I will have a look at this in detail tomorrow.
Pleasure, the discussion of basics is always important. The derivation by Marion and Thornton is the one that is always used, but that does not mean that it is correct. On going from their equation (4.55) to their equation (14.56) their bold u dot bold u is the product of velocity along the direction of the force, so they get equation (14.56). The Lorentz factor is defined as gamma = (1 – (u / c) squared) power minus half, where u squared is a scalar. I will go through this derivation in due course but today I will write up UFT401. In problem 14-38 Marion and Thornton give the magnitude of the relativistic Newtonian force as F = gamma cubed m du / dt . Since gamma cubed is a scalar it is clear that F bold = gamma cubed m d u bold / dt.
Date: Wed, Feb 14, 2018 at 7:47 PM
Subject: Re: Definition of Derivative with Respect to a Vector
To: Myron Evans <myronevans123>
OK, many thanks for clarification, with these definitions the Lagrange equations can be written in this vector form.
Meanwhile I came across another problem in M&T and note 377(1). The time derivatve of gamma is computed via the substitution
d gamma/dt = d gamma / dv0 * dv0 /dt.
Here v0 is the modulus of the velocity bold v0. I think it is not admissible to replace the term dv0/dt by a vector term in the result
F = m gamma^3 dv0/dt
in order to obtain the threedimensional result. Instead one had to do a different substitution for each component, for example
d gamma/dt = d gamma / dv0_x * dv0_x /dt
etc. Because of
v0^2 = v0_x^2 + v0_y^2 + v0_z^2
this gives a different result with cross-terms of velocities.
Horst
Am 14.02.2018 um 14:30 schrieb Myron Evans:
After some discussions with Horst the derivative of a scalar with respect to a vector is defined as the well known vector gradient (Google "derivative with respect to a vector") so that is how the Euler Lagrange equation (4) of the note should be defined. I exemplify this definition by deriving Eq. (16) of UFT377. I also give an example of a lagrangian differentiated with respect to bold r dot from Appendix 21 of Atkins, "Molecular Quantum Mechanics", third edition and also show how this equation can be interpreted. So in vector algebra
del phi : = partial phi / partial bold rThat is what I meant in UFT377. The important result is that the relativistic Newton equation gives retrograde precession. Discussions such as this are very important in order to check fundamentals. The definition (1) of this note can be extended to n dimensions, notably four dimensions, and the Lagrange method is important throughout modern physics.