Jet Bundle - Jet Prolongation

Jet Prolongation

A local diffeomorphism ψ: Jr(π) → Jr(π) defines a contact transformation of order r if it preserves the contact ideal, meaning that if θ is any contact form on Jr(π), then ψ*θ is also a contact form.

The flow generated by a vector field Vr on the jet space Jr(π) forms a one-parameter group of contact transformations if and only if the 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 obtain

where we have expanded the exterior derivative of the functions in terms of their coordinates. Next, we note that

and so we may write

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 D_i.

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.