ABSTRACT
Introduction Many nonlinear boundary value problems can be reduced to solve their Euler-Lagrange equation
∇J(u) = 0 (1) where ∇J is the gradient of a C1 energy function J on a Banach space B. A solution u∗ ∈ B to (1) is called a critical point of J . u∗ is said to be nondegenerate if J ′′(u∗) is invertible. The first candidates for a critical point are the local extrema to which the classical critical point theory was devoted in calculus of variation. Most conventional numerical algorithms focus on finding such stable solutions. Critical points that are not local extrema are unstable and called saddle points. In physical systems, saddle points appear as unstable equilibria or transient excited
states. For example, a common and important problem in computational chemistry and condensed matter physics is the calculation of the rate of transition states.
