Newton's Laws as Axioms and Inertial Frames

The Problem of the Observer

Imagine you are sitting in a train moving at a constant speed. If you toss a ball straight up, it lands back in your hand. To you, the ball moves in a vertical line. To a person standing on the platform, the ball follows a parabolic arc. Both descriptions are correct, yet they look different. This raises a fundamental question: which observer is "right"? In Newtonian mechanics, there is no single privileged observer. Instead, we define a special class of observers, called inertial frames, in which the laws of motion are simplest. In these frames, an object with no net force acting on it continues in a straight line at constant speed. If you are in a frame where a force-free object accelerates, your frame is not inertial; it is accelerating relative to the true inertial class. The core task of this lecture is to define this class mathematically and to show that Newton's laws are invariant under transformations between any two such frames.

Defining the Inertial Class

An inertial frame is not defined by what it contains, but by how it moves relative to other frames. We postulate that if one frame is inertial, then any other frame moving at a constant velocity relative to it is also inertial. This set of all such frames forms the "inertial class." Mathematically, the transformation between two inertial frames \(S\) and \(S'\) is a Galilean transformation. It consists of a spatial translation \(\mathbf{d}\), a fixed rotation \(R\), and a constant velocity boost \(\mathbf{V}\). Crucially, time is absolute: \(t' = t\). Because the relative velocity \(\mathbf{V}\) is constant, the relative acceleration is zero. This zero relative acceleration is the key property that preserves the form of Newton's second law.

Here, \(\mathbf{r}(t)\) is the position in frame \(S\), and \(\mathbf{r}'(t)\) is the position in frame \(S'\). The rotation \(R\) is time-independent, meaning the axes of \(S'\) do not rotate relative to \(S\). This restriction is vital; if the axes were rotating, fictitious forces would appear, violating the simple form of \(\mathbf{F} = m\mathbf{a}\).

Galilean Invariance of the Second Law

We now demonstrate that Newton's second law is invariant under these transformations. Let us differentiate the position transformation twice with respect to time. Since \(\mathbf{d}\) is constant and \(R\) is fixed, their derivatives vanish. The first derivative gives the velocity relation: \(\mathbf{v}' = R\mathbf{v} + \mathbf{V}\). The second derivative gives the acceleration relation: \(\mathbf{a}' = R\mathbf{a}\). This result shows that acceleration transforms as a vector under rotations but is unaffected by the constant velocity boost \(\mathbf{V}\). This is the mathematical heart of Galilean relativity: uniform motion is indistinguishable from rest when measuring acceleration.

Now consider the force. In Newtonian mechanics, forces are typically functions of relative positions and relative velocities (e.g., gravity depends on \(\mathbf{r}_1 - \mathbf{r}_2\), and drag depends on \(\mathbf{v}_1 - \mathbf{v}_2\)). Under a Galilean transformation, relative positions rotate by \(R\), and relative velocities also rotate by \(R\) (since the constant \(\mathbf{V}\) cancels out in the difference \(\mathbf{v}_1' - \mathbf{v}_2' = R(\mathbf{v}_1 - \mathbf{v}_2)\)). Therefore, the force vector transforms as \(\mathbf{F}' = R\mathbf{F}\). Substituting these into Newton's second law in the primed frame:

Since \(R\) is a linear operator, we can factor it out: \(R(\mathbf{F} - m\mathbf{a}) = 0\). Because \(R\) is a rotation matrix (and thus invertible), this implies \(\mathbf{F} - m\mathbf{a} = 0\), or \(\mathbf{F} = m\mathbf{a}\). Thus, if the law holds in one inertial frame, it holds in all inertial frames. This is the content of derivation D-021.

The Role of Mass and Force

The invariance argument above relies on two assumptions: that mass \(m\) is invariant (the same in all inertial frames) and that forces are constructed from relative quantities. Mass invariance is a postulate of Newtonian mechanics; it is not derived from the laws themselves but is required for consistency. If mass varied with velocity, the simple form of the second law would break down under boosts. Similarly, the definition of force as a function of relative state variables ensures that the "interaction" between two particles looks the same to all inertial observers. If a force depended on a velocity measured relative to “absolute space” rather than relative to other bodies (such as the air for drag), different inertial observers would assign it different values and predict different accelerations, violating the principle of relativity. Therefore, the structure of Newtonian dynamics is tightly coupled to the definition of the inertial frame.