4 ms·

I love how someone in one of the answers has an intuitive feeling that the phase space dimensionality can be reduced, and asks "does this hold generally", and t

I love how someone in one of the answers has an intuitive feeling that the phase space dimensionality can be reduced, and asks "does this hold generally", and then Terence Tao comes along in a comment and explains why it indeed does hold. Math Overflow must be one of the most extreme examples of successful communication on a narrow topic between people at widely different levels of fame/skill.

Sharlin 10y ago
Terry Tao is also the author of the top answer, mind.
- mirimir 10y ago
  Wow, I could actually follow that explanation, and it's been decades since I thought much about mechanics. Very intuitive. However, I'm left a little confused at the end, by this: > The process then repeats itself (imagine a marble rolling frictionlessly between two equally tall hills, starting from a position very close to the peak of one of the hills). But it's not really just two hills, I think. It must be two hills in a closed space, because you can get from one peak to the other in either direction. Yes? Also, back in physical reality, I wonder whether the periodic axis rotation continues in the same direction, once a small perturbation gets it started. Or whether the direction of axis rotation randomly changes. Anyone know?
  - ecma 10y ago
    The Lagrangian answer covers this a bit more obviously by stating a solution which results in a closed curve of states around some equilibrium point(s) for certain initial variables. I think that matches your understanding of this as two peaks in a closed circuit.
    - mirimir 10y ago
      Thank you. When I saw this thread, I was sure that it would be something about orbital mechanics or the Coriolis effect. But no, it seems.
  - Double_Cast 10y ago
    Given Tao's rotating frame: * For each M-mass, the centrifugal-vector lies along the x-axis. * For each m-mass, the centrifugal-vector lies along the z-axis. * For each centrifugal-vector, the magnitude is proportional to "the point-mass's distance from the disk's axis-of-rotation (the y-axis)". Since the m-masses move along a unit circle in the yz-plane, their magnitudes vary. E.g. an m-mass which starts at (0, .9, .1) will accelerate towards (0, 0, 1) then deccelerate towards (0, -.9, .1). Then it will retrace its path. This results in oscillation between (0, .9, .1) and (0, -.9, .1) along a semi-circle in the yz-plane. N.b. no oscillation will occur if an m-mass starts at exactly (0, 1, 0).

Sharlin 10y ago

Terry Tao is also the author of the top answer, mind.

mirimir 10y ago
Wow, I could actually follow that explanation, and it's been decades since I thought much about mechanics. Very intuitive. However, I'm left a little confused at the end, by this: > The process then repeats itself (imagine a marble rolling frictionlessly between two equally tall hills, starting from a position very close to the peak of one of the hills). But it's not really just two hills, I think. It must be two hills in a closed space, because you can get from one peak to the other in either direction. Yes? Also, back in physical reality, I wonder whether the periodic axis rotation continues in the same direction, once a small perturbation gets it started. Or whether the direction of axis rotation randomly changes. Anyone know?
- ecma 10y ago
  The Lagrangian answer covers this a bit more obviously by stating a solution which results in a closed curve of states around some equilibrium point(s) for certain initial variables. I think that matches your understanding of this as two peaks in a closed circuit.
  - mirimir 10y ago
    Thank you. When I saw this thread, I was sure that it would be something about orbital mechanics or the Coriolis effect. But no, it seems.
- Double_Cast 10y ago
  Given Tao's rotating frame: * For each M-mass, the centrifugal-vector lies along the x-axis. * For each m-mass, the centrifugal-vector lies along the z-axis. * For each centrifugal-vector, the magnitude is proportional to "the point-mass's distance from the disk's axis-of-rotation (the y-axis)". Since the m-masses move along a unit circle in the yz-plane, their magnitudes vary. E.g. an m-mass which starts at (0, .9, .1) will accelerate towards (0, 0, 1) then deccelerate towards (0, -.9, .1). Then it will retrace its path. This results in oscillation between (0, .9, .1) and (0, -.9, .1) along a semi-circle in the yz-plane. N.b. no oscillation will occur if an m-mass starts at exactly (0, 1, 0).