PS: Re: Fwd: relativistic Hamilton equations

This is true if p = gamma m r dot, but the Hamilton equation

r dot = partial H / partial p

is worked out in previous notes and papers using

r dot = partial H / partial v sub N partial v sub N / partial p

This was first done in Note 425(2) to obtain v sub N = r dot.

Relativistic Hamilton equations

PS: you defined the relativistic gamma factor by the momentum p. In M&T always the velocity u is used which corresponds to v_N in your notation. However, p is defined by the gamma factor itself so that we obtain a recursion. On the other hand there are no velocity variables in the Hamilton mechanism. When using p throughout the equations, the appearance of the recursion may be avoided. Only at the end of the calculation one can rewrite the results to ordinary coordinates.
Since p implicitly contains the gamma factor, I expect that the results will be different from an Euler-Lagrange calculation for example, where the velocities are used in the gamma factor. We should clarify this discrepancy.

Horst

Am 29.12.2018 um 08:36 schrieb Myron Evans:

OK thanks. I would suggest computing the Hamilton equations on the Newtonian level first, to gain experience, then proceed to special relativity and then to m theory. The results are well known so this would test the code. On the Newtonian level the various choices of p and q are given in note 426(2). The equations are:

p dot = – dH / dq
q dot = dH / dp

where
p = p; q = r
H = p squared / 2 – mMG / r

So the first Hamilton equation gives

p dot = – mMG / r squared

and the second Hamilton equation gives

r dot = p / m

So combining the last two equations gives

r double dot = – MG / r

which is the right result in the inertial frame. It gives the Newtonian ellipse. You have already produced many renowned results by using Maxima to integrate equations such as the above. So you have already integrated the Hamilton equations and the results are already well known throughout the world. In special relativity

H = gamma m c squared – mMG / r

where

gamma = ( 1 – p squared / m squared c squared)) minus half

and

p = gamma m r dot, q = r

So the first Hamilton equation gives

p dot = – mMG / r squared

i.e.

d(gamma m r dot) / dt = -mMG / squared

and as shown in Note 425(2) the second Hamilton equation gives

r dot = p / m

As in previous UFT papers special relativity gives a precessing ellipse. That is already regarded as a classic result by our vast readership.
So the code is working fine and the above is already known to be the way of reducing the Hamilton equations to equations that have already been integrated to give orbits.
The m theory has given many spectacularly interesting results this year, the role of the Hamilton equations is to produce results such as those in UFT425 and notes for UFT426 for dm(r1) / dr1 amd d(mr1) / dv1. The power of the Hamilton equations begins to reveal itself through the Hamilton Jacobi equations.

I tried to compute the Hamilton equations by computer in coordiantes (r, phi) directly. I used the gamma factor

.

The velocity is described by the derivatives of q’s, not the p’s. The Hamiltonian is according to M&T and some last notes:

.

The results are not the correct ones, at least for q_r dot and q_phi dot. A factor of 2 appears, and the factor q_r squared is missing for p_phi dot. Have you an idea what is still wrong?

Horst

Am 28.12.2018 um 06:52 schrieb Myron Evans:

non-relativistic Hamilton equations

They look right. Agreed, the Hamilton equations need the canonically conjugate p and q generalized coordinates. Usually p is found from the Lagrangian method, or by inspection. The advantage of the Hamilton method is that it can be extended to many areas of physics using first order equations, and can be extended to the Hamilton Jacobi formalism. The Lagrange and Hamilton equations can be used together, then they give new equations of motion as in UFT425. I inferred another new equation of motion in Note 426(1). As you infer, the first order equations are easier to integrate. In m theory the combined use of the Hamilton and Lagrange equations has already revealed a new equation for dm(r) / dr. This is the first time that it has been shown how the Hamilton equations work in special relativity. This is by no means easy to see.


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