Example 1. Prolongation of Jet Spaces
Define the jet space where with coordinates
Display the jet coordinates, the coordinate vector fields, the 1-forms, and the contact 1-forms.
Prolong the jet space to .
Again display the jet coordinates, the coordinate vector fields, the 1-forms, and the contact 1-forms.
Example 2. Prolongation of Vector Fields
Define the jet space where with coordinates .
Define an arbitrary point vector field on .
Prolong to order 1--this agrees with the standard prolongation formula found in all texts.
Define the infinitesimal generator for a rotation in the - plane.
Prolong to order 1.
Prolong to order to 2--we can achieve the same result by prolonging .
Define the jet space , where with coordinates .
Define a vector field whose flow simultaneously scales the coordinates .
Example 3. Prolongation of Transformations
Define the jet space where with coordinates .
Define a projectable transformation from to .
Prolong phi1 to order 2.
Define a differential substitution from J to
Prolong phi2 to order 2.
Example 4. Prolongation of Differential Equations
Define a second order ode on (coordinates
Calculate the second order prolongation of DE1. Note that the list of jet variables to be solved for is also prolonged.
Define a system of over-determined partial differential equations in 2 independent variables and 1 dependent variable .
The second prolongation of DE2 is an overdetermined system of Frobenius type--all the 4-th order derivatives of which can be solved for.