Force" is applied.

2) 2nd Law specifies that when 1) b) applies, then the "change in Momentum" [or "Acceleration"] shall be proportional to the "Force" applied. That is, the relation is linear, which is not specified in 1)

The 3rd law says that for any system, relative to its center of gravity, Forces vectorially add up to $0$, but that's another subject for another time.

Your argument is that one can infer the 1st Law from the 2nd Law. You could make that argument, but for historical reasons, Newton felt it necessary to make it an axiom that any object, either at rest or in motion, will not change vector velocity without external force. Up until Newton's time, there was still confusion or were opinions held that, for example, cannons shot into the sky could simply "tire" of motion and fall out of the sky--with no need for air resistance "to make speeding cannonballs tire". This is the reason, Newton tried to argue, why planets can continue to "fall' forever in an orbit about the sun, since nothing is inducing the planets to change their free fall motion.

[It needs to be said here that military people before Newton's time were well aware that cannonballs eventually hit the ground because of gravity, which is the whole idea, but lesser known to the general public is that the same ballistics experts then were aware that cannonballs seemed to "suddenly quit" and drop out of their arcs prematurely. We know now that the reason for this is air resistance.]

Note that the 2nd Law doesn't say that vector velocity of inertial frames only change if there's force. It simply says that if there is force, then acceleration of such inertial frames shall be proportional to it. But, by itself, the 2nd Law leaves open the possibility of vector velocity of inertial frames changing for reasons other than force--however outlandish that now seems. It didn't seem so outlandish at one time.

- 6 years ago