  In this paper we study the geometrical structures on the cotangent bundle
using the notions of adapted tangent structure and regular vector fields. We
prove that the dynamical covariant derivative on $T^{*}M$ fix a nonlinear
connection for a given $\mathcal{J}$-regular vector field. Using the Legendre
transformation induced by a regular Hamiltonian, we show that a
semi-Hamiltonian vector field on $T^{*}M$ corresponds to a semispray on $TM$ if
and only if the nonlinear connection on $TM$ is just the canonical nonlinear
connection induced by the regular Lagrangian.
