In reply to the second remark, Note 377(1) uses exactly the same method as Marion and Thornton, third edition,Eqs. (14.53) to (14.58), the derivation of the relativistic kinetic energy from the work integral. The relativistic Newtonian force is used in Eq. (14.54). Eq. (3) of the note is Eq. (14.56) of Marion and Thornton. It is seen that dt cancels out in Eq. (2), giving Eq. (3). nothing else is assumed. Integrating Eq. (3) by parts gives the well known relativistic energy T = (gamma -1) m c squared.

OK thanks. I will reply to these two interesting remarks in two postings. In point (1) I used the expression for the Newtonian force given by Marion and Thornton, third edition, problem (14.38). As you know, the relativistic Newtonian force is the differential of the relativistic momentum with respect to the time t in the lab frame. The Minkowski force is the differential of the relativistic momentum with respect to the proper time tau. In component form the relativistic Newton force is given by Eqs. (21) and (22). These two simultaneous equations when integrated numerically give retrograde precession, as you showed. The retrograde precession comes directly from the relativistic Newton equation. The Lagrangian (11) with proper Lagrange variable bold r gives the relativistic Newton equation directly. As you know there is freedom to chose the proper lagrange variable, the choice bold r leads directly to the well known relativistic Newton equation, so it is a valid choice. The lagrangian (11) is also correct. The rules of the Hamilton / Lagrange dynamics are also obeyed, the kinetic energy is a function only of dr(bold) / dt, and the potential energy is a function only of r bold. The forward precession is obtained by rewriting the lagrangian (13) as the lagrangian (1). The latter is used in the Euler Lagrange equations (5) and (6) with two proper variables X and Y. The key point is that choice of proper variables is different for forward and retrograde precession. The choice for forward precession also obeys the rules of the Hamilton / Lagrange dynamics because the kinetic energy is a function only of X dot and Y dot, and the potential energy is a function only of X and Y. It is well known that different choices of proper Lagrange variables leads to different dynamics. They must be chosen by experience or "by inspection" as they say in mathematics. The conservation of relativistic angular momentum can be demonstrated y by a choice of Lagrange proper variables r and phi of the plane polar system. As shown in Note 401(1), Relativistic angular momentum is always conserved and the relativistic hamiltonian is always conserved

Equations for forward and backward precession

To: Myron Evans <myronevans123>

I came across the following two problems.

1.

In UFT 377, eq,(16), the relativistic Newtonian force is derived from the expression

d bold p / dt = gamma m d bold v / dt. (1)

This gives

gamma^3 m r dot dot (3)

by evaluating the above expression (1). The result is equated to the result from the Euler-Lagrange equations by definition but both results are probably not the same. This then is the reason for giving forward and backward precession.

2.

When evaluating (1) in note 377(1), a substitution is made by the

d gamma/dt = d gamma/dv0 * dv0/dt.

However in the gamma factor there is not v0^2 but the modulus of v in form of v_x ^2 + v_y ^2. When inserting this term, the differentiation gives a tensor expression, see protocol. I am not sure if this can be further simplified, but it is quite different from the simple result (3).

Horst

405-1.pdf