I don’t think your characterization of the vector field is quite correct; I would recommend Cartan (1922, §§184-185), Godbillon (1969, Proposition XI.3.7), or Souriau (1970, Theorem 7.29). Or this answer conveniently linked in this page’s “related” list $\to$.
Francois Ziegler
- 31.5k
- 6
- 121
- 176