In this nice paper the author proves that all natural symplectic forms on the tangent bundle of a pseudo-Riemannian manifold are pull-backs of the canonical symplectic form on the cotangent bundle with respect to some diffeomorphisms which are naturally induced by the metric.
@article{701583,
title = {Natural symplectic structures on the tangent bundle of a space-time},
booktitle = {Proceedings of the 15th Winter School "Geometry and Physics"},
series = {GDML\_Books},
publisher = {Circolo Matematico di Palermo},
address = {Palermo},
year = {1996},
pages = {[153]-171},
mrnumber = {MR1463517},
zbl = {0908.53017},
url = {http://dml.mathdoc.fr/item/701583}
}
Janyška, Josef. Natural symplectic structures on the tangent bundle of a space-time, dans Proceedings of the 15th Winter School "Geometry and Physics", GDML_Books, (1996), pp. [153]-171. http://gdmltest.u-ga.fr/item/701583/