It is usually extremely difficult to solve a nonlinear differential equation A[u]=0 directly but it may be much easier to discover the minimum or maximum points of an appropriate energy functional I(u) where A=I’.
Such variational principle is advocated by Max Plank as a fundamental law in nature (e.g., relativity theory follows a geodesic path derived from this principle in space-time). Its connection to functional approximation is also interesting – it might be very difficult to obtain the complete knowledge of some function f but much easier to work with its minimum and maximum.
It should be noted that how nonlinearity and its associated variational principle is in contrast with more conventional reductionism approach. In linear systems, reduction works just fine; it is the nonlinearity in nature which calls for a holistic solution. The universality of variational principle lies its capability of handling a wide range of phenomenons involving complex interaction among a large number of individual units.
Another closely related technique is fixed-point methods – intuitively, minimum and maximum of a function are fixed points robust to local perturbations. The desirable properties of nonlinear mapping such as contraction and compaction often require strong assumption about the regularity of the solution (e.g., global Lipschitz) which makes it practically less useful.
