4 ms·

It has the same “units” (if you mean “energy”) as mc2 as well and that doesn’t suggest anything to me… Your intuition is much better than mine - or it’s informe

• vinnyvichy 2y ago
  Sorry! I meant they have the same form as used in the energy differential (1-form), but I had thought "units" would make more sense. In fact, this comparison was how I came to the intuition, although, as you coyly suggested, I did do a check with my earlier intuitions.. https://physics.stackexchange.com/questions/415943/why-does-the-internal-energys-total-differential-behave-like-this https://physics.stackexchange.com/questions/415943/why-does-...
  • kgwgk 2y ago
    I agree that thermodynamic relations - and Legendre transformations - are fascinating. I don’t think I ever fully understood them though - at least not to the point where they became “intuitive” :-)
    • vinnyvichy 2y ago
      Erm sorry again to have implied they were intuitive, all I meant was that it was relatively intuitive --maybe i should have said "retrievable in a high-pressure concept-doodling game" --compared to a wall of text..
      • kgwgk 2y ago
        No need to apologize! I was joking, I think I get what you mean.
    • vinnyvichy 2y ago
      If you let me flash you my (still indecent) state of intuition.. "convex conjugates" (more precisely but limited sense "momentum maps") are delimited continuations in a optimization algorithm. https://en.wikipedia.org/wiki/Convex_conjugate https://en.wikipedia.org/wiki/Convex_conjugate https://en.wikipedia.org/wiki/Delimited_continuation https://en.wikipedia.org/wiki/Delimited_continuation
      • 082349872349872 2y ago
        Delimited continuations are to exponentials as convex conjugates are to implications?
        • vinnyvichy 2y ago
          I'm pretty sure I don't understand the possible meanings of what you said there either so let's try :) <layman-ish op-research lingo> I meant that the tangent to the convex conjugate ("momentum") provides bounds on what the values returned by the dual step in a primal-dual algo should be. I don't know which meaning of "exponential" I should focus on here (the action perhaps? A power set? A probability distribution?), but "implications" seem to refer to a constraint on outputs contingent on the inputs so I will go with that. Delimited continuations seem to be the closest thing I found in the PL lit, aka wikipedia, feel free to suggest something less kooky :) </lol>
          • 082349872349872 2y ago
            That makes much more sense than my flash, which had been following a spark in the other direction: Delimited continuations are functions, and as such (in a world of algebraic types where we can take sums and products of types) exponentials of types, ran^dom. [in particular, with the substitution of isomorphism for equality they follow the normal K-12 rules: C^(A+B) ~= C^A * C^B, etc.] I'd just been glancing at https://en.wikipedia.org/wiki/Convex_conjugate#Examples https://en.wikipedia.org/wiki/Convex_conjugate#Examples and the pattern by which single-branched f(x) seems to often become a multibranch f(x) reminded me of how logic-reversing functions in general and logical implication in particular "adds branching": if we wish to establish x'<=5 then if x is already <= 5 we may `skip` but otherwise we must calculate x-5 (and then subtract it off); similarly an implication x->y may be satisfied on one branch by not x but on the other requires y. [and on the general topic: I like to think of temperature as tying together energy and entropy, where positive temperatures yield the familiar relationships but negative temperatures "unintuitive" ones]
            • vinnyvichy 2y ago