Toby on The Hardest Soviet Exam and Jade on Non-deterministic Newtonian Mechanics

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I had a copy of Mechanics, which was translated by John Bell:

 I couldn't get past page one:

It was the final paragraph that I didn't understand. I wanted to know whose experience of what produced the knowledge that the state of the system is completely determined by the coordinates and velocities simultaneously specified. I guess I was doing an exam in Theology, or Zoology:

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Avshalom Elitzur on physics and zombies and things

For more about the Two State Vector Solution, see  Jenn Grant - The Bells Are Ringing.

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You can listen to the whole of his IAI talk Consciousness and Material Reality on his YouTube channel:

8:12 He says "physics is full" by which he means that if you introduce any non-material, non-energetic agency into the laws of physics then you have to abandon the conservation laws.

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Jade has been working on this video for at least six months:

See Picard–Lindelöf theorem and Lipschitz continuity.

In that paragraph of Landau & Lifshitz' Mechanics they state that mathematically, this determinism means that if the co-ordinates and velocities of the particles are given at some instant, then the accelerations are uniquely determined. So they are implicitly making a connection between the mathematical statement of the Newtonian laws and some unspecified empirical observation. Jade mentions (at 18:47) that there is no actual instant when this instantaneous version of Newton's law is broken:

If you want to go into the details there is this video on Lipshitz Continuity from The Bright Side of Mathematics:

And for the topological details, see Misha Rudnev's notes https://people.maths.bris.ac.uk/~maxmr/opt/closedandconvex.pdf:

13:12 She does not mention the peculiar shape of the dome, which causes the equations of motion to have this specific form. One might otherwise wonder why this whole argument does not follow with an ordinary spherical dome. The shape of the dome Norton constructs is such that the particle at rest at the apex has zero instantaneous velocity and acceleration, and indeed (as observed in this blog post by Gruff Davies) zero instantaneous jerk (rate of change of acceleration with time) but the snap (rate of change of rate of change of acceleration) is non-zero (1/6 to be precise). The topological details become important when you consider the time-reversal of the spontaneous non-equilibrium solutions. In this case the trajectories are those where the initial position and velocity are such that the ball comes to rest at exactly the apex of the dome, but in those cases the ball takes an infinite amount of time to get there. Hence the Lipschitz condition on the domain being an open set that contains an epsilon-ball becomes important and this is the essence of Norton's argument about there being no first instant of motion. See also Zeno's paradoxes.

Norton's  2006 follow-up is available here: The Dome: An Unexpectedly Simple Failure of Determinism.

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The question Elitzur raises about non-material, non-energetic causal agency being contrary to conservation laws is also connected with this question of the domain of the equations of motion, because of the way Gravitational energy is defined as the work done by gravity moving an object "from infinity" towards the centre of mass.  This motion too is something that has no initial instant, because "at infinity" there is no gravitational force.