New Formulation of the Hamilton Equations in Special Relativity

Many thanks for this meticulous check. Eq. (19) is the result of the choice of the canonically conjugate generalized coordinates p and q in Eq. (20), and in Hamiltonian dynamics p and q are independent by definition (Marion and Thornton). So partial p / partial q = 0 and Eq. (19) follows. This equation implies new equations of motion Eq. (22) and Eq. (23). In Eq. (22) the r dependence reappears. Eq. (32) holds because gamma does not have a direct dependence on t. To be strictly accurate partial gamma / partial t = 0 as you write. The middle term of Eq. (37) comes from the well known result for the relativistic Newton equation d (gamma m v sub N ) / dt = m gamma cubed d v sub N / dt = – mMG / r squared. This has been used in several previous papers and notes. Eq. (39) is the direct result of the choice of p and q as canonically conjugate. So all looks OK. This note interrelates the Hamilton and Euler Lagrange dynamics to give the new equations of motion (22) and (23). It shows that Hamilton’s dynamics contain information that is not present in the Lagrange dynamics, i.e. that p and q are canonically conjugate and independent. This point is clearer when we consider canonically conjugate L and phi, because partial L / partial phi = 0.

New Formulation of the Hamilton Equations in Special Relativity

Some details of th note are not understandable to me although the overall result is clear.
Why vanishes the partial derivative of v_N (eq.19) although v_N depends on r by definition of (14)?
Why should eq. (32) hold? What is true is

partial gamma / partial t = 0

because gamma does not explicitly depend on time. However r, r dot etc. depend on time therefore the total derivative should not vanish.

In eq.(37) the middle term should be omitted, then the equation is identical to that derived in earlier papers.

For eq. (39) see discussion of eq.(19).

The action S_phi should be determinable by

S_phi = integral gamma m r^2 phi_dot^2 dt

where phi dot = d phi/dt has been used.

Horst

Am 03.01.2019 um 14:13 schrieb Myron Evans:

New Formulation of the Hamilton Equations in Special Relativity

This uses the choice (20) of canonically conjugate generalized coordinates and the hamiltonian (33). The choice (20) means that there appear two new equations of motion, Eqs. (22) and (23), which may be solved numerically with the Evans Eckardt equations (25) and (26). It is shown that the Hamilton equations produce the force equation (37) of special relativity, used in previous UFT papers to produce a precessing orbit. So both the Hamilton and Euler Lagrange equations produce the same force equation. In addition, the Hamilton equations give the new equations of motion (22) and (23). To deduce these analytically requires the complete consideration of Euler Lagrange Hamilton dynamics. So the Hamilton equations produce new information not given by the Euler Lagrange equations. Finally the choice of canonically conjugate generalized coordinates (4). when used in the Hamilton equation (41) shows that the angular momentum is a constant of motion. In the Hamilton Jacobi formalism it is therefore defined by the action equations (46) and (47).The Hamilton Jacobi formalism therefore defines the action. The three complete systems of dynamics were traditionally the Euler Lagrange, Hamilton and Hamilton, but a fourth system has just been inferred, the Evans Eckardt system of dynamics. All four should be used together for any problem being considered. Their power is greatly amplified by the ability of the Maxima code written by Horst Eckardt to integrate them.


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