Lie derivative of any contact form θ preserves the contact ideal. Let us begin with the first order case. Consider a general vector field V1 on J1(π), given by We now apply to the basic contact forms and expand the exterior derivative of the functions in terms of their coordinates to obtain: Therefore, V1 determines a contact transformation if and only if the coefficients of dxi and in the formula vanish. The latter requirements imply the contact conditions The former requirements provide explicit formulae for the coefficients of the first derivative terms in V1: where denotes the zeroth order truncation of the total derivative Di. Thus, the contact conditions uniquely prescribe the prolongation of any point or contact vector field. That is, if satisfies these equations, Vr is called the r-th prolongation of V to a vector field on Jr(π). These results are best understood when applied to a particular example. Hence, let us examine the following. Example Consider the case (E, π, M), where E ≅ R2 and M ≃ R. Then, (J1(π), π, E) defines the first jet bundle, and may be coordinated by (x, u, u1), where for all p ∈ M and σ in Γp(π). A contact form on J1(π) has the form Consider a vector V on E, having the form Then, the first prolongation of this vector field to J1(π) is If we now take the Lie derivative of the contact form with respect to this prolonged vector field, we obtain Hence, for preservation of the contact ideal, we require And so the first prolongation of V to a vector field on J1(π) is Let us also calculate the second prolongation of V to a vector field on J2(π). We have as coordinates on J2(π). Hence, the prolonged vector has the form The contact forms are To preserve the contact ideal, we require Now, θ has no u2 dependency. Hence, from this equation we will pick up the formula for ρ, which will necessarily be the same result as we found for V1. Therefore, the problem is analogous to prolonging the vector field V1 to J2(π). That is to say, we may generate the r-th prolongation of a vector field by recursively applying the Lie derivative of the contact forms with respect to the prolonged vector fields, r times. So, we have and so Therefore, the Lie derivative of the second contact form with