newtonian mechanics - Intuition as to why the orientation (of a 3D object) is not a conserved quantity?

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,

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  you can't change your linear momentum.

  
  


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

  
  


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  you can't change your angular momentum.

  
  


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  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?

Answer

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.