Otdfbt

A well known derivation of the free-space Lagrangian in Special Relativity goes as follows:

• The action $mathcalS$ is a functional of the path taken through
  configuration space, $mathbfq(lambda)$, where $lambda$ is the
  path parameter




• The action can be thought of as the total 'cost' of this path through configuration space. The path which is chosen is the
  'cheapest' of these paths (i.e. the one which minimises the action)



• 'Valid physics' can be retrieved by correctly assigning each point along the path a `cost', to do so we invoke a function called the Lagrangian, $mathcalL$, such that:

$$mathcalS[mathbfq] = int_lambda_1^lambda_2 mathcalL(mathbfq(lambda), dotmathbfq(lambda), lambda) mathrm d lambda tag1$$

• The extremal $mathcalS$ is given when $mathcalL$ satisfies the Euler-Lagrange equations.




• In free space (assumed to be homogeneous and isotropic), the `cost' of each point along the path cannot be determined by either the position along the path, or the position in configuration space, as this would violate our free-space assumptions.




• The only determining factor that can be allowed to influence the total cost of each point in space is the infinitesimal path length at each point, up to a dimensional constant $alpha$. Therefore:
  $$ mathcalS[mathbfq] = alpha int_mathbfq mathrm d s tag2$$



• Using $mathrm d s^2 = mathrm d t^2 - mathrm d mathbfx^2 $, this gives:
  $$ mathcalS[mathbfq] = alphaint sqrt1 - dotx^2 mathrm d ttag3$$
  


• We choose $alpha = - m c^2$ as the simplest invariant quantity that has the correct dimensions. Therefore if our path parameter is the coordinate time $t$, we have:
  $$ mathcalL = - m c^2 sqrt1 - dotx^2 tag4$$
  

This proof is found in many different sources (probably most notably in Landau-Lifshitz Volume 2, Chapter 2). This idea generalises into General Relativity, where the free-space Lagrangian is:
$$ mathcalL propto sqrtg_mu nu dotx_mu dotx_nu tag5$$
However. If we try to insert the Newtonian Euclidean 3-metric, it seems that we don't get the expected result: $mathcalL = frac12 m v^2 $. If we insert the Euclidean metric into the Landau-Lifshitz general definition, we find:
$$ mathcalL propto |mathbfv| tag6$$
The equations of motion predicted by the normal Lagrangian are a statement of Newton's I axiom ($p = $ const in free space), but the result of this Lagrangian is:
$$ textsgn(v_i) = textconst tag7$$
This isn't wrong, but it clearly does not contain all the information we expect the Lagrangian to contain!

Why does this approach (which has such resounding success in the relativstic case!) fail so badly when applied to the (supposedly simpler) Newtonian case? I know that under certain circumstances we can square the Lagrangian and retain the same equations of motion, but those proofs all relied on affine parameters etc., which seems like overkill for a Newtonian mechanics problem.

Am I missing something obvious? It seems like it should be trivial to recover classical mechanics from this method, when it is so 'easy' to get relativstic mechanics from it....

The problem with your approach is that your proposed action
$$S = int |mathbfv| , dt$$
is not invariant at all. While Landau's action is invariant under Lorentz transformations, and in fact completely coordinate independent, yours is not invariant under even Galilean transformations, which add a constant to $mathbfv$. The space-only analogue of a Lorentz transformation is not a Galilean transformation, it's a rotation. Your result is invariant under rotations, but that's not enough.

Taking the nonrelativistic limit is not as simple as forgetting the time component. Parametrizing by time, the relativistic Lagrangian can also be written as
$$mathcalL = sqrtdott^2 - dotmathbfx^2 = sqrt1 - mathbfv^2.$$
We can arrive at your answer "getting rid of the one", but that's not the right way to take the limit. Instead the one becomes more important, and we have to keep it and Taylor expand about it, giving
$$mathcalL approx 1 - fracmathbfv^22$$
which recovers the usual nonrelativistic action. The argument for why we have to square $mathbfv$ is also given in volume 1 of Landau and Lifshitz. Perhaps a simpler way to say this is that Galilean physics isn't simply relativity with the time dimension removed.

1. A non-relativistic square root Lagrangian (6) has an issue with null-velocity vectors. It is not differentiable at zero velocity, which is unphysical from the perspective of Newtonian mechanics, cf. this related Phys.SE post.




2. We should point out that the relativistic square root Lagrangian (5) has a similar shortcoming with null/lightlike vectors. It is only applicable to timelike vectors corresponding to massive point particles, cf. e.g. this Phys.SE post.