The inverse challenge of the calculus of adaptations was once first studied by way of Helmholtz in 1887 and it's totally solved for the differential operators, yet just a couple of effects are recognized within the extra normal case of differential equations. This ebook appears at second-order differential equations and asks in the event that they may be written as Euler-Lagrangian equations. If the equations are quadratic, the matter reduces to the characterization of the connections that are Levi-Civita for a few Riemann metric.To clear up the inverse challenge, the authors use the formal integrability concept of overdetermined partial differential platforms within the Spencer-Quillen-Goldschmidt model. the most theorems of the e-book provide a whole representation of those options simply because all attainable events look: involutivity, 2-acyclicity, prolongation, computation of Spencer cohomology, computation of the torsion, and so forth.
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