From: ttt_heg@web.de   
      
   Am Freitag000009, 09.01.2026 um 07:34 schrieb Ross Finlayson:   
   > On 01/08/2026 07:55 PM, Ross Finlayson wrote:   
   >> On 01/08/2026 05:55 PM, Thomas 'PointedEars' Lahn wrote:   
   >>> Ross Finlayson wrote:   
   >>>> The idea that everything physics is always parameterized   
   >>>> by time or 't' is often formalized "the Lagrangian",   
   >>>   
   >>> No, the parametrization by time is a concept in Lagrangian _mechanics_   
   >>> which   
   >>> is based on the _principle of stationary ("least") action_.  The   
   >>> action is   
   >>> defined as   
   >>>   
   >>>    S[x(t)] = ∫ dt L[x(t), dx(t)/dt, t],   
   >>>   
   >>> where x may be a vector (field), and L is the Lagrangian (function).   
   >>> [Both S and L are *functionals*: they depend on a function, x(t);   
   >>> hence the customary notation with rectangular brackets.]   
   >>>   
   >>> In special relativity, one finds from the Minkowski metric   
   >>>   
   >>>    ds^2 = c^2 dτ² = c^2 dt^2 - dx^2 - dy^2 - dz^2   
   >>>                   = c^2 dt^2 [1 - (dx/dt)^2 - (dy/dt)^2 -   
   (dz/dt)^2]   
   >>>                   = c^2 dt^2 (1 - V^2/c^2)   
   >>>   
   >>> that   
   >>>   
   >>>    S[x] = -m c ∫ ds = -m c ∫ dt c √(1 - V^2/c^2)   
   >>>                     = ∫ dt [-m c^2 √(1 - V^2/c^2)],   
   >>>   
   >>> where the prefactor -m c is introduced so as to produce a quantity with   
   >>> dimensions of action (energy × time, cf. ℎ and ℏ) and the correct   
   >>> canonical   
   >>> momentum [*], and in the integrand one with dimensions of energy; so the   
   >>> relativistic non-interacting Lagrangian is   
   >>>   
   >>>    L = -m c^2 √(1 - V^2/c^2) = -m c^2 √[1 - (dX/dt)^2/c^2].   
   >>>   
   >>> It turns out that this leads to the correct energy--momentum relation,   
   >>> as I pointed out earlier.   
   >>>   
   >>>    [*] For example, the canonical 3-momentum is, from the Euler--   
   >>> Lagrange   
   >>>        equations   
   >>>   
   >>>          0 = d/dt ∂L/∂(dX/dt) - ∂L/∂X = d/dt ∂L/∂V -   
   ∂L/∂X = d/dt ∂L/∂V   
   >>>   
   >>>          P = ∂L/∂V   
   >>>            = -m c^2 ∂/∂V √(1 - V^2/c^2)   
   >>>            = -m c^2/[2 √(1 - V^2/c^2)] ∂/∂V (1 - V^2/c^2)   
   >>>            = -m c^2/[2 √(1 - V^2/c^2)] (-2 V/c^2)   
   >>>            = m V/√(1 - v^2/c^2)   
   >>>            = γ(v) m V.   
   >>>   
   >>>    [It is interesting to note that this way the relativistic/exact   
   >>> 3-momentum   
   >>>     for a massive particle can be derived purely from the Minkowski   
   >>> metric,   
   >>>     without a Lorentz transformation (but the Minkowski metric is   
   >>> Lorentz-   
   >>>     invariant, somewhat by design [I showed before that you do not even   
   >>>     need to assume Lorentz invariance to derive it, just a constant   
   >>> speed   
   >>>     with which information propagates in space)].   
   >>>   
   >>> Since from the above follows that ds = c dτ, one can also write   
   >>>   
   >>>    S[x(τ)] = -m c ∫ dτ c = -m c^2 ∫ dτ.   
