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Say you start off floating in space, in a fixed position and orientation, with zero linear and angular velocity, with no external forces. So you are a closed mechanical system. By twisting your body around,

  • you can't change your linear momentum.

  • you can't change your position (center of mass).

  • you can't change your angular momentum.

  • you can change your orientation (i.e. rotation)!

The fact that you can change your orientation comes as a surprise to me-- why isn't it conserved like the other three quantities? It's a familiar fact-- cats do it all the time in order to land on their feet, and you can find videos of astronauts doing it on the international space station. See the videos linked from https://space.stackexchange.com/questions/2954/how-do-astronauts-turn-in-space . But it still seems counterintuitive to me that they can do this while not being able to change the other three quantities. Is there some intuitively clear explanation as to why?

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    $\begingroup$ Your question is good and subtle, so I only have a hunch that the following may be relevant. Are you aware of Richard Montgomery's "Gauge Theory of the Falling Cat"? It's kind of another way of expressing the same idea as @Christoph - "orientation" is only meaningful as long as you are talking about the same point in the space of cat shapes (or configuration of a system of masses). Or, said backwards, we can condense a full configuration space of a system of masses into equivalence classes of configurations where our equivalence relation ... $\endgroup$ – WetSavannaAnimal Feb 25 '15 at 11:29
  • $\begingroup$ ...is "can be mapped from one to the other by an isometry". Montgomery summarises these ideas pithily by making the orientation a gauge field on the space of cat shapes (a fiber bundle with the base space as shapes and orientations as fiber) $\endgroup$ – WetSavannaAnimal Feb 25 '15 at 11:30
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    $\begingroup$ Random idea: may orientation be not a conserved quantity somehow because rotations in different planes don't commute?.. $\endgroup$ – Ruslan Feb 25 '15 at 11:34
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    $\begingroup$ @Ruslan Orientation is not conserved even if movement is confined in a single plane. $\endgroup$ – JiK Feb 25 '15 at 11:46
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    $\begingroup$ It might be helpful to think about how reaction wheels accomplish their effects. $\endgroup$ – zwol Feb 25 '15 at 21:47
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It is because the moment of inertia is not a conserved quantity.

The statement that an isolated body can't change its position is more precisely the statement that an isolated body cannot change the position of its centre of mass. The position of the centre of mass, ${\bf R}$, is given by:

$$ {\bf R} = \frac{1}{M}\sum m_i {\bf r}_i $$

where $M$ is the total mass and the $m_i$ are the masses of the individual elements of our system. Mass is a conserved quantity, so all the masses in our equation are constants and if we differentiate with respect to time we get:

$$ \dot{\bf R} = \frac{1}{M}\sum m_i \dot{\bf r}_i = \frac{\bf P}{M} $$

where ${\bf P}$ is the total momentum. Since momentum is conserved the total momentum must be a constant and if we differentiate again we get $\ddot{\bf R} = 0$, so the acceleration of the centre of mass must always be zero.

Now let's try and apply the same argument to the angular equivalent of the centre of mass. By analogy with the centre of mass we can define a centre of angle as:

$$ \Theta = \frac{1}{I}\sum I_i \theta_i $$

The next step is to try and differentiate $\Theta$ twice with respect to time in the hope of obtaining $\ddot{\Theta} = 0$. The problem is that neither the total moment of inertia nor the moments of the individual elements are constants, but instead can be functions of time. In general our result will be:

$$ \ddot{\Theta} \ne 0 $$

which means that $\Theta$ is not a constant.

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    $\begingroup$ I like this answer. I think it captures the essence of why things break down, and also gives a hint as to how to accomplish zero-gravity rolls and cat flips: it's easy to change one's moment of inertia, e.g. by extending one's arms or retracting them. $\endgroup$ – Don Hatch Feb 25 '15 at 12:57
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    $\begingroup$ Comment to the answer (v1): It should be stressed that an angle is born as a multivalued function. A single-valued definition of an angle depends on a choice of branch, which cannot be chosen to be continuous everywhere. This somewhat jeopardizes the notion of a center of angle. $\endgroup$ – Qmechanic Feb 25 '15 at 13:15
  • $\begingroup$ @Qmechanic, maybe this could be repaired by considering the angle as a normalized vector, $(\cos \theta, \sin \theta)$? There's a related problem and solution in directional statistics. $\endgroup$ – A. Donda Feb 26 '15 at 3:31
  • $\begingroup$ @A.Donda there would still be a large class of objects whose "centre of angle" vector was the (directionless) zero vector despite having a recognizable orientation (ie. no circular symmetry). $\endgroup$ – Hugh Allen Feb 26 '15 at 17:22
  • $\begingroup$ A very good answer. $\endgroup$ – Yashbhatt Feb 28 '15 at 7:48
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John correctly stated that this is possible because re-configuring our bodies allows us to change our moment of inertia, but not our mass.

As the question was about an intuitive explanation, consider adding a series of floating weights to get an analogous situation for translational motion:

The astronaut stretches their arms above the head, grabs a weight, moves it along their body and releases it at the waist. Doing this repeatedly will allow the astronaut to change their position.

