Prolongation
A prolongation of a differential equation on a jet bundle is a higher-order differential equation obtained by adjoining differential consequences of the original equations. If E subset J^kE, its rth prolongation E^((r)) subset J^(k+r)E requires the original equations and all of their total derivatives through order r to vanish. A local section solves the original differential equation exactly when its prolonged jets satisfy every finite prolongation.
The infinite prolongation E^((infty)) is the inverse limit of the finite prolongations. It carries the restricted Cartan distribution, whose integral manifolds describe the prolonged graphs of solutions, and is the geometric object underlying a diffiety.
See also
Cartan Distribution, Diffiety, Integral Manifold, Jet Bundle, Partial Differential EquationExplore with Wolfram|Alpha
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References
Saunders, D. J. The Geometry of Jet Bundles. Cambridge, England: Cambridge University Press, 1989.Vinogradov, A. M. "Local Symmetries and Conservation Laws." Acta Appl. Math. 2, 21-78, 1984.Cite this as:
Weisstein, Eric W. "Prolongation." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/Prolongation.html