Re: Poincaré

On 10/07/2024 10:27 AM, Richard Hachel wrote:

We do not realize the immense flash of genius brought to humanity by
Poincaré.
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His transformations are as brilliant as they are magnificent. Lorentz
had searched for them for years, without ever succeeding.
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But Poincaré was an immense mathematician, the greatest of his time.
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Coupled with philosophical reflections and very advanced physical
knowledge.
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It is very regrettable that out of hatred, out of self-loathing (the
French people are in the entire history of peoples the ones who are the
most self-loathing), we have come to adulate instead a 27-year-old
cretin copyist whose closest friends said that he was incapable of
dividing by 2.
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It was his friend Grossmann who had to do his calculations.
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But that's life.
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We will return to the transformations given by Poincaré in June 1905
(copied in September 1905 by Einstein).
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They are magnificent, and open to a lot of conclusions, such as, for
example, the elasticity of lengths
by change of reference frame. I wrote: elasticity, not contraction.
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We have a bar of length l in R, what will be its length in R'?
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What does Poincaré tell us?
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x'=(x-Vo.To)/sqrt(1-Vo²/c²)
y'=y
z'=z
To'=(To-x.Vo/c²)/sqrt(1-Vo²/c²)
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Well understood, these transformations give us the length of the rod,
and we have directly in R':
l'=l.sqrt(1-Vo²/c²)/(1+cosµ.Vo/c)
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This is simply a mathematical deduction (in case the angle of sight
towards A and towards B is small enough to be considered almost zero, µ
being on the other hand the angle between the projected sight and the
direction of the mobile studied).
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You can do the calculation by yourself, it is at the level of a good
middle school student.
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We can therefore see that setting l'=l.sqrt(1-Vo²/c²) is totally
inappropriate, and that it only works if the rod passes into an
orthogonal position relative to the observer in R'.
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This is enough for physicists, but the concept still seems quite
different to me.
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Poincaré, if we understand him well, indicates, by his transformations,
that the length depends on the relative speed (gamma factor) but ALSO on
the position of the observer, and that is incredibly hard to swallow.
Yet that is what Poincaré says in his transformations.
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Do some numerical tests. For example, set a rod A(12.9)-B(24.18) in R.
Length?
l=15.
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Let's change the frame of reference. Will we have a smaller rod by
contraction of x (y invariant)?
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That's what physicists say.
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That's not what the TLs say.
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No, we'll have a much longer stem in R'.
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A(40,9)-B(80,18)
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l'=41.
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You can check for yourself.
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R.H.

Poincare is pretty great.
He gets into the Dirichlet problem, and also,
there's the Analysis Situs, which later has
that Lefshetz writes also an Analysis Situs,
and so it's kind of good to have analysis
that kind of spurns Gauss, then also there's
that Poincare has some notions of topology,
like for example a notion of a "rough" line
or "rough" plane to complement the "smooth"
line and "smooth" analysis, as with regards
to the definitions of differentiability and
pretty much how the entire school sort of
gets hand-held through Riemann differentiable
and measurable out through Lebesgue differentiable
and measurable, yet, those are both after the
Dirichlet differentiable (and measurable,
after Fejer, sort of) as with regards to Dirichlet,
and Poincare, great analysts, and Poincare
a great geometer.
Dirichlet -> Jordan (atoms, bullets, infinitesimals)
Dirichlet -> Riemann/Lebesgue, Stietljes, ... (fluents)
Dirichlet -> Poincare (Fourier)
These three sorts of approaches to measure theory
after differentiability and for continuity, has
that these days there's quasi-invariant measure
theory, and the measure problem is much or mostly
associated with the harmonic and up after Chladni functions
the Dirichlet problem, yet most are entirely down
the path past Lebesgues, which of course is great,
though not complete.
Yeah the quasi-invariant measure theory is a thing,
to both measure theory and invariant theory,
sort of like the pseudo-differential,
or for example the fiber space, Hermann's topology's.
This is because there are more law(s) of large numbers
and infinities that mostly physics has none.