Can relativistic momentum and its conservation be derived fromconservation of Newtonian momentum and the Lorentz transformations?

On Apr 26, 8:09 am, Daryl McCullough wrote:
wrote:

The conservation of momentum (in either Newtonian or Minkowski
space-time) is an easy consequence of the classical Noether
theorem and the homogeneity of space(time).

That's an incomplete answer. If you know the Lagrangian,
then you can apply Noether's theorem to prove that the
momentum (defined as the variation of the Lagrangian with
respect to velocity) is conserved. But first you have to
find the Lagrangian. How do you derive a Lagrangian?

Since the first equation to start the derivation of the Lagrangian
method is:

** Action = integral(t1, t0)[L dt]

So, all you have to do is to look for something that will fit in.
shrug

Luckily, the Lorentz transform can be written into just one single
equation.

** c^2 dt”^2 – ds”^2 = c^2 dt^2 – ds^2

Where

** ds”^2 = dx”^2 + dy”^2 + dz”^2
** ds^2 = dx^2 + dy^2 + dz^2

If “ frame is observing itself, the above equation can be simplified:

** c^2 dt”^2 = c^2 dt^2 – ds^2

Or

** dt” = sqrt(1 – (ds/dt)^2 / c^2) dt

So, the Lagrangian is very obviously to be:

** L = sqrt(1 – (ds/dt)^2 / c^2)

Where

** Action = Time elapsed at “ frame

In the case of GR, this method of finding the Lagrangian also works.
Of course, this is the Lagrangian of geodesics. It should not be
confused with the Lagrangian that Hilbert pulled out of his ass to
derive the field equations. To this date, there is no understanding
of how Hilbert’s Lagrangian can be derived and how the Eisntein-
Hilbert action really means and why it has to be extremized. It is
all bull****. shrug