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

compare also
http://sirius.ucsc.edu/demoweb/cgi-bin/?mechan-no_rot-2nd_law and Arons

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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

1 Answer

There are two different ideas in the "superposition of motion", one which is kinematics, and the other dynamics. The kinematic law is a trivial decomposition of vectors--- the velocities form a vector space, and you can add them. This is also true in relativity, if an object A is moving with velocity v, and another object B is moving u faster than v, in that it covers u more distance per unit time (where distance and time are in the stationary frame), then v+u is the velocity of object B.

But in relativity, the difference velocity u is not the velocity of object B as measured in the frame of object A, because the A frame has different time and space axes. But velocities still form a vector space, only the symmetry of changing frames to moving with velocity v does not correspond to the trivial addition of vector velocities as it does in Galilean kinematics.

The second question, the one about forces, is dynamical. You are asking why do separate forces produce separate motions, and are there any cases where this fails. The answer is no, because there is a conservation law working here--- the conservation of momentum. When you apply a force F, you are adding F units of momentum to an object per unit time. When you apply a second force F', you are adding F' units of momentum to the object. The two forces add because momentum is a vector conserved quantity--- it's separate components are separately conserved, and the components of the forces tell you how much of each momentum component is coming in.

Conserved quantities are those that add up to a constant no matter what happens, an its always pure addition, even when the dynamics are nonlinear. So there is no case where two external forces leads to anything other than two additive momentum changes, and when the momentum is entirely contained in moving particles, this means that two forces on a moving particle produce additive changes in velocity, additive accelerations.

The justification of the Newtonian picture from conservation of momentum (and angular momentum) is more fundamental.

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