Lagrangian

working draft

Background : I eschew infinities. Physically, infinities are nonsense for any local process or observation, where local can be as large as the observable universe. Nature has a physical implementation, it is geometrically beautiful, and it prevents infinities and singularities.

In physics, Lagrangian mechanics is a formulation of classical mechanics founded on the least-action principle Lagrangian mechanics describes a mechanical system with a pair (M,L), consisting of a configuration space M and a smooth function L called a Lagrangian. By convention, L = T – V, where T and V are the kinetic and potential energy of the system, respectively.

The least-action principle requires that the action functional of the system derived from L must remain at a stationary point (a maximum, minimum, or saddle) throughout the time evolution of the system. This constraint allows the calculation of the equations of motion of the system using Lagrange’s equations.

Wikipedia

How shall we approach the Lagrangian? My intuition says that solving the general cases of two opposite point charges or two like point charges with arbitrary path history at T0 would, by the principle of superposition, be the most general solution. As we bootstrap from first principles, proving the general solution in fundamental absolute math could springboard into optimized mathematics and simulation for assemblies and reactions of assemblies. We’ll start by looking at cases where the two point charge path histories are entirely |v| < @, i.e., sub-field speed. The cases where |v| = @ and |v| > @ will be considered later. There are special cases that are point, line, and planar as well as the general time + 3D space case.

  • time + 1D point solution :
    • opposites or likes at the same absolute location with path history in that spot forever. : No motion, no force, no energy, nothing happens.
  • time + 2D line solution :
    • opposites with separation r and no transverse path history and speed s at T0. : Force of attraction on each translates to work done and acceleration. PE and KE are trading.
    • likes with separation r and no transverse path history and speed s at T0. : Force of repulsion on each translates to work done and acceleration. PE and KE are trading.
  • time + 2D plane solution :
    • opposites with separation r and no non-planar path history and speed s at T0. : Hyperbolic acceleration apart.
    • likes with separation r and no non=planar path history and speed s at T0. : Elliptical orbit forms. Identical path.
  • time + 3D space solution : |v| < @, i.e., sub-field speed :
    • opposites with separation r and path history and speed s at T0. : Do PE and KE trade between the charges to arrive at a common orbital plane?
    • likes with separation r and path history and speed s at T0. : Force of repulsion on each translates to work done and acceleration. PE and KE are trading.

Looking at the 2D linear cases they are simple linear systems of attraction or repulsion where action occurs on a 1/r Dirac sphere head on with no shear from the emitter or receiver. Here are some scenarios.

  • opposite charges large r with a path history and potential stream
  • like charges start at small r with a path history and potential stream
  • and so on, calculating T and V at all points along the path.

The 2D orbiting dipole should be modeled over the full range from 1Hz to Planck frequency. In isolation, this would be an unchanging elliptical orbit. For illustration purposes and first pass math, circular orbits are shown. For the most general case, the orbits are assembly-native wave equations. At this point we are only examining isolated two-charge systems since we can arrive at any assembly level behaviour via superposition.

