In J.D. Jackson's(adsbygoogle = window.adsbygoogle || []).push({}); Classical Electrodynamics, an argument is made in support of the assertion that the relativistic Lagrangian [itex]\mathcal L[/itex] for a free particle has to be proportional to [itex]1/\gamma[/itex]. The argument goes something like this:

From this, it is (according to Jackson) "obvious" that the relativistic Lagrangian for the free particle has to be

- [itex]\mathcal L[/itex] must be independent of position and can therefore only be a function of velocity and mass.
- [itex]\gamma \mathcal L[/itex] must be a Lorentz scalar.
- The only available Lorentz invariant function of the 4-velocity is [itex]c^2 = v_\mu v^\mu[/itex].

[tex]

\mathcal L = -mc^2 / \gamma.

[/tex]

I guess I can see why this should be the case, given that the Euler-Lagrange equations need to be satisfied and that the Lagrangian needs to have the appropriate units. What I *don't* get is where (2) and (3) come from. Can someone please explain?

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# The relativistic Lagrangian

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