In detail, starting with retracted arms in case of rotations and raised arms in case of translations:

\begin{array}{l|l} \textbf{rotation} & \textbf{translation} \\ \hline \text{spread your arms} & \text{pick up weight} \\ \text{to increase moment of inertia} & \text{to increase mass} \\ \hline \text{twist your body} & \text{lower arms} \\ \text{to change orientation} & \text{to move COM of body} \\ \hline \text{retract arms} & \text{drop weight} \\ \text{to decrease moment of inertia} & \text{to decrease mass} \\ \hline \text{untwist body to get back} & \text{raise arms to get back} \\ \text{into initial body configuration} & \text{into initial body configuration} \\ \end{array}

The last step will counter the rotation / forward motion, but as the moment of inertia / mass will be less than in step 2, there's a net change in orientation / position.

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  • $\begingroup$ I must confess I don't quite see the connection between this description and the question. Perhaps add some more clarifying material, pointing out the correspondence between this translational case and the rotational case, and where/why the correspondence breaks down? $\endgroup$ – Don Hatch Feb 25 '15 at 21:13
  • $\begingroup$ @DonHatch: see edit $\endgroup$ – Christoph Feb 25 '15 at 21:38
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It seems helpful to consider an extremely simple scenario. Suppose an astronaut is floating near two balls of lead; in this case the closed system consists of the astronaut together with the balls. She can pull the balls together without changing the momentum or angular momentum of the system. She can then rotate them in the center with almost no change, and separate them again. If that little twist in the center bothers you, you can imagine that she instead has three lead bars, pulls them together so that one slips between the other two, and then pulls them apart along a different axis. The actual mechanics of human and cat motion are more complex, of course, but you can think of movements like raising and lowering the arms and swinging them forwards and backwards as essentially similar.

Imagine that your whole body is held rigid and straight except that you can swing your arms at your shoulders. Start with your arms at your sides. Now lift them up and forwards as though you were bumping a ball in volleyball, until they are perpendicular to your body. Your body will tilt forwards. Now pull your arms apart, to the left and right. You will again tilt forward. Finally, push your arms back down to your sides. You will not tilt at all, but will return to your original body shape, only tilted forwards relative to your original orientation.

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  • $\begingroup$ +1, but perhaps a simpler example of turning yourself would be to just swing your both hands in constant circular motion: while you keep swinging, your body is rotating in opposite direction (you start exactly the same way as you described, but instead of stopping and pulling your arms apart, you just continue swinging). $\endgroup$ – kristjan Feb 25 '15 at 18:28
  • $\begingroup$ @kristjan, while that motion is simpler, it seems slightly harder to analyze, because the arms do not swing about the center of mass. $\endgroup$ – dfeuer Feb 25 '15 at 18:42
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    $\begingroup$ @dfeuer, this is a good explanation of how orientation can be changed, but that has been extensively analyzed elsewhere... lots of papers about how cats right themselves, and questions answered about that on this forum, in fact. My question is more about trying to gain intuition about why certain quantities are conserved and others aren't, counter to my intuition that these are all similar kinds of quantities. $\endgroup$ – Don Hatch Feb 25 '15 at 19:30
  • $\begingroup$ There's a good overview with video links in the answers to space.stackexchange.com/questions/2954/… $\endgroup$ – Don Hatch Feb 25 '15 at 20:03
  • $\begingroup$ ... and I just now added that link to the question $\endgroup$ – Don Hatch Feb 25 '15 at 20:18
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Here is a simpler answer: if something can change shape, then it doesn't really have an orientation.

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  • $\begingroup$ Consider a colorful rubber ball with sand. It can change shape, but you can tell its orientation at any time by looking at its colors. $\endgroup$ – Ruslan Feb 25 '15 at 12:23
  • $\begingroup$ it can change shape and if it does it may end up in a different orientation. $\endgroup$ – Smooth space Feb 25 '15 at 12:28
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    $\begingroup$ Yes, but you assert that it doesn't have an orientation in your answer. $\endgroup$ – Ruslan Feb 25 '15 at 12:28
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    $\begingroup$ It's not obvious to me that a non-rigid shape has no meaningful orientation. In the case of interest, you do some contortions and then relax back to some canonical shape. In that case your overall orientation change is well defined, just as it is for a rigid body. (It can be quantified as a matrix or quaternion.) What bothers me is that after contortions are done, you find that you were unable to achieve any velocity or position change or to set yourself spinning... but somehow you were able to change your orientation. Isn't that surprising? $\endgroup$ – Don Hatch Feb 25 '15 at 12:31
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    $\begingroup$ Right, orientation can be changed and the others are conserved... but I don't see that you're saying anything more here than what was observed in the question. The question is why orientation is changeable within a closed system when the other three aren't-- that seems really surprising to me. $\endgroup$ – Don Hatch Feb 25 '15 at 12:54
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An even simpler answer than the others given here is that rotation of an object by an integer number of turns leaves it in the same apparent rotational state as it had previously. If one has an object in space with two coaxial parts whose moments have e.g. an x:y ratio and the parts rotate relative to each other, one part will make y rotations for every x rotations of the other. If e.g. x is 1.1 and y is 1.0, then if x makes one full revolution y will make 1.1 rotations. Although those numbers will balance as suggested by their moments, x will appear to be in its original orientation and y will appear to have rotated by 0.1 turns.

All of the other examples involving changing moments effectively involve having some parts of the system make a complete rotation relative to other parts and then end up in the same apparent orientation.

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protected by Qmechanic Feb 25 '15 at 20:12

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