  • Does this task require an electrostatics or electodynamics solution?
    • It’s kind of both in a way. I think careful study is required.
    • Each point charge always experiences a constant and unchanging Dirac sphere stream from it’s partner, less any external perturbations.
    • In this sense the position is static from the point of view of calculating potential energy, right?
    • But since the point charges really are moving physically there will be shear in the Dirac sphere stream both at the emitter and at the receiver.
    • The shear affects the shape of the perceived potential field and therefore the forces and therefore the steady state at that frequency.
    • But does the shear affect the kinetic energy?
    • I guess it comes down to asking whether we need to tally up based on all the forces or do we tally up only after the superposition. I should probably know all this. Doh.
  • V = Potential energy
    • We know the closest orbit radius and therefore we can calculate the attractive potential energy.
    • How much potential energy is in the attractive bond?
      • Do we consider the energy on the basis of each point charge or the bond?
      • The energy should be considered at the points of action, which are at each point charge.
      • We can aggregate later to the bond if that is convenient and well defined.
    • Distinguish between attractive and repulsive potential energy?
      • How to model attractive and repulsive potential energy?
      • We might consider using sign.
        • Is attractive potential energy positive or negative and why?
        • Is this convention helpful for superposition or does that not make sense like it does with the potential field. I can’t think of why it would make sense, but let’s consider a while longer.
  • T = Kinetic energy
    • We know the frequency and the radius, therefore we can calculate the kinetic energy of each point charge.
    • We don’t need to worry about centripetal force, right?
  • The dipole changes state with energy.
    • State includes frequency and radius.
    • Each cycle presumably corresponds to h-bar j-s.
  • With L = T – V, kinetic energy minus potential energy, how is energy conserved?
  • I’ve never quite understood that when it comes to Lagrangians.
  • Kinetic and potential are trading off, but are we guaranteed T + V is a constant in classical mechanics?
  • When you think about potential energy of point charges, there is attractive potential energy and repelling potential energy.
  • Yet I don’t see that in the equations for potential energy.
  • Is it not necessary to distinguish attractive vs. repelling potential energy?
  • I’m wrestling with how to define the mass of a point charge or an orbiting dipole.
  • In my theory there is apparent energy which maps to mass, and energy shielded by superposition which is not included in mass.
  • When giving an assembly (particle) a group velocity, the Noether core reveals more shielded energy as velocity increases thus increasing apparent energy and mass.
  • As v approaches c many assemblies would decay before reaching c.
  • Thus the Lorentz factor doesn’t blow up to infinity for those assemblies — they can’t reach v = c (or field speed).
  • This is important because with a physical implementation reactions will occur near some of these boundary conditions — which is unlike pure math — which I think is a good thing for my theory.
  • Alas, energy at this decay is too high to be testable.
  • I presume an individual point charge reveals all its energy as mass.
  • I’m wrestling with whether an isolated orbiting dipole reveals all it’s energy as mass.
  • I’m also wondering if I add T + V to get apparent energy.
  • There are a lot of things to consider because the design must also map to the effective theories of present day physics.
  • Attractive vs. repelling potential energy is generally handled as a sign convention. E.g., for the Newtonian 2-body problem you have the idea of orbital energy which is conserved (loosely speaking the combined kinetic and potential energy of the 2 celestial bodies, intentionally neglecting the rest mass of those bodies)
  • Typically, the reference 0 gravitational potential energy is chosen to be at radius r=infinity, with the usual limit procedure.
  • Since kinetic energy has to be provided to convert from an elliptical to a hyperbolic orbit, empirically attractive potential energy has a negative sign, while repulsive potential energy has a positive sign.
  • With this empirical sign convention: elliptical orbits have a negative orbital energy, while hyperbolic orbits have a positive orbital energy. (Parabolic orbits have exactly 0 orbital energy, which suggests they are numerically unstable and are easily perturbed into one of the other geometric categories.)
  • Yes, total energy is conserved in classical mechanics (both Newtonian and Special Relativistic), and is defined as the sum of kinetic and potential energy. Changes in apparent rest mass (either from motion, or from changes in temperature) mostly show up on the kinetic energy side, with a lesser effect on the potential energy side.
  • Once you start accounting for causality-speed c lags, you’re close to General Relativity territory. (E.g., when Special-Relativity modeling the solar system the direction of the gravitational pull for Jupiter on Earth is along the light-rays from Jupiter to Earth, not the coordinate-now position of Jupiter. Systematically doing this causes some non-conservation issues, and ends up approximating both gravitational waves and General Relativistic orbital precession.)
Interlocutor on PBS Space Time Discord

“changes in temperature” — it is really interesting that you mentioned that. I realize that in the back of my mind there is a little voice saying, “hey what about the energy of the six personality charges in the Noether core poles?” I’ve been repressing that voice because I would think it minor in comparison to the Noether core dipole energies. But, if I am going to be true to my “we don’t need renormalization assertion” then we really can calculate all this stuff exactly down to h-bar in the Noether core and then whatever T and V the personality charges have. It also fits with the idea that the personality charges in the poles can be transducers of energy between external stimuli and the Noether core. Ahh, so much to think about.

The animation covers all the fermions, so that includes the electron and the quarks that make protons and neutrons. So pretty much your every day mass. It also covers neutrinos, but I think neutrinos are really a special case of a less energetic oscillating precursor to a photon. Thus neutrinos are part way between Bose-Einstein 2D form and Fermi-Dirac 3D form and the oscillation of the assembly also oscillates the proportions of apparent (massy) and shielded energies.

It still kind of blows my mind that there is not much online about orbits of point charges. They are so much cooler than gravitational orbits. There is a lot more going on! And everyone knows Coulomb’s law is dualistic with the law of gravitation. Likewise, Conway’s game of life seems like it would be a LOT more fun with point charges and continuous motion. Can’t find anything on that either. Why is that? It really makes me wonder about human creativity that this isn’t a well explored subject area.

Even a high energy reaction could throw off products with super low energy that would quickly join the spacetime aether. I’m talking products here. Regarding ephemeral ghost particles near standard matter, we already know that the fields of high energy tri-dipoles in standard matter excite the nearby aether, which makes total sense considering the vortices from each dipole.