The Schwarzian derivative and Euler-Lagrange equations (Q2085435)

The Schwarzian derivative of a (sufficiently regular) function \(u : \mathbb R \to \mathbb R\) is \[ S(u) = \frac{u'''}{u'}- \frac{3}{2}\left( \frac{u''}{u'}\right)^2. \] The author shows \begin{itemize} \item[1)] that \(S(u)\) is a constant of the motion for the Euler-Lagrange equation \(\mathrm{EL}\) associated to the functional \[ \mathcal I_L[u] = \int \left(\frac{u''}{u'} \right)^2 dt; \] \item[2)] that the solutions of the equation \(S(u) = 0\) are critical points of the functional \[ \mathcal I_S [u] = \int S(u)dt \] with respect to a certain subclass of variation vector fields. \end{itemize} This gives a variational flavour to the Schwarzian derivative. In the last part of the paper, the author computes the Wünschmann invariants of \(\mathrm{EL}\), and studies the geometry of the space of solutions \(\mathcal M\), showing that the projectivized tangent bundle of \(\mathcal M\) is canonically equipped with a field of curves projectively equivalent to certain models depending on whether \(S(u)\) is greater, smaller or equal to \(0\).