by Joseph Grifone (Université Paul Sabatier, France) & Zoltán Muzsnay (Kossuth Lajos University, Hungary)

The inverse problem of the calculus of variations was first studied by Helmholtz in 1887 and it is entirely solved for the differential operators, but only a few results are known in the more general case of differential equations. This book looks at second-order differential equations and asks if they can be written as Euler–Lagrangian equations. If the equations are quadratic, the problem reduces to the characterization of the connections which are Levi–Civita for some Riemann metric.

To solve the inverse problem, the authors use the formal integrability theory of overdetermined partial differential systems in the Spencer–Quillen–Goldschmidt version. The main theorems of the book furnish a complete illustration of these techniques because all possible situations appear: involutivity, 2-acyclicity, prolongation, computation of Spencer cohomology, computation of the torsion, etc.

• An Introduction to Formal Integrability Theory of Partial Differential Systems
• Frölicher–Nijenhuis Theory of Derivations
• Differential Algebraic Formalism of Connections
• Necessary Conditions for Variational Sprays
• Obstructions to the Integrability of the Euler–Lagrange System
• The Classification of Locally Variational Sprays on Two-Dimensional Manifolds
• Euler–Lagrange Systems in the Isotropic Case