Thoughts on Independent Motions

We seem to make a big deal in intro mechanics that x- and y- motions are “independent”. There certainly are in a limited number of cases (constant forces, linear drag forces, and linear restoring forces, for example) in which the force dynamics described among cartesian coordinates are separate. But this condition isn’t true in most situations (gravitational forces, electric forces, magnetic forces, non-linear drag, non-linear restoring forces). Still, in many situations, we can still find a coordinate system in which the forces are separable. For example, all the non-linear restoring forces (gravity, electric, rubberbands) are separable in (r,θ) coordinates, but the magnetic and non-linear drag forces I don’t think are separable, at least not generally so.

I could be wrong, but I think all (or most) force (fields) that can be described by the gradient of a scalar function will have at least one coordinate choice for which the force dynamics separate. This requires some further inspection, but it has to do with curl being zero. Linear drag forces are, of course, non-conservative; but they “nicely” separate due to a cancellation of factors that would otherwise couple them. This nice cancellation doesn’t occur with v² dependent drag forces, so non-linear drag forces are not separable.

I’m not sure, butI think when we teach this notion of “independent motions”, we are confusing independence with orthogonality, or perhaps separability with orthogonality. X and y cartesian coordinates are certainly orthogonal, but motions described using x and y coordinates are only independent (or separable) under circumstances in which they aren’t coupled.

Anybody know the history of teaching “horizontal and vertical forces” are independent? My suspicion is that it is an over-generalization from projectile motion.

7 thoughts on “Thoughts on Independent Motions

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  1. Hi Brian,

    I’m not sure who your “we” is referring to; some of us are pretty careful not to make this mistake. I agree there are two different concepts here that are often confused.

    But I would guess that your speculation is correct, that this comes from over-generalizing the constant force case. Of course, there’s no reason you have to use orthogonal vectors as a basis for your vector space. It would be … interesting … to have students work out kinematic equations for motion under constant acceleration in non-orthogonal coordinates (with x and y at a 45 degree angle to each other, for example). Wouldn’t work in intro physics, but it would be a nice mind-bending exercise for students in advanced mechanics.

  2. Yeah, I should be careful with “we”. I think I mean “we” to capture that it was both how I was taught and how most instructors I have seen since teach it. We also suggests that it is common, which I think it is. But “we” certainly doesn’t mean every individual.

    1. That makes me think that another possible cause (and I agree that it is common, I really have to be careful if I want to avoid this error) is that this is just sloppy expert talk. Experts often talk in shorthand and don’t literally mean what they say. They know that saying x and y are independent is subject to certain caveats and that exactly what “independent” means may depend on context.

      This also reminds me of an interesting talk I saw Mazur give a couple years ago that touched on the misuse of “conserved” and “constant” in many texts when discussing conservation of energy and momentum. It’s a similar sort of issue.

  3. I think there are a couple different things going on.

    Linear forces should always be able to be separated in the sense that the operator has an eigenbasis.

    As for things like gravity, those are all central forces, and so we already know that the force is purely in terms of r, it’s separable due to the fact that its a function of a single coordinate.

    To that end, since the magnetic force is linear, if I’m right it should be separable, and it is.

    if we start with

    m vdot = v x B

    you’ll note that the x, y, and z coordinate equations are coupled, but..

    now consider a coordinate system where z points along the magnetic field direction, and then form the combinations
    chi = x + i y
    eta = x – iy

    then the dynamics will
    m chidot = -i B chi
    m etadot = i B eta

    the two coordinates decouple.

    Interesting to think about. On the whole, I think what you are interested in is the dynamics decoupling, that is whether there is a basis in which the dynamics is uncoupled. Then it becomes clear that linear forces should always admit an orthogonal basis, and the nonlinear cases you mention are trivial in that they depend only on the separation, hence depend on a single coordinate to begin with.

    Fun stuff.

    1. I think this assumes a constant B, am I right? This isn’t usually the case, which emphasizes my point once again, that there are only a limited number of idealized situations in which they equations decouple. We can also find electric fields that decouple… fields between a capacitor… field away from an infinite line… etc.

      1. Technically, I believe that even if the magnetic field was position dependent, you could decouple the equations of motion by switching to the coordinates that follow the magnetic field, it would be a rather nasty coordinate system, but it would work.

        For instance for the magnetic field around a straight wire, you could use the coordinates:

        Z = theta
        X = r + i z
        Y = r – i z

        I agree with you on the whole that decoupled equations of motion are a rather limited case, but it remains that most of the intro physics deals with linear forces. If you want to naturally introduce coupled equations, you ought to introduce some interesting nonlinear interactions. You create more ‘realistic’ coupled equations naturally by considering more ‘realistic’ physics.

  4. Okay, I might be missing the point here, but I’ve been thinking about how we talk about this in my (first-try-at-physics high school) classes recently because it is just starting to come up.

    With balanced forces, we talked about how if the forces are balanced overall, then they must be balanced in any given direction.

    With unbalanced forces, we talk about wanting to put one of our axes in the same direction as the acceleration so that the forces are unbalanced completely in the direction of one of our axes (and therefore balanced in the perpendicular direction).

    And we talked just a little bit about if we tilt our x-axis, we’d tilt our y-axis to be perpendicular to it because we’ll make sure to cover everything if our axes are perpendicular. But we didn’t really think too hard about it. Having x- and y-axes that are perpendicular feels so obvious to them because of math classes. And they very quickly come to the idea that if the acceleration is completely in the x- or y- direction, then the other direction would be balanced (none of the acceleration is in that direction). Not a lot of thinking or prodding.

    Anyway, we only consider relatively simple cases where the forces usually stay constant the entire time.

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