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How can I show that the most convenient way to move the arms while walking is swinging them back and forth, alternatively?

To pose the question in another way: can I prove, starting from the conservation of momentum and angular momentum, that given a rigid solid body moving at constant speed and with two appendices on the side, this appendices will move as our arms do when we walk "freely"?

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    $\begingroup$ The mechanics of walking are quite complex. A Google should find you articles on the subject. Basically, swinging the arms in opposition to the legs reduces the torque on the body and therefore makes it twist less. $\endgroup$
    – John Rennie
    Jun 24, 2014 at 11:37
  • $\begingroup$ @JohnRennie minor nitpick: I'd prefer to say "complicated" (not "complex") so as to avoid suggesting the solution requires going into the complex plane. $\endgroup$
    – Carl Witthoft
    Jun 24, 2014 at 11:45
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    $\begingroup$ @JohnRennie "...reduces the torque on the body and therefore makes it twist less". Is there a way to show this formally, without relying on experimental data? $\endgroup$
    – usumdelphini
    Jun 24, 2014 at 12:16

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This is a responce to usumdelphini's comment:

Is there a way to show this formally, without relying on experimental data?

I'm not going to attempt a deep analysis of walking, but it's fairly straightforward to show that using your arm reduces twisting. Suppose you're looking down on the person walking from above, and suppose they're a cylinder $^1$. Then you'd see something like:

Leg

When you move your right leg forward this takes some force $F$, and by Newton's third law there is an equal and opposite force on the point where the leg is attached to your body (the green dot). This causes a torque on your body of:

$$ T = Fr $$

and as a result your body will tend to twist clockwise.

Now suppose you move your left arm forward at the same time:

Leg and arm

Moving your arm creates a torque of $T_{arm} = fr$ in the same way, but the torque is in the opposite direction. So the net torque on your body is:

$$ T = Fr - fr = (F - f)r $$

and the result is that your body will twist less.

$^1$ traditionally we assume everything is spherical, but I'll make an exception in this case

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