As in Marion and Thornton pp. 213 ff. of the third edition the Lagrange equations can be derived from the Newtonian p = partial T / partial r dot (Eq. 6.102) so they are not independent of this equation. This is also eq. (14.107) of Marion and Thornton and they show that it leads to the relativistic lagrangian. So this equation is not independent. Finally the relativistic momentum is given by p rel sub X = partial lagrangian / partial X dot and the Y component, as in Eqs. (15) to (19). The origin of relativistic momentum is discussed in Section 14.7 of Marion and Thornton. It is conservation of linear momentum. Finally the equation p sub i = partial lagrangian / partial q dot sub in generalized coordinates is eq. (6.151) of Marion and Thornton. From eq. (6.151), the Lagrange equations follow immediately by differentiating both sides, to give Eq. (6.152). This is elegant, and the easiest way of seeing that Eq. (2) of Note 402(2) is a well known equation. It simply merges Eqs. (6.151) and (6.152) but uses the slightly unfamiliar differentiation with respect to a vector. So p = partial lagrangian / partial q dot is an equation from which the Euler Lagrange equations can be derived. They can also be derived from Hamilton’s Principle of Least Action as is well known. The equivalence of the Lagrange and Newton equations for rectangular coordinates is discussed in Section 6.6. The Lagrange and Hamilton equations are more general than the Newton equations if generalized coordinates are used. I know that we have showed M and T can go wildly wrong, when dealing with general relativity, but the foregoing basics are well established. It is very good to discuss these basics. Does anyone else have a comment? This kind of free and collegial discussion is essential, it may sometimes lead to new discoveries. We have made more discoveries in the past decade than needles on a hedgehog.
Date: Mon, Feb 19, 2018 at 12:17 PM
Subject: Self Consistency problem with relativistic Newtonian force
To: Myron Evans <myronevans123>
To my understanding the resulting inconsistency is quite clear. Lagrange theory gives relativistic equations of motion from the relativistic Lagrangian. If the equations of motion are derived from another equation directly, in this case
F = m d(p_rel)/dt = gamma^3 m dv/dt,
this is an INDEPENDENT approach, and it cannot be expected that both methods give the same result a priori. For the results to be identical, it must be
p_rel = partial L / partial bold r dot,
and this is obviously not the case. Nevertheless the Lagrangian gives the relativistic angular momentum as a constant of motion. But there is no prescription that the above first equation contains the same p_rel as obtained formally from the Lagrangian. To my knowledge the approach
p_rel = gamma m v
comes from generalization of relativistic dynamics based on the Lorentz transform which only holds for constant relative motion. Perhaps this is the problem.
Horst
Am 18.02.2018 um 13:21 schrieb Myron Evans:
There is freedom of choice of proper Lagrange variables, but the review just sent over seems to be the only way to achieve complete and rigorous self consistency, and in this sense the method is unique. The formal Euler Lagrange equation using a proper Lagrange variable vector r is rigorously correct but if and only if it is correctly interpreted and correctly expressed in any given coordinate system. The formal equation to my mind is elegant and economical.