The Meaning of Newton's Second Law of Motion Being Invariant Under Certain Transformations

Question

What do we mean when we say that Newton's Second Law of Motion is invariant under Galilean transformations?

Does it mean that the value of a force measured in one reference frame is the same when measured in another reference frame, which are moving with constant velocity relative to each other? Or that the form of the equation remains the same? I do not really understand what we mean by the latter. Let's take the example of three reference frames $S$, $S'$, and $S''$. $S'$ moves with instantaneous velocity $u$ relative to $S$ along the positive $x$-axis, $S''$ with $v'$ relative to $S'$, and $S''$ with $v$ relative to $S$. Let's say $S'$ and $S''$ are accelerating with $S''$ having a greater acceleration than $S'$ as measured in $S$, which is an inertial frame. We can write for the acceleration of $S''$ as measured in $S'$:

$$\frac{dv}{dt} = \frac{du}{dt} + \frac{dv'}{dt}$$ $$\frac{dv'}{dt} = \frac{dv}{dt} - \frac{du}{dt}$$

If $S''$ is attached to a mass $m$, the force on the mass as measured in $S'$ is given as

$$m\frac{dv'}{dt} = m\frac{dv}{dt} - m\frac{du}{dt},$$

which is less than the force on the mass as measured in $S'$ by $m\frac{du}{dt}$. With the same form of equation: (mass)x(acceleration of that mass), we get different values for the force on the mass. So we might say that Newton's Second Law is not invariant under a transformation if the reference frame is non-inertial, but we say that because the value of the force is not same, what about the form?

EDIT: How do you show that the 'form' of the equation, whatever they mean by it, changes when you transform from an inertial frame to a non-inertial frame?

Answer 1

What does it mean by saying that Newton's laws are invariant under Galilean transformations?

Simply speaking, the structure of the Newton's law (or the dynamical equation) is preserved under a Galilean transformation. You can see from these transformation equations that the acceleration of a particle as measured from all inertial frames will be the same. As per the Newtonian concept, mass is treated as absolute, in the sense that it does not depend on the relative motion between the observer and what is being observed. Hence, according to the Newton's law, which states that $\vec{F}=m\vec{a}$, the force acting on a particle as measured from all inertial systems will be the same. Usually the theory of relativity is concerned about the co-variance symmetry associated with every physical law. You may see from the principle of relativity that its the laws of physics that are the same as seen from all inertial frames, not the values of any physical quantities. However, in the case of Newton's law, the values of the force as measured from two inertial frames should tally for the structure of the law to become invariant. For example, if the momentum of a certain physical system is found to be invariant from an IF, it should be so in all other IF, even though the values of momentum as measured from each frame can depend on the relative motion between the two frames.

Now what do we mean by the statement that the form of a law is invariant?

Take the example of Newton's second law. It tells us that a system (with a finite mass) will have an acceleration if there is a nonzero force acting on it and the acceleration for a given force depends on the mass of the system. That mean, if the acceleration as measured from an inertial system is zero, then we can arrive at the conclusion that the net force acting on it is zero. It's a law. This law for that very system holds for all other inertial systems too. This is possible only if the way in which the law connects the force and the acceleration should remain invariant. This is what we mean by the invariance of a law.

How do you show that the 'form' of the equation, whatever they mean by it, changes when you transform from an inertial frame to a non-inertial frame?

If you observe a physical system from an accelerating frame, then this acceleration will induce an additional force acting on the particle. So you will see that the particle is under a force. Suppose an observer from an inertial system measures that the particle is moving at constant velocity, i.e., its not accelerating. The same particle as observed from an inertial system will appear to be accelerating due to the acceleration of the frame itself. Hence, an observer from a non-inertial reference frame will see a force acting on the particle. That means, if you take the observation from an inertial system to a non-inertial one, it violates the Newton's law of motion. Hence Newton's laws are not invariant under transformations between non-inertial frames. To preserve, the structure o the law has to be modified by introducing a fictitious force on the particle due to the acceleration of the frame. This will change the structure of Newton's law.

Answer 2

Newton's law are invariant only when transformed from one inertial frame to another. Galilean transformation connect inertial frames, which would mean (if I understand your notation right) that $du/dt=0$.

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Well... one of those derivatives will be 0. I don't quite understand why you need three accelerating frames... but Galilean transformations do not transform quantities between accelerated frames.