Correction and extension: Equations of motion from dH/dt=0, dL/dt=0 and Lagrangian

Correction and extension: Equations of motion from dH/dt=0, dL/dt=0 and Lagrangian

This is a very important result and major progress using computer algebra. The equations dH / dt = 0 and dL / dt = 0 are very fundamental and lead to the equations of motion in Note 420(6a). For the purposes of distinguishing them they can be referred to as the Evans Eckardt equations of motion of classical dynamics. This distinguishes them from the Hamilton canonical equations of motion and the Newton equations. They can be integrated by computer using Horst’s code. The latter will eventually be made publicly available in a user friendly way. The new equations of motion will also lead to many new insights in quantum mechanics because the latter is based on the hamiltonian as is well known. For example they will lead to a quantum force equation. I think that Note 420(6a) should be the main theme of UFT420. This protocol shows the importance of computer algebra in being able to solve equations that are far too complicated to deal with by hand. It will be very interesting to find what insights the Evans Eckardt equations give. The previous equations of motion in UFT416 were based on the lagrangian method and give results that are different from dH / dt = 0 and dL /dt = 0. I will work on Eq. (6) of the protocol 420(6a). I can see that Eq. (6) of the hamiltonian protocol 420(6a) reduces to the correct Newtonian result when gamma = 1, dm(r1) /dr1 = 0 and dphi /dt = 0. We then get r double dot = – MG / r squared, the Newtonian result. The dH / dt = 0 method is entirely new, and will lead to a new quantum mechanics. The fact that the lagrangian method gives different results is a fundamentally new insight in mathematics, another major discovery. In the next note for UFT420 I will go ahead and develop the 420(6a) protocol to isolate the vacuum force, and write out the protocol in the Note. Finding the right lagrangian is important of course, but that is a very interesting problem in mathematics rather than physics. I think that dH / dt = 0 and dL / dt = 0 should be adapted from now on as the equations of motion of classical dynamics and orbit theory, giving an entirely new subject.

Correction and extension: Equations of motion from dH/dt=0, dL/dt=0 and Lagrangian

There was an error in the L definition. Here the new calculations:

420(6a): dH/dt, dL/dt
420(6b): Lagrangian

Results: see last 2 equations of both protocols.
There are indeed differences between both calculations. For phi dot dot,
there is a factor of 1/2 in the first term of the Lagrange solution. For
r dot dot, there is an additional term in the Hamiltonian solution
besides the factor of 1/2.
We can now try to modify the Lagrangian to find the same results.

Horst

Am 01.12.2018 um 21:50 schrieb Horst Eckardt:
> I used eqs.(21) and (24) of note 420(3) to compute the expressions
>
> dH/dt = 0
> dL/dt = 0
>
> in polar coordiantes of r_1 space. I used the form m(r_1)
> consequently, not the form m(r). It is not clear if always the correct
> distinction of r and r_1 is made in the note. This may be a source of
> confusion.
> The final results o17, o18 in the protocol are the combined equations
> of motion. There is no classical term
>
> – phi dot r_1 dot / r_1
>
> for phi dot dot. Please check if the initial approaches are correct. I
> gues that anything went wrong.
>
> Horst
>
>

420(6a).pdf

420(6b).pdf

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