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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...

A collection of lectures on functional analysis and applications. Areas addressed include: sets, spaces and functions; integrals; topological spaces; spaces of operators and functionals; linear operators; spectral theory of linear operators; and nonlinear problems of functional analysis.