# [Physics] Is Newton’s first law something real or a mathematical formalism

calculusinertial-framesnewtonian-mechanics

Why do objects always 'tend' to move in straight lines? How come, everytime I see a curved path that an object takes, I can always say that the object tends to move in a straight line over 'small' distances, but as you take into account the curvature of the path, a force acting on the particle appears. I mean, I can always take a small enough portion of the curve, zoom in enough, and conclude that the object is moving in a straight line, but then as I zoom out I find out that a force is acting on the particle. The force of gravity is everywhere and, no matter how weak it is, it will make the particle take a path which is different from a straight line. This is my question: since particles are, in reality, never moving in straight lines, is Newton's first law a mathematical formalism or some true property of material objects?

Nice question! The answer to this depends on the version of Newton's first law you use.

In the Principia, the statement of the first law, as translated by Machin, is:

Every body perseveres in its state of rest, or of uniform motion in a right line, unless it is compelled to change that state by forces impressed thereon.

This is immediately followed by a series of examples:

Projectiles persevere in their motions, so far as they are not retarded by the resistance of the air, or impelled downwards by the force of gravity. A top, whose parts by their cohesion are perpetually drawn aside from rectilinear motions, does not cease its rotation, otherwise than as it is retarded by the air. The greater bodies of the planets and comets, meeting with less resistance in more free spaces, preserve their motions both progressive and circular for a much longer time.

Of the three examples, not one involves motion in a straight line! Since the first law is stated in the Principia in words rather than equations, there's a lot of room for ambiguity. Keep in mind as well that scientists reading the Principia in that era didn't know calculus, and vectors weren't invented until centuries later. Newton had to write in language his contemporaries would understand, even if it was at the cost of precision.

There are many different ways in which the first law has been stated over the years, as described in this question: History of interpretation of Newton's first law .

You can modify it to be a statement that if you choose a specific axis $x$, then the absence of any forces in the $x$ direction gives $dv_x/dt=0$ at that instant in time. This is probably the interpretation that's most directly suggested by Newton's three examples.

You can modify it to be a statement about objects that are acted on by zero total force.

As described in the other question, it's now popular (probably due to the influence of the analysis in Mach 1919) to describe it as a statement about the existence of inertial frames.

Gravity does present some unique issues, since it's a long-range force and can't be shielded against. Mach 1919 gave a very thorough and insightful critique of the logical basis of Newton's laws. Here is my own presentation of the question of what the first law really means and some experimental tests. In general relativity, we define a free-falling frame as an inertial frame, so that the motion of a projectile is defined to be "straight."

Ernst Mach, "The Science Of Mechanics," 1919, http://archive.org/details/scienceofmechani005860mbp