426(4): Hamilton Jacobi equation on the Newtonian Level

Many thanks again! The essence of the HJ method is to find constants of motion through transformations which are derived from the Hamilton Principle of Least Action and the Hamilton equations. The Hamilton Jacobi equation is H + partial S / partial t = 0, where S is the action, or Hamilton’s principal function. In quantum mechanics the quantum of action as you know is h bar, the reduced Planck constant. Then a complete separability of action is looked for, as in Eq. (10) of Note 426(3). An important equation of the Hamilton Jacobi dynamics is p = partial S / partial q. The example in Eq. (14) ff of this note makes the method clear. This example can be found on the internet. It leads to two constants of motion and two differential equations (17) and (18) which give trajectories that are difficult to find in any other way. Moreover other constants of notion can be defined. The trajectories in this particular problem are x and z for motion in the xz plane. This note then applies the HJ method to central motion defined by the hamiltonian (25). This is transformed to the HJ equation (28) with actions for r and phi separated as in Eq. (29). The angular momentum L is then identified as a constant of motion and used to derive Eq. (34). This can be integrated by computer to give S sub r, p sub r and q sub r. This equation quantizes to the Schroedinger equation. This note is consolidated in Note 426(4), where the classical action can be found from Eq. (11) by computer integration. Eq. (11) gives the elliptical orbit. The next step is to proceed to the use of the relativistic Hamilton Jacobi method because the relativistic level is needed for the m theory.


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