A well known derivation of the free-space Lagrangian in Special Relativity goes as follows:
The action $\mathcal{S}$ is a functional of the path taken through configuration space, $\mathbf{q}(\lambda)$, 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, $\mathcal{L}$, such that:
$$\mathcal{S}[\mathbf{q}] = \int_{\lambda_1}^{\lambda_2} \mathcal{L}(\mathbf{q}(\lambda), \dot{\mathbf{q}}(\lambda), \lambda) \mathrm d \lambda \tag{1}$$
The extremal $\mathcal{S}$ is given when $\mathcal{L}$ 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: $$ \mathcal{S}[\mathbf{q}] = \alpha \int_\mathbf{q} \mathrm d s \tag{2}$$
- Using $\mathrm d s^2 = \mathrm d t^2 - \mathrm d \mathbf{x}^2 $, this gives: $$ \mathcal{S}[\mathbf{q}] = \alpha\int \sqrt{1 - \dot{x}^2} \mathrm d t\tag{3}$$
- 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: $$ \mathcal{L} = - m c^2 \sqrt{1 - \dot{x}^2} \tag{4}$$
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: $$ \mathcal{L} \propto \sqrt{g_{\mu \nu} \dot{x}_\mu \dot{x}_\nu} \tag{5}$$ However. If we try to insert the Newtonian Euclidean 3-metric, it seems that we don't get the expected result: $\mathcal{L} = \frac{1}{2} m v^2 $. If we insert the Euclidean metric into the Landau-Lifshitz general definition, we find: $$ \mathcal{L} \propto |\mathbf{v}| \tag{6}$$ 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: $$ \text{sgn}(v_i) = \text{const} \tag{7}$$ 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....
