Scientists find an effective solution for the three-body problem

Take a look at this (not the same link above I target a single comment):

https://www.reddit.com/r/Physics/comments/mn11r/the_dome_a_s...

I'll have you know you're debating with higher powers here. Some people here are not your typical people who just took a physics class.

Either way intuition without the math is enough to break your brain.

Imagine the opposite scenario you flick a ball up the dome with just the right amount of force that it comes to rest right at the apex. One property of Newtonian mechanics is that motion is time reversible. Meaning that the opposite motion here should be valid. And the opposite motion of the scenario I described is indeed what occurs. The particle is at rest on the dome and arbitrarily just rolls off randomly.

Note that this is caveated on the site with the fact that this intuition doesn't work with hemispheres. Apparently with a hemisphere you can flick the ball with a certain amount of force and it takes infinite time to reach the apex of the dome. So if you time reverse that it means any ball resting on a hemispherical dome can roll off of the apex arbitrarily but it takes an infinite amount of time. The un-determinism only works for the special dome here called "Nortons Dome."

For example, the standard way to make an electronic thermometer is by, more or less, measuring the current across a semiconductor diode at a given voltage. This current is an exponential function of the ratio between the voltage and a "threshold voltage" or "thermal voltage" Vt multiplied by an "ideality factor" n: I = Is (exp(V/(nVt)) - 1).

The threshold voltage Vt varies linearly with temperature (it's kT/q, depending only on Boltzmann's constant and the charge of the electron, about 25 mV at room temperature), so in a sense the current at a given voltage is a measurement of the temperature. But the ideality factor n depends on the purity of the semiconductor material (generally in the range 1.0 to 2.0), and the saturation current Is depends on the physical size of the diode junction. Moreover, the ideality factor can change over time as the diode ages. So we're in the position of simultaneously measuring the temperature, the size of the diode, and the quality of its aged semiconductor material.

The solution usually taken, as I understand it, is to measure the current through the same diode at two given voltages, one after the other, and to use a standard value for n which is good enough. Then the ratio of the two voltages tells you nVt (as long as the "- 1" is too small to matter) and from that you can calculate the temperature. In theory, by taking three or more measurements at different points in the I-V curve, you could correct for unknown n as well, but I haven't read of anyone doing this; instead, for high-precision thermometry, they use an RTD.

(Actually, you measure the voltage at two given currents, because that way you don't burn up your temperature sensing diode if the temperature is a little higher than you expected; the power dissipated then varies logarithmically with temperature rather than exponentially. But it comes to the same thing in the calculations.)

It's actually even worse than it sounds, because in fact when you measure a voltage, you're always measuring it with respect to some reference voltage, so your actual measurement is a function of the temperature, the saturation current Is, the ideality factor n, and your reference voltage Vref, which is typically subject to an error of around 2%. But you will note that the ratiometric approach described above cancels out any errors due to Vref, because the ratio of the two voltages will be unaffected by a wrong reference voltage, as long as it's the same wrong reference voltage. So you stick a big capacitor on it and take the measurements in quick succession.

All of this is, from a certain point of view, in the service of making the number you finally produce very sensitive to the temperature of the diode and very insensitive to other factors, like the battery voltage, the temperature of the rest of the thermometer circuit, the age of the components, the humidity in the air, and so on. But all of the actual physical quantities being measured on the diode are the complex mix of factors described above.

MIMO antennas or phased-array receiver antennas or microphones are another example: the signal at each antenna/microphone is a linear superposition of all the differently-phase-shifted source signals, and you process that data to get independent measurements of all the original source signals.

I wouldn't be surprised if chaotic metrology offered new ways to measure very tiny differences, but I suspect it will take a lot of time to figure out the math to make that work.

This is not about success, but about being realistic about what you can expect from your life. Everyone is going to be unhappy from time to time and everyone will have downs ; this is just not what you usually see on social media. So when people say this, it's to reduce people's feeling of being inadequate, not to take away success.

If you think about it, it's actually two sides of the same coin: People only see humongous companies and insane salaries, but not the years of hard work that went into getting there. Similarly, they only see happy faces on social media, but not the bad sides that everyone has.

What you meant to say is there is no solution at the singularity. I never said that you didn't say this. I said that your statement implies that past the singularity the solution is nondeterministic.

>For the three body problem to be non-deterministic, one initial condition would have to lead to multiple valid trajectories, which is never the case.

I never concluded this. I concluded that we don't know if this is never the case for the 3 body problem. For classical mechanics in general I concluded that non-determinism is the case.

However let's reiterate what came from your initial post: "You think Newtonian physics is non-deterministic?"

I have proven this statement to not be something that I think, but something that is completely true and thus I have validated what you "thought" was my initial point. That should provide some partial conclusions to your initial points and you have admitted that you agree.

>What there does not apply to all initial conditions with distinct mass positions?

The theory does not apply at the singularities. You stated this yourself, no solution exists.

>The conclusion you have arrived at is wrong, and I urge you to reconsider.

I am in the process of reconsidering by simply speaking to you. That is the purpose of this entire endeavor.

Can you prove to me that past the point of collision the solution is deterministic? Obviously classical mechanics is still highly applicable outside of the the singularity. So after the given initial conditions and up to the point of collision the solution is deterministic.

During the collision the model fails to describe anything.

After the collision there are multiple outcomes that can occur depending on factors you assume and make up in your regularization scheme. All of these possible paths are governed by classical mechanical laws as they are outside of the singularity but we have several possibilities here. Or maybe not. Who knows. My claim is nobody knows. But we do know the system is still ruled by physical laws so the space of possible solutions is bounded. Basically I'm asking you to prove to me that within this space there is one deterministic unique solution.

>As I noted above, this is a residue of the idealization of the problem and could be removed by modeling with hard spheres instead of point masses.

Sure. But that's not the point. If you want to talk about non ideal cases things become weird even when you get really close to the singularity.

It can happen with a time-dependent force, i.e. not Newtonian dynamics.

Or if a particle can change its velocity without a force being applied to it, i.e. not Newtonian mechanics.

It is more straightforward to see using Lagrangian or Hamiltonian mechanics, which are better suited to this kind of constrained problem, if one is so inclined.