Note 420(6): Complete Agreement between the Hamiltonian and Lagrangian m Theory

This computer algebraic check is most valuable as usual. This factor half looks like a very interesting development, ultimately giving new physics. On the classical and special relativistic levels there is exact agreement between the hamiltonian and lagrangian methods. This was my "baseline" calculation in Note 420(3). The classical hamiltonian method is given in Eqs. (35) to (37) of Note 420(3) in Cartesian coordinates, giving exactly the same result as the lagrangian method. The special relativistic hamiltonian calculation is given in Eqs. (41) to (45) in Cartesian coordinates, again giving the right result and exactly the same result as the lagrangian method. The hamiltonian equation of motion dH / dt = 0 is very powerful in my opinion, and can be quantized directly. The hamiltonian method in m theory is developed in Note 420(5), in the (r1, phi) coordinate system. Start with the hamiltonian (1) defined in UFT416 ff. This is given in Eq. (1) of Note 420(5) and is known from UFT417 ff. to give correct conservation of energy and angular momentum. The equation of motion (3) gives Eq. (4). This leads to Eq. (8), and the first thing to do is to check equation (8) by computer algebra. The Leibniz theorem gives equation (9). Now use Eq. (5) to give the result (10). This again can be checked by computer algebra. Eq. (11) follows. Eq. (12) has been used in previous UFT papers and Eq. (12) can also be checked by computer algebra. Eq. (13) follows from Eq. (12). Finally use eq. (14) to find Eq. (17). All these steps can be checked by computer algebra but it is quite simple linear algebra. Eq. (17) contains the vacuum force (18), which is of course missing from all theories with the exception of m theory. Eq. (17) correctly reduces to ECE2 relativity because in the latter case dm(r1) / dr1 = 0 and m(r1) = 1. All these hand calculations (which should always be regarded as preliminary as you know) can be checked as usual with computer algebra to eliminate human error entirely. The end result of this quite famous methodology by now is that we know that our theories are always technically correct. The lagrangian method is given in UFT417(1) and can also be checked by computer algebra. The lagrangian method gives a vacuum force that is half that given by the hamiltonian method, so in my opinion the lagrangian must be chosen to give the results of the very fundamental hamiltonian method. This is the awkward bit in lagrangian calculations as you know, the lagrangian is not known a priori and has to be guessed, or as teh mathematicians say derived by inspection. So in a revised Note 420(6) I will give the general equations for the adjustment of the lagrangian. The new lagrangian must be L = L0 + L1

where L0 is the old lagrangian of Note 417(1) and L2 a lagrangian defined by:

dL1 / dr1 = – (mc squared / 2) gamma dm(r1) / dr1
dL1 / dr1 dot = 0

These last two equations can be solved by computer algebra to give L1 and to give the results of the hamiltonian method. They impose a constraint on dm(r1) / dr1 and this seems to be the new physics. This constraint is not given by the hamiltonian method so using both the lagrangian and hamiltonian methods gives new physics in the most general spherical spacetime defined by m(r1). There are other possibilities, for example the lagrangian may need undetermined multipliers chosen to give the hamiltonian results. This is another awkward bit in lagrangian theories, the undetermined multipliers are not known a priori. However it could be that they can be found by adjusting them to give the hamiltonian results. Finally, the problem is solved by dm(r1) / dr1 = 0 but then we lose the vacuum force entirely. As you showed in UFT419, the constant m(r1) method gives a completely new orbit, a non precessing ellipse which is not Keplerian or Hooke / Newton, to me one of many amazing results of m theory.

The new Lagrangian (9) of the note gives an additional term at the rhs of (13) and the factor 1/2 remains, see my computer calculation.
Are you shure that the calculation of dH/dt is correct? In which note did you this? I can double check.


Am 30.11.2018 um 11:23 schrieb Myron Evans:

Note 420(6)

Note 420(6): Complete Agreement between the Hamiltonian and Lagrangian m Theory

Complete agreement is obtained by using the lagrangian (8), which reduces to Eq. (11) when m(r) = 1, giving the correct limit.


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