The projective Cartan-Klein geometry of the Helmholtz conditions
The conditions under which a linear second order ODE is the Euler-Lagrange equation of a variational problem are classically given by the Helmholtz conditions. Since the latter depend on the existence of solutions to other ODE's, more practical necessary conditions have been introduced in the 80'ies by Sarlet, Engels and Bahar. The authors associate to a linear second order ODE the complete projectivisation of a fundamental set of its solutions, which gives a curve in the half-Grassmannian, called the Jacobi curve. Now, for each (non-degenerate) curve in the half-Grassmannian, a complete invariant, called the Jacobi endomorphism, has been constructed in an earlier paper of the first author and \textit{J. C. Álvarez Paiva} [Adv. Appl. Math. 42, No. 3, 290--312 (2009; Zbl 1167.53016)], generalising the classical Schwarzian of complex analysis. The main theorem of the present paper states that the Sarlet-Engels-Bahar conditions can be obtained as successive derivatives of the Jacobi endomorphism of the associated Jacobi curve, so the conditions can be expressed in terms of the local geometry of the Jacobi curve. The authors explicitly show the projective nature of the Helmholtz condition while additionally giving a Cartan-Klein-geometric interpretation of the Sarlet-Engels-Bahar conditions. By the invariance of the Helmholtz condition under a group acting on the coefficients of a differential equation, the above is related to the conditions under which curves in the half-Grassmannian \(\mathsf{Gr}(n,2n)\) are congruent under the action of \(\mathsf{GL}(2n)\). This problem, together with its relation to the above, is also considered in this paper.
scientific article; zbMATH DE number 7003613
scientific article