# Some questions about the logics of the principles of independence of motion and composition of motion

In high-school level textbooks* one encounters often the principles of independence of motion and that of composition (or superpositions) of motions. In this context this is used as "independence of velocities" and superposition of velocities (not of forces).

This is often illustrated by the example of the motion of a projectile, where the vertical and horizontal motions are said to be independent and that the velocities add like vectors.

Now, if $x\colon \mathbb{R} \to \mathbb{R^3}$ describes the motion of the of the considered object, it is clear that one can decompose the velocity $v = \dot x$ arbitrarily by $v = v_1 + v_2$ where $v_1$ is arbitrary and $v_2 := v - v_1$.

This leads me to my first question: Am I correct that this is purely trivial math and contains no physics at all? If so, it would not deserve to be called "principle of composition of motions" or something like that and said to be fundamental.

However it seems that one could interpret the decomposition above such that $v_1$ is the velocity of the object with reference to a frame of reference moving with $v_2$. If so, how can one see, that this goes wrong in the relativistic case?

Now suppose you have two forces $F_1$ and $F_2$ which you can switch on and off, suppose that $F_i$ alone would result in a motion $x_i$ ($i=1,2$). Newtonian Mechanics tells us the principle of superposition of forces, i.e. if you turn both forces $F_1$ and $F_2$ on, the resulting motion $x$ is the solution of the differential equation $\ddot x = \frac1m F(x, \dot x, t)$ (where m is the Mass of our object) with $F = F_1 + F_2$.

One might interpret the principle of composition of motions such that always $\dot x = \dot x_1 + \dot x_2$ holds true. This is clearly the case if $F_i$ depends linearily on $(x,\dot x)$. However I think that it doesn't need to be true for nonlinear forces. This leads me the my third question: Is there any simple mechanical experiment where such nonlinear forces occur, which shows that in this case the "principle of composition of motions" doesn't hold?

*I have found this in some (older) german textbooks, for example: Kuhn Physik IIA Mechanik, p. 107, Grimsehl Physik II p.16,17

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To your third question: Consider the motion of a projectile in consideration of air resistance $~v^2$. –  student Mar 10 '12 at 16:08