425(4): Interpretation of the Hamilton Equations

This note prepares the groundwork for the application of the Hamilton equations in special relativity and m theory by explaining their interpretation in classical orbital theory. The canonically conjugate generalized p and q must first be found by inspection or by using the Euler Lagrange equations, then the Hamilton equations implemented in any frame of reference. The Hamilton equations are powerful first order equations which can be used in many fields of physics, and can be developed into the Hamilton Jacobi equation. The Hamilton equations are derived as shown in this note and the Evans Eckardt equations derived from the Hamilton equations. Complete self consistency is demonstrated between the Euler Lagrange and Hamilton equations, and the hamiltonian derived from the Euler Lagrange equations. There are two sets of canonically conjugate variables, the momentum p and position r, and the angular momentum L and the angle phi of the plane polar coordinate system. The Hamilton equations of 1834 were first derived by Lagrange in 1809. Poisson derived similar equations in 1809. Neither realized that they had derived fundamental equations of motion. In 1831 Cauchy realized that they were fundamental equations of motion. In 1834 Hamilton derived them from the Hamilton Principle of Least Action. The final version of UFT425 will be based on this note. The relativistic Hamilton Jacobi equation was used in “The Enigmatic Photon” in the field of electrodynamics.


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