Jet bundle

It makes it possible to write differential equations on sections of a fiber bundle in an invariant form.

Historically, jet bundles are attributed to Charles Ehresmann, and were an advance on the method (prolongation) of Élie Cartan, of dealing geometrically with higher derivatives, by imposing differential form conditions on newly introduced formal variables.

Since the early 1980s, jet bundles have appeared as a concise way to describe phenomena associated with the derivatives of maps, particularly those associated with the calculus of variations.

⁠ be a multi-index (an m-tuple of non-negative integers, not necessarily in ascending order), then define: Define the local sections σ, η ∈ Γ(p) to have the same r-jet at p if The relation that two maps have the same r-jet is an equivalence relation.

The integer r is also called the order of the jet, p is its source and σ(p) is its target.

The annihilator of the Cartan distribution is a space of differential one-forms called contact forms, on Jr(π).

is a contact form if and only if for all local sections σ of π over M. The Cartan distribution is the main geometrical structure on jet spaces and plays an important role in the geometric theory of partial differential equations.

The dimension of the Cartan distribution grows with the order of the jet space.

Hence, θ = b(x, u, u1)θ0 must necessarily be a multiple of the basic contact form θ0 = du − u1dx.

Proceeding to the second jet space J2(π) with additional coordinate u2, such that a general 1-form has the construction This is a contact form if and only if which implies that e = 0 and a = −bσ′(x) − cσ′′(x).

In local coordinates, every contact one-form on Jr+1(π) can be written as a linear combination with smooth coefficients

, is A vector field is called horizontal, meaning that all the vertical coefficients vanish, if

A vector field is called vertical, meaning that all the horizontal coefficients vanish, if ρi = 0.

A section is called a vector field on E with and ψ in Γ(TE).

A solution is a local section σ ∈ ΓW(π) satisfying

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

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

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.

satisfies these equations, Vr is called the r-th prolongation of V to a vector field on Jr(π).

Hence, from this equation we will pick up the formula for ρ, which will necessarily be the same result as we found for V1.

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.

to preserve the contact ideal, we require And so the second prolongation of V to a vector field on J2(π) is Note that the first prolongation of V can be recovered by omitting the second derivative terms in V2, or by projecting back to J1(π).

is the equivalence class of sections of π that have the same k-jet in p as σ for all values of k. The natural projection π∞ maps

into p. Just by thinking in terms of coordinates, J∞(π) appears to be an infinite-dimensional geometric object.

, being born as a direct limit, carries an additional structure: it is a filtered commutative algebra.

, so it is a smooth function on the finite-dimensional manifold Jk(π) in the usual sense.

Enhance I(E) by adding all the possible compositions of total derivatives applied to all its elements.

The submanifold E(∞) of J∞(π) cut out by I is called the infinite prolongation of E. Geometrically, E(∞) is the manifold of formal solutions of E. A point

of E(∞) can be easily seen to be represented by a section σ whose k-jet's graph is tangent to E at the point

at the point p. Most importantly, the closure properties of I imply that E(∞) is tangent to the infinite-order contact structure