TL;DR

How to show that ##L=-m \, ds/dt## for a free particle.

In Dirac's "General Theory of Relativity" (p. 52), he postulates that the action for a free particle of mass ##m## is $$I=-m \int ds$$ hence the Lagrangian is $$L=-m\frac{ds}{dt} = -m\frac{\sqrt{\eta_{\mu\nu}dx^\mu dx^\nu}}{dt}
\, .$$ To confirm that ##-m## is the correct coefficient, he assumes flat spacetime (special relativity) and calculates $$\frac{\partial L}{\partial \dot{x}^k} = -m\frac{\partial}{\partial \dot{x}^k}\left( \frac{ds}{dt}\right) = m\frac{ \dot{x}^k }{ ds/dt } = m \frac{dx^k}{ds}$$ which is the correct formula for relativistic 4-momentum ##p^k##. ("As it ought to be", says Dirac.)

##\qquad## But doesn't this assume that $$p^k = \frac{\partial L}{\partial \dot{x}^k} \quad ?$$ This is true for the non-relativistic Lagrangian for a free particle $$L=T-U = \frac{1}{2}mv^2 \qquad (*)$$ but is it true for a relativistic particle?

##\qquad## (Landau-Lifshitz give what seems to be a more convincing confirmation. They also postulate that ##L= \kappa \, ds/dt##, and assume flat spacetime (special relativity) where ##ds^2 = \eta_{\mu\nu}dx^\mu dx^\nu##. Then $$L=\kappa \frac{ds}{dt}=\kappa \sqrt{1-v^2} \, .$$ Hence, for velocities ##v \ll 1##, we have $$L = \kappa - \frac{1}{2}\kappa v^2 + O(v^4) \, .$$ The nonrelativistic Lagrangian is shown above in ##(*)##, so to get the correct kinetic energy term, we must have ##\kappa = -m##.)

##\qquad## I'd like to understand Dirac's confirmation that ##\kappa = -m##. How does he know in advance that $$p^k = \frac{\partial L}{\partial \dot{x}^k} $$ for a relativistic particle? It seems like he's using circular reasoning.

##\qquad## I think what Dirac forgot to add was "Since ##p^k = \partial L / \partial \dot{x}^k## in the case ##v \ll 1##, we see that ##\kappa = -m##."

I think it's circular reasoning because ##p^k = \partial L / \partial \dot{x}^k## is derived for a non-relativistic free particle, where ##L=mv^2/2##. But, otherwise, ##p^k \equiv \partial L / \partial \dot{x}^k## is the definition of the "conjugate momentum". We cannot simultaneously define ##p^k \equiv \partial L / \partial \dot{x}^k## and ##p^k \equiv m\, dx^k/ds = mv^k##.

I think Dirac needs to reduce to the non-relativistic case (like L-L) in order to determine ##\kappa##.

##\qquad## I think what Dirac should have done is this: Consider the case ##v \ll 1##, where ##L = T = mv^2/2##. Then we confirm by calculation that $$\frac{\partial L}{\partial \dot{x}^k} = \frac{m}{2}\frac{\partial v^2}{\partial \dot{x}^k}=m\dot{x}^k \,. $$ Comparing this with $$\frac{\partial L}{\partial \dot{x}^k} = -\kappa \frac{dx^k}{ds} $$ and using ##ds=dt## when ##v \ll 1##, we obtain ##\kappa = -m##.

Kostik said:

TL;DR Summary: How to show that ##L=-m \, ds/dt## for a free particle.

But doesn't this assume that pk=∂L∂x˙k? This is true for the non-relativistic Lagrangian for a free particle

This is also true for SR Lagrangian. For SR Lagrangian pk has the factor [tex]\frac{1}{\sqrt{1-v^2/c^2}}[/tex] as we see in experiments.　Analytical mechanics holds also in special relativity only by changing the Lagrangian functions.

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Demystifier said:

In physics there are (at least) two notions of momenta, canonical momentum and kinetic momentum. The kinetic momentum is defined as ##m\dot{x}##, while the canonical momentum is defined as ##\partial L/\partial \dot{x}##. In general, they do not need to be identical. But physicists sometimes use a sloppy language and call them both simply "momentum". From the context it is usually clear which momentum they have in mind.

And in SR, one ends up being a vector and the other a covector. It can be confusing ...