On inverse variational problem: examples

Inverse problem of the calculus of variations is a vast subject with many results. I will discuss some examples related to ODEs, making an emphasis on parametrized vs unparametrized problems.
The simplest and most studied case is about systems of second order differential equations, also known as path geometries. Here I will mention some results joint with Vladimir Matveev arXiv:2203.15029.
Then I will discuss recent results joint with Vladimir Matveev and Wijnand Steneker arXiv:2412.04890 about variationality of so-called conformal geodesics. This system is given by third order differential equations, which makes it rather unconventional for traditional approaches. I will also mention an on-going project using the invariant variational bicomplex.