426(2): New information about m(r1) from the new equation of motion
Agreed with this, the Hamilton equations give new information when used with the Lagrange equations, and your code for them will be very useful.
426(2): New information about m(r1) from the new equation of motion
There seems to be a typo in eq. (8), it should read partial r dot / partial L at the rhs.
It is important to know that this generalized momentum p_phi=L is different from that in eq. (9) which is that of a general coordinate system (r-phi). If p_phi=L, one has to transform p_phi to a linear momentum having the same physical dimensions as p_r:
p_phi –> p_phi/r
or better:
p_phi –> p_phi/q_r.
This has confused me in some previous notes and in the computer-based calculation of Hamilton equations which is under way.
In eq.(15/16) the middle part is missing a time derivative. The rhs of (16) should have an additional "m" in the denominator to describe a momentum, similarly in (24).
Eq. (30) seems to be valid in the inertial system only because v_N (in 31) has no angular component. Perhaps it would be recommendable to write out the Lagrange and Hamilton equations with coordinate indices as in M&T to avoid ambiguities.
Horst
Am 28.12.2018 um 15:38 schrieb Myron Evans:
426(2): New information about m(r1) from the new equation of motion
The new information is dm(r1) / dv1 = 0. This adds to the information about dm(r1) / dr1 in UFT425. This note explains the discussion with Horst this morning through the vector Hamilton equations (12 and (13) and shows that it is possible to choose p = m v sub N as the canonical momentum whee v sub N is the complete Newtonian velocity . The next stage is the development of the Hamilton Jacobi formalism of m theory. This can be used for the whole of physics and not just dynamics. The quantization of m theory takes place of course through the hamiltonian and that can be tied up with the well known Lamb shift theory of vacuum effects in the H atom. The Lamb shift can be understood as an effect of spherically symmetric spacetime.