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?

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 $du/dt=0$.


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.

  • $\frac{du}{dt}$ is the acceleration of $S'$ as measured in $S$. $S'$ and $S''$ are non-inertial frames. – Omar Abdullah Feb 17 '17 at 22:35
  • What do you mean 'invariant'? The equations give the same value of a quantity? – Omar Abdullah Feb 17 '17 at 22:39
  • Yes. Up to rotations of the axes, the force and acceration will have the same numerical value in two inertial frames. If they are not inertial, anything goes and Newton's 3rd law will not yield the same numerical values for the forces and accelerations. – ZeroTheHero Feb 17 '17 at 22:41
  • 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? – Omar Abdullah Feb 17 '17 at 22:41
  • 1
    The forces in the two frames would not be the same, i.e. $F'\ne F$. $F=ma$ is an operational definition of force so if $a'\ne a$ then $F'\ne F$. That's invariance: i.e. numerical values do not change. – ZeroTheHero Feb 17 '17 at 22:50

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