Relativistic Hamilton equations
This is all excellent work, and any method is equally valid mathematically. It will be interesting to see which method is the most computationally efficient, and whether any new information emerges, such as that in UFT425, using a combination of Euler Lagrange and Hamilton. I think that these nineteenth century methods can be made considerably more powerful with the computers now available, from desktops to supercomputers. The Hamilton Jacobi method leads to differential equations which were often impossible to solve in the nineteenth and early twentieth centuries, but which are now easily soluble by computer. It will be interesting to see whether it gives new information about m theory. The aim is to find which combination of methods is the most powerful. For example a combination of Lagrange and Hamilton gives dm(r1) / dr1. The Evans Eckardt equations of motion could be combined with the Hamilton equations or Hamilton Jacobi equations. Your algorithm for the Hamilton equations is new and original, and could well lead to very interesting new results. The Hamilton Jacobi equation for a central potential gives the Schroedinger equation and is a direct route to quantization. The subjects regarded as "complete dynamics" currently include Euler Lagrange, Hamilton and Hamilton Jacobi. However we now have a new complete dynamics, the Evans Eckardt dynamics. The great power of the EE dynamics emerges in m theory. In the Newtonian dynamics and special relativity there are advantages of Euler Lagrange, Hamilton and Hamilton Jacobi, but they are well known. The startling progress has been made with m theory in the year 2018.
Relativistic Hamilton equations
My intention was to write the Hamiltonian directly in a predefined frame of reference by canonical coordiantes without transforming the frame. If frame transformation is required it should be done for the canonical coordinates p_i, q_i directly. The question is if this is possible without knowing the tranformation in a more convenient coordinate set.
However I will try your method below, it is a way to arrive at coordinates in the desired frame. Probably they can then be rewritten to canonical coordinates in that frame.
Horst
Am 29.12.2018 um 15:23 schrieb Myron Evans:
Relativistic Hamilton equations
These results look interesting and can be integrated with Maxima, giving a lot of new techniques. The Hamilton and Hamilton Jacobi equations can be used in any frame of reference. The rule for going from the inertial frame to any other is as follows. In the inertial frame
r double dot = – mMG / r squared
To transform to plane polars use
a bold = (r double dot – r phi dot squared) e sub r
+ (r phi double dot + r phi double dot + r dot phi dot) e sub phiso we get two equations as in several UFT papers:
r double dot – r phi dot squared = – mMG / r squared
and
dL / dt = 0
The extension to special relativity and m theory is given as you know in UFT415 onwards. . Having used the Hamiltonian method to get the first equation above we know that all is OK. Your previous use of the inertial frame in several papers is also correct. The most powerful equations are our own new equations, dH / dt = 0 and dL / dt = 0. This is because the code can integrate them to give any kind of result.
(r double dot –