   >>>   
   >>> The physical paths of free motion, which (one can prove) are spacetime   
   >>> geodesics, are those where the action S[x(t)] is minimal (stationary in   
   >>> general).  From the form above one can see that those are the   
   >>> trajectories W   
   >>> along which the elapsed proper time ∆τ = ∫_W dτ is maximal, which is   
   >>> another   
   >>> way of describing "time dilation" when there is relative motion, and   
   >>> finally   
   >>> explaining the "twin paradox" as nothing more than a consequence of   
   >>> different elapsed proper times along different worldlines.   
   >>>   
   >>> One can also see here that mass arises naturally from assuming the   
   >>> principle   
   >>> of stationary action.   
   >>>   
   >>>> sort of like "the Machian" is a usual notion of far-field.   
   >>>   
   >>> No, nonsense.   
   >>>   
   >>>> [pseudo-scientific word salad]   
   >>>   
   >>> You are a hopeless case.   
   >>>   
   >>   
   >> So, parameterized by time then, like I said,   
   >> like Lagrange says.   
   >>   
   >> You mention least action and it's a pretty reasonable   
   >> principle, where the theory is sum-of-histories sum-of-potentials   
   >> least-action least-gradient a continuum mechanics, that   
   >> obviously enough it's a field theory.   
   >>   
   >> You know, momentum isn't very much conserved in kinematics.   
   >> It sort of adds up for each of the ideal equal/opposite   
   >> inelastic interactions, yet any sort of rotation loses it.   
   >>   
   >>   
   >>   
   >> Much like "whatever satisfies the _Lorentzian_ is a model   
   >> of relativity", there's that "whatever satisfies the   
   >> _Lagrangian_ is a model of relativity with a clock hypothesis".   
   >>   
   >>   
   >> Perhaps you might be familiar with the notion of "implicits",   
   >> for example that "x" is "x(t)" and forces are always implicitly   
   >> functions of time, t, and so on.   
   >>   
   >> Forces are functions of time, ....   
   >>   
   >> Then, besides that logic demands a temporality else   
   >> it's readily demonstrable as false, time the usual   
   >> parameter t is an implicit.   
   >>   
   >> Implicits may remind   
   >> of "running constants", then for example about notions   
   >> like the monomode process, since usually accounts as   
   >> after the _Laplacian_, the sum of 2'nd order partials,   
   >> the _Lorentzian_, the sum of 2'nd order partials x +- t,   
   >> and whether that's zero or off-zero, non-zero.   
   >>   
   >> The differential d and partial-differential little-greek-d   
   >> are two different things, your Lagrangian L is already   
   >> second-order in d^2 t while velocity V is only first   
   >> order, then taking their partials w.r.t. each other,   
   >> finds that now what was taken as the root of the square,   
   >> gets issues with the nilpotent and nilsquare, about   
   >> the off-zero case, helping explain why what falls out   
   >> as a linear expression or in simple terms,   
   >> ignores part of its own derivation there.   
   >>   
   >> Otherwise it's quite plainly Galilean, one may note.   
   >> (Eg, any "unboundedness as infinity".)   
   >>   
   >> Meeting the form, ....   
   >>   
   >>   
   >>   
   >> Yeah, it seems quite so that the larger reasoners   
   >> very well appreciate the contents of that "T-theory,   
   >> A-Theory, theatheory" thread.   
   >>   
   >> Including its logical elements, its mathematical elements,   
   >> and otherwise its canonical and novel elements, so relevant.   
   >>   
   >> Then also for physics.   
   >>   
   >>   
   >>   
   >> It seems the action S is simply contrived to dump out   
   >> the usual definition, as it is, "timeless", and absent   
   >> moment, of momentum the linear since Lagrange.   
   >> Being that it's just "defined".   
   >>   
   >>   
   >>   
   >> "Implicits" is what's involved, since whatever then   
   >> results in the derivations cancelling themselves away,   
   >> perfectly model Lagrangians, Lorentzians, ..., Laplacians,   
   >> a hollow shell.   
   >>   
   >>   
   >   
   >   
   > When encountering various fields of mathematics,   
   > when the only tool there is is a hammer then   
   > everything looks like a nail, yet, in a world of   
   > nails, many varieties of hammers will do.   
   >   
   > So, when learning about things like "the operator calculus"   
   > and "functional analysis" it's a pretty great thing,   
      
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