425(3): Rigorous Self Consistency of m Theory
This note is a detailed demonstration of the rigorous self consistency of m theory in the elegant Euler Lagrange Hamilton dynamics. This is the first time that detailed consideration has been made of Hamilton’s equations in the UFT series. The self consistent choice of the Hamilton canonical variables is given in Eqs. (27) and (28). This gives the Einstein energy equation (36) in m space, first derived in UFT424. Eq. (47) is obtained in a rigorously self consistent manner from both Eqs. (42) and (43). This checks the starting equation of Note 425(2). The static solution is rigorously equivalent to the rest energy (56) of m theory. The static solution is for a particle m at rest attracted gravitationally by a particle M at rest in m space. The general solution is Eq. (57) and in the inertial frame gives the remarkable result (68) for the hamiltonian, reducing it to classical format in m space. The first Evans Eckart equation reduces this to the force equation in m space in an inertial frame, Eq. (76), giving a new definition of vacuum force, Eq. (80). The hamiltonian can be transformed to plane polar coordinates (r1, phi) giving Eq. (88). Finally the solution obtained in Note 425(2), and checked by computer algebra, is given in Eq. (93) in plane polar coordinates (r1, phi). This related dm(r1) / dr1 to M(r1). This is very complicated equation but can be solved as discussed by Horst this morning. The final note for UFT425 will be considered in the next and final note, using the second Hamilton equation. This is an entirely new classical dynamics valid for any m space.