Among the exercises in the first chapter of Goldstein's book "Classical Mechanics", it appears the lagrangian $$ L\left(x,\frac{dx}{dt}\right) = \frac{m^2}{12}\left(\frac{dx}{dt}\right)^4 + m\left(\frac{dx}{dt}\right)^2\times V(x) -V^2(x). $$
For a related answer, and the solution of the Euler-Lagrange equation see What does a Lagrangian of the form $L=m^2\dot x^4 +U(x)\dot x^2 -W(x)$ represent?
Apart from its solution, my question is: does this Lagrangian correspond to a real physical system, or is it just a mathematical construction? Is the fact that one recovers Newton's second law a coincidence? (see link above for the derivation)
Moreover, assume a more general Lagrangian of the form $L = \alpha \left(\frac{dx}{dt}\right)^4 + \beta \left(\frac{dx}{dt}\right)^2 -V^2(x) $, where $\alpha$ and $\beta$ are just constants. Are there general considerations (e.g. about energy conservation, etc.) that can be made?
