Abstract: Lifts of Frobenius on p-adic formal schemes are functions that may be viewed as analogues in arithmetic geometry of vector fields on manifolds. In particular, vector fields on the tangent bundle of a manifold, appearing for instance in Hamiltonian mechanics, have as arithmetic analogues lifts of Frobenius on the first jet space of a scheme. In this talk, I will discuss my study of these functions and some resulting comparisons and discrepancies discovered between the arithmetic and geometric tangent bundles.