On globally invariant Euler-Lagrange equations for curves

Invariant Lagrangians yield invariant Euler-Lagrange equations and local methods for computing these are well-established, starting with Anderson and Griffiths. We focus on global algebraic invariants, using an invariant version of variational bicomplex or, more generally, C-spectral sequence. One motivation is the question, posed by Kogan and Olver, whether invariant variational problems with only singular extremals can exist. We show that the example of conformal geodesics answers this question positively and motivates the need for global invariant methods. We then discuss how to compute invariant Euler-Lagrange equations using global invariants and how this can be applied in practice, both as a supplementary tool for existing local methods, as well as in a purely global setting. We demonstrate these principles with some examples, all for systems of ODEs (unparametrized curves). 
This talk is based on joint work with Boris Kruglikov and Eivind Schneider (Tromsø).