“…This defines a canonical projection τ : T * (L × ) → J 1 (L * ), which in the adapted coordinates (s, q i , p, p j ) in T * L × and the adapted coordinates (z, q i , p j ) in J 1 (L * ) (so that j 1 (S)(q) = (S(q), q i , ∂S ∂q j (q)) reads τ (s, q i , p, p j ) = (p, q i , p j /s) and has fibers being orbits of the R × -action. Hence, M = J 1 (L * ) = T * (L × )/R × , and τ is exactly the projection P → P/R × = M. It is easy to see that the R × -action and the multiplication by reals in the vector bundle T * (L × ) commute, so that we get a double principal-vector bundle structure on T * (L × ) (see [42]), This is a particular example of a weighted principal bundle (more precisely, a VB (Vector Bundle)-principal bundle) defined in [36].…”