Hamiltonian Dynamics in m Theory
Hamiltonian dynamics introduce new information into m theory, via Eq. (26) of Note 425(3). This uses the conjugate generalized coordinates of Hamiltonian dynamics. If the choice (27) and (28) is made we obtain the result (39) which leads to Eq. (44). This is obtained self consistently in two ways, from Eqs. (42) and (43). This analysis leads to Eq. (48), which was the starting point of Note 425(2). The analysis holds in any coordinate system and gives information on dm(r1) / dr1 and m(r1), so they are no longer empirical. In Newtonian dynamics and special relativity this type of analysis does not give new information, but in m theory it gives new information which is just starting to be developed. In Note 425(3) I assumed that partial v1 squared / partial r1 is zero in Eqs. (59) and (60). However Eq. (44) means that partial (m gamma v1 squared) / partial r1 = 0, so the correct result is Eq. (93). This was first derived in Note 425(2) and has been checked by computer algebra. The choice of generalized conjugate coordinates in Eqs. (27) and (28) is the only self consistent choice, and this is by no means obvious. This is one of the things that have to be found by experience or inspection. Note carefully that Lagrangian and Hamiltonian dynamics use generalized coordinates (q dot, q, t) and ( p, q, t) respectively. They are conjugate and independent, so partial q dot / partial q = 0 in Lagrangian dynamics, and partial p / partial q = 0 in Hamiltonian dynamics. The second Hamilton equation (49) of Note 425(3) has not yet been used, and inputs more information. The vector Hamilton equations can also be used. Eq. (93) can be crunched out on a desktop, mainframe or supercomputer, using iterative methods described in a note from Horst yesterday. I checked in Note 425(2) that the Hamilton equations work in special relativity. This is by no means obvious. They also work on the Newtonian level. I first studied these methods as a second year undergraduate in mathematics and wondered why the Euler Lagrange and Hamilton equations were used if they provided the same information as the Newton equations. The answer is that the Euler Lagrange and Hamilton equations use generalized coordinates, and are much more powerful than Newton for this reason. In m theory this is clear, new equations emerge by use of the full range and power of the Euler Lagrange Hamilton dynamics. The Hamilton Principle of Least Action is also very powerful, and the hamiltonian is the basis of quantum mechanics. Anyone who has studied mathematics as an undergraduate should be able to understand what is going on. Those with no experience of mathematics can follow the main arguments and graphics by Horst Eckardt. These are very valuable because they reduce a maze of complicated mathematics to an understandable level.