425(2): New Equation for dm(r1) / dr1

OK many thanks! This is true, but there is another solution: m(r1) gamma = 1 / gamma, so m(r1) = 1 / gamma squared. This is the v sub N = 0 solution for a particle at rest. In general the Hamilton and Euler Lagrange equations hold for any coordinate system, so I proceeded with the plane polar system in Eqs. (58) and (59) as you note. This leads to a solution for an orbit. Eq. (57) is the fundamental Eq. (55) of Euler Lagrange Hamilton dynamics, so is always valid in any coordinate system.
425(2): New Equation for dm(r1) / dr1

I encountered a problem here: Eqs. (56,57) hold in the inertial coordinate system where gamma does not depend on r1. Therefore the rhs of both equations should be zero. Furthermore, it follows from the lhs that

partial m(r1) / partial r1 = 0.

The situation is different in the (r1, phi) system. Eqs. (60/61) are correct but the question is if you can use (57) in this case. Maybe you can. Then the subsequent calculation is valid. I will try to obtain the result (70) by computer.

Horst

Am 16.12.2018 um 13:56 schrieb Myron Evans:

425(2): New Equation for dm(r1) / dr1

The use of the Hamilton canonical equations of motion is shown in this note to lead to a new expression of dm(r1)/ dr1 in terms of the angular momentum L in Eq. (70).

dm(r1) / dr1 = – L squared / (c squared gamma squared m squared r1 cubed)

so m(r1) is constant unless there is an angular momentum present, another amazing result of m theory, although I say it myself. So the power of the Hamilton canonical equations becomes immediately apparent. It is clear that dm(r1) / dr1 is related to the centrifugal force L squared / (m r1 cubed). Hamilton derived his Hamiltonian dynamics in 1833 from the earlier Lagrangian dynamics, based in turn on work by Euler. Although he is always known as Rowan Hamilton in Trinity College Dublin, where I am sometime Visiting Academic, he became Sir William Rowan Hamilton, Irish Astronomer Royal, and a predecessor as Civil List Pensioner. He was offered F. R. S. but could not afford the fees. Clearly, he was not terribly interested in paying the fees, and the fact that he was not an F. R. S. does not mean a thing, the important things are the Hamilton dynamics, the hamiltonian, the Hamilton Principle of Least Action, his discovery of quaternions, and many other major discoveries. He was also a poet, not the greatest who ever lived, but nevertheless a poet. The entire subject of quantum mechanics is based on the hamiltonian, as is very well known.


%d bloggers